(*Date Sun Aug 23 04:49:00 GMT-0600 1998
*)

(*:Title: SURFS.m *)

(*:Author: Alfred Gray *)

(* Copyright 1994-1998 by Alfred Gray *)

(*:Package Version: 1.0 *)

(*:Mathematica Versions: 2.2, 3.0 *)

(*:Summary: 
This package consists of definitions of parametrizations of surfaces.
*)

(*:Keywords:
agnesirev, apple, astell, astroidmincurve, bohdom, bour, bourmincurve, bourpolar, boy, cardioidmincurve, catalan, catalandef, catalanmincurve, catenoid, catenoidasym, catenoidiso, catenoidmincurve, cayleysexticmincurve, circleinvolutemin, circleinvolutemincurve, circularcone, circularcylinder, circularmembrane, circularmembranemovie, circularmembraneplot, cissoidmincurve, clothoidmin, clothoidmincurve, cnccosurfrev, cnccotcheb, cnchyprofilemincurve, cnchysurfrev, cnchytcheb, cossurface, cossurfaceimplicit, costa, costamincurve, cpcprofilemincurve, cpcsurfrev, crosscap, crosscapimplicit, deltoidinvolutemin, deltoidinvolutemincurve, deltoidmin, deltoidmincurve, dini, dinitcheb, earth, eighteight, eightsurface, ellhypcyclide, ellipsemin, ellipsemincurve, ellipsoid, ellipsoidimplicit, ellipticalhyperboloidminus, ellipticalhyperboloidplus, ellipticcoor, ellipticparaboloid, ellipticparaboloidimplicit, enneper, ennepercatenoidmincurve, ennepercatenoidpolar, ennepermincurve, enneperpolar, epicycloidmincurve, equihom1implicit, equihom2implicit, equihom3implicit, equihom4implicit, equihom5implicit, equihom6implicit, exptwist, exptwistasym, flattorus, funnel, funnelasym, gencube, genhypparab, genoctahedron, goursatimplicit, handkerchief, heart, helicoid, helicoidprin, heltocat, henneberg, hennebergmincurve, hyperbolamincurve, hyperbolicparaboloid, hyperbolicparaboloidimplicit, hyperboloid, hyperboloidimplicit, hypocycloidinvolutemin, hypocycloidinvolutemincurve, hypocycloidmincurve, hy2sheet, hy2sheetimplicit, invertedcircularcone, invertedcircularcylinder, invertedtorus, jorgemeeksmincurve, jorgemeeksmincurveprime, jorgemeekspolar, kidney, kleinbottle, kleinbottlebis, kleinbottlewri, klein4, kuen, kuentcheb, kazoolaimplicit, kummerimplicit, lemniscatemin, lemniscatemincurve, logmincurve, logminpolar, logspiralmincurve, menn, mercatorellipsoid, mercatorsphere, mincurvs, moebcirc, moebiusstrip, monkeysaddle, monkeysaddlepolar, nephroidmincurve, nielsenspiralmincurve, nodoidnds, parabolamin, parabolamincurve, paraboliccoor, paraboliccyclide, paraboloid, pear, pearbis, perturbedms, perturbedmspolar, pillow, plane, planepolar, planetocat, plucker, pluckerpolar, pseudocrosscap, pseudosphere, pseudospherebis, pseudospheretcheb, rconoid, richmond, richmondmincurve, richmondpolar, roman, rosemin, rosemincurve, scherk, scherkimplicit, scherktowermincurveprime, scherk5, scherk5implicit, scmincurve, shoe, shoeasym, sievert, sinovaloid, sinsurface, sinsurfaceimplicit, snail, sphere, spherehelix, sphereimplicit, squaremembrane, squaremembranemovie, squaremembraneplot, stereographicellipsoid, stereographicsphere, stereographicspherepolar, swallowtail, tetrahedral, thomsen, torus, torusasym, torusbis, torusimplicit, tractrixmincurve, twiflat, twisphere, twocuspsimplicit, veronese, wallis, whitneyumbrella, wrinkles
*)

BeginPackage["SURFS`","CURVES`","CSPROGS`","PLTPROGS`","NumericalMath`BesselZeros`"]

    surfs:=Print[Select[Names["SURFS`*"],#!="surfs"&]]

(* Parametrically Defined Surfaces *)

    agnesirev::usage="{u,v}->agnesirev[a][u,v] is a parametrization of the surface of revolution formed from a rotated witch of Agnesi.  To plot try agnesirev[1][0,2Pi,-1,1]."

    apple::usage="{u,v}->apple[u,v] is a parametrization of the surface of revolution that resembles an apple.  To plot try apple[0,3Pi/2,-Pi,Pi]."

    astell::usage="{u,v}->astell[a,b,c][u,v] is a parametrization of an astroidial ellipsoid with axes of lengths a, b and c.  To plot try astell[1,1,1][0,2Pi,-Pi/2,Pi/2]."

    bohdom::usage="{u,v}->bohdom[a,b,c][u,v] is a parametrization of a Bohemian dome. It is formed by moving an ellipse along a circle in a perpendicular plane so that the ellipse remains parallel to a plane.  To plot try bohdom[4,4,4][0,2Pi,-Pi,Pi]."

    bour::usage="{u,v}->bour[u,v] is a parametrization of Bour's minimal surface.  To plot try bour[-1,1,-1,1]."

    bourpolar::usage="{r,theta}->bourpolar[n][t][r,theta] is the polar parametrization of a 1-parameter family of minimal surfaces such that {r,theta}->bourpolar[n][0][r,theta] is Bour's minimal surface of degree n. {r,theta}->bourpolar[m,n][t][r,theta] is the polar parametrization of a 1-parameter family of minimal surfaces such that {r,theta}->bourpolar[m,n][0][r,theta] is Bour's minimal surface of degree {m,n}.  To plot try bourpolar[2,1][0][0,1.7,-Pi,Pi]."

    boy::usage="{u,v}->boy[a,b,c][u,v] is a parametrization of Boy's surface.  To plot try boy[4,4,1][-Pi/2,Pi/2,-Pi/2,Pi/2]."

    catalan::usage="{u,v}->catalan[a][u,v] is a parametrization of Catalan's minimal surface.  To plot try catalan[1][-2Pi,2Pi,-Pi/2,Pi/2]."

    catalandef::usage="{u,v}->catalandef[a][t][u,v] is a 1-parameter family of minimal surfaces connecting Catalan's minimal surface to its conjugate.  To plot try catalandef[1][Pi/4][-2Pi,2Pi,-Pi/2,Pi/2]."

    catenoid::usage="{u,v}->catenoid[a][u,v] is the minimal surface of revolution generated by t->catenary[a][t].  To plot try catenoid[1][0,3Pi/2,-1.5,1.5]."

    catenoidasym::usage="{p,q}->catenoidasym[a][p,q] is a parametrization of a catenoid by asymptotic curves.  To plot try catenoidasym[1][-Pi,Pi,-Pi,Pi]."

    catenoidiso::usage="{u,v}->catenoidiso[a][u,v] is an isothermal parametrization of a catenoid.  To plot try catenoidiso[1][-Pi,Pi,-Pi,Pi]."

    circleinvolutemin::usage="{u,v}->circleinvolutemin[1,a][t][u,v] is a 1-parameter family of minimal surfaces such that {u,v}->circleinvolutemin[1,a][0][u,v] is a minimal surface containing the first involute of a circle as a geodesic.  To plot try circleinvolutemin[1,1][1][0,Pi,-1,1]."

    circularcone::usage="{u,v}->circularcone[a,b][u,v] is a parametrization of a circular cone of radius a and slope b/a.  To plot try circularcone[1,0.5][-Pi,Pi,-1,1]."

    circularcylinder::usage="{u,v}->circularcylinder[a][u,v] is a parametrization of a circular cylinder of radius a.  To plot try circularcylinder[1][-Pi,Pi,-4,4]."

    clothoidmin::usage="{u,v}->clothoidmin[a][t][u,v] is a 1-parameter family of minimal surfaces such that {u,v}->clothoidmin[a][0][u,v] is a minimal surface containing a clothoid as a geodesic.  To plot try clothoidmin[1][0][-5,5,-0.3,0.3]."

    cnccosurfrev::usage="{u,v}->cnccosurfrev[a,b][u,v] is a parametrization of a surface of revolution of constant negative curvature -1/a^2 of conic type.  To plot try cnccosurfrev[1,0.83][-Pi,Pi,-0.6,0.6]."

    cnccotcheb::usage="{u,v}->cnccotcheb[a,b][u,v] is a Tchebyshef parametrization of a constant negative curvature surface of revolution of conic type.  To plot try cnccotcheb[1,1.2][-3.5,3.5,-Pi/1.2,Pi/1.2]."

    cnchysurfrev::usage="{u,v}->cnchysurfrev[a,b][u,v] is a parametrization of a surface of revolution of constant negative curvature -1/a^2 of hyperboloid type.  To plot try cnchysurfrev[1,0.83][-Pi,Pi,-0.6,0.6]."

    cnchytcheb::usage="{u,v}->cnchytcheb[a,b][u,v] is a Tchebyshef parametrization of a constant negative curvature surface of revolution of hyperboloid type.  To plot try cnchytcheb[1,0.8][-3.5,3.5,-Pi/0.8,Pi/0.8]."

    cossurface::usage="{u,v}->cossurface[a][u,v] is the surface {u,v}->a*{Cos[u],Cos[v],Cos[u + v]}. The implicit equation is (z - a*x*y) == a^2*(1 - x^2)*(1 - y^2).  To plot try cossurface[1][-Pi,Pi,-Pi,Pi]."

    costa::usage="{u,v}->costa[u,v] is a parametrization of Costa's minimal surface. To plot try costa[0.001,1.001,0.001,1.001]."

    cpcsurfrev::usage="{u,v}->cpcsurfrev[a,b][u,v] is a parametrization of a surface of revolution of constant positive curvature 1/a^2. There are 3 cases: football type (a>b), sphere (a=b), barrel type (a<b).  To plot try cpcsurfrev[1,0.5][-Pi,Pi,-1,1]."

    crosscap::usage="{u,v}->crosscap[a][u,v] is a parametrization of a cross cap in R^3.  To plot try crosscap[1][-Pi/2,Pi/2,-Pi/2,Pi/2]."

    deltoidinvolutemin::usage="{u,v}->deltoidinvolutemin[a][t][u,v] is a 1-parameter family of minimal surfaces such that {u,v}->deltoidinvolutemin[a][0][u,v] is a minimal surface containing the involute of a deltoid as a geodesic.  To plot try deltoidinvolutemin[1][0][-2Pi,2Pi,-0.3,0.3]."

    deltoidmin::usage="{u,v}->deltoidmin[a][t][u,v] is a 1-parameter family of minimal surfaces such that {u,v}->deltoidmin[a][0][u,v] is a minimal surface containing a deltoid as a geodesic.  To plot try deltoidmin[1][0][-Pi,Pi,-0.3,0.3]."

    dini::usage="{u,v}->dini[a,b][u,v] is the standard parametrization of Dini's surface of constant negative curvature -1/a^2. It is the generalized helicoid of slant b generated by a tractrix. The case b=0 is the standard parametrization of a pseudosphere.  To plot try dini[1,0.2][0,4Pi,0.05,1]."

    dinihelicoid::usage="{u,v}->dinihelicoid[pm,a,b,c][u,v] is a generalized helicoid of constant negative curvature -1/a^2 and slope c. The parameter pm is plus or minus 1."

    dinitcheb::usage="{u,v}->dinitcheb[a,b][u,v] is a Tchebyshef principal patch and {p,q}->dinitcheb[a,b][p + q,p - q] is a Tchebyshef asymptotic patch for Dini's surface of constant negative curvature -1/a^2.  To plot try dinitcheb[1,0.2][-3,3,0,6Pi]."

    earth::usage="{u,v}->earth[u,v] is a parametrization of the earth (with equatorial diameter 12756.4 kilometers and polar diameter 12713.2 kilometers).  To plot try earth[-Pi,Pi,-Pi/2,Pi/2]."

    eighteight::usage="{u,v}->eighteight[u,v] is a parametrization of the generalized surface of revolution formed by moving a figure eight about a figure eight.  To plot try eighteight[-Pi,Pi,-Pi/2,Pi/2]."

    eightsurface::usage="{u,v}->eightsurface[u,v] is a surface of revolution generated by t->eight[t].  To plot try eightsurface[-Pi,Pi,-Pi/2,Pi/2]."

    ellhypcyclide::usage="{u,v}->ellhypcyclide[a,c,k][u,v] is a parametrization of the cyclide of Dupin of radius k whose focal sets are the ellipse u->{a*Cos[u],Sqrt[a^2 - c^2]*Sin[u],0} and the hyperbola v->{c*Sec[v],0,Sqrt[a^2 - c^2]*Tan[v]}.  To plot try ellhypcyclide[8,3,5][-Pi,Pi,-Pi,Pi]."

    ellipsemin::usage="{u,v}->ellipsemin[a,b][t][u,v] is a 1-parameter family of minimal surfaces such that {u,v}->ellipsemin[a,b][0][u,v] is an elliptical catenoid and {u,v}->ellipsemin[a,b][Pi/2][u,v] is an elliptical helicoid. The patch {u,v}->ellipsemin[a,b][0][u,v] contains the ellipse t->ellipse[a,b][t] as a geodesic.  To plot try ellipsemin[2,1][0][-Pi,Pi,-1,1]."

    ellipsoid::usage="{u,v}->ellipsoid[a,b,c][u,v] is a parametrization of an ellipsoid with axes of lengths a, b and c. {u,v}->ellipsoid[a,c][u,v] is a parametrization of an ellipsoid with axes of lengths a, a and c.  To plot try ellipsoid[1,2,3][-Pi,Pi,-Pi/2,Pi/2]."

    ellipticalhyperboloidminus::usage= "{u,v}->ellipticalhyperboloidminus[a,b,c][u,v] is a parametrization of a hyperboloid of one sheet as a ruled surface.  To plot try ellipticalhyperboloidminus[1,2,3][-Pi,Pi,-Pi/2,Pi/2]."

    ellipticalhyperboloidplus::usage= "{u,v}->ellipticalhyperboloidplus[a,b,c][u,v] is a parametrization of a hyperboloid of one sheet as a ruled surface.  To plot try ellipticalhyperboloidplus[1,2,3][-Pi,Pi,-Pi/2,Pi/2]."

    ellipticcoor::usage="{u,v,w}->ellipticcoor[d,e,f][a,b,c][w][u,v] is an elliptic coordinate patch on R^3. The parameters d, e and f are plus 1 or minus 1.  To plot try ellipticcoor[1,1,1][12,5,1][-1][1,5,5,12]."

    ellipticparaboloid::usage="{u,v}->ellipticparaboloid[a,b][u,v] is the surface {u,v}->{u,v,u^2/a^2 + v^2/b^2}.  To plot try ellipticparaboloid[1,2][-1,1,-1,1]."

    enneper::usage="{u,v}->enneper[u,v] is a parametrization of Enneper's minimal surface.  To plot try enneper[-1.5,1.5,-1.5,1.5]."

    ennepercatenoidpolar::usage="{r,theta}->ennepercatenoidpolar[n,a,b][t][r,theta] is the polar parametrization of a 1-parameter family of minimal surfaces constructed from the minimal curve whose Weierstrass representation is given by f[z_]:=2 and g[z_]:=a/z + b*z^n. For t=0 and a=b=1 the surface has a catenoid-like bottom and an enneper-like top with n + 1 lobes.  To plot try ennepercatenoidpolar[1,1,1][0][0.40,1.08,-Pi,Pi]."

    enneperpolar::usage="{r,theta}->enneperpolar[n][t][r,theta] is the polar parametrization of a 1-parameter family of minimal surfaces such that {r,theta}->enneperpolar[n][0][r,theta] is Enneper's minimal surface of degree n.  To plot try enneperpolar[1][0][0,2,-Pi,Pi]."

    exptwist::usage="{u,v}->exptwist[a,c][u,v] is a parametrization of a helicoid-like surface whose twisting varies exponentially.  To plot try exptwist[1,0.2][0,4Pi,-1,1]."

    exptwistasym::usage="{p,q}->exptwistasym[a,c][p,q] is the parametrization by asymptotic curves of {u,v}->exptwist[a,c][u,v].  To plot try exptwistasym[1,0.3][-Pi,2Pi,-Pi,2Pi]."

    flattorus::usage="{u,v}->flattorus[a,b][u,v] is a parametrization of a torus into R^4 as the product of two circles. {u,v}->flattorus[a,b,c,d][u,v]} is a parametrization of an elliptical torus into R^4 as the product of two ellipses. The Gaussian curvature is zero."

    funnel::usage="{u,v}->funnel[1,1,c][u,v] is a parametrization of the surface of revolution formed by rotating the graph of t->c*Log[t].  To plot try funnel[5,1,2][-Pi,Pi,0.01,1]."

    funnelasym::usage="{p,q}->funnelasym[c][p,q] is the parametrization by asymptotic curves of the surface of revolution formed by rotating the graph of t->c*Log[t].  To plot try funnelasym[1][-Pi/2,Pi/2,0.0,Pi]."

    gencube::usage="{u,v}->gencube[n,a,b,c][u,v] is a generalized cube. {u,v}->gencube[1,a/Sqrt[2],a/Sqrt[2],a][u,v] is a sphere. {u,v}->gencube[2,a,a,a][u,v] is a parametrization of an ordinary cube. {u,v}->genoctahedron[3,a,a,a][u,v] is an astroidial cube.  To plot try gencube[2,1,1,1][-Pi/2,Pi/2,-Pi,Pi]."

    genhypparab::usage="{u,v}->genhypparab[a,b][u,v] is a parametrization of a hyperbolic paraboloid as a ruled surface.  To plot try genhypparab[1,1][-1,1,-1,1]."

    genoctahedron::usage="{u,v}->genoctahedron[n,a,b,c][u,v] is a generalized octahedron. {u,v}->genoctahedron[1,a,b,c][u,v] is an ellipsoid. {u,v}->genoctahedron[2,a,a,a][u,v] is a parametrization of an ordinary octahedron. {u,v}->genoctahedron[3,a,b,c][u,v] is an astroidial ellipsoid.  To plot try genoctahedron[2,1,1,1][-Pi/2,Pi/2,-Pi,Pi]."

    handkerchief::usage="{u,v}->handkerchief[a][u,v] is a parametrization of a handkerchief shaped surface.  To plot try handkerchief[1][-1,1,-1,1]."

    heart::usage="{u,v}->heart[a][u,v] is a surface of revolution generated by t->cardioid[a][t] with the axis of revolution passing through the cusp.  To plot try heart[1][-Pi,Pi/2,-Pi,0]."

    helicoid::usage="{u,v}->helicoid[a,c][u,v] is the standard parametrization of a circular helicoid of radius a and slant c. {u,v}->helicoid[a,b,c][u,v] is the standard parametrization of an elliptical helicoid of slant c.  To plot try helicoid[1,1,0.3][0,4Pi,-1,1]."

    helicoidprin::usage="{u,v}->helicoidprin[b][u,v] is a parametrization of a circular helicoid of radius b and slant b by principal curves. The parametrization is also isothermal.  To plot try helicoidprin[1][-Pi/2,Pi/2,-Pi/2,Pi/2]."

    heltocat::usage="{u,v}->heltocat[t][u,v] is a 1-parameter family of minimal surfaces connecting a helicoid to a catenoid.  To plot try heltocat[Pi/4][0,4Pi,-1.5,1.5]."

    henneberg::usage="{u,v}->henneberg[u,v] is a parametrization of Henneberg's minimal surface.  To plot try henneberg[0.3,1,-Pi,Pi]."

    hyperbolicparaboloid::usage="{u,v}->hyperbolicparaboloid[u,v] is a parametrization of a hyperbolic paraboloid.  To plot try hyperbolicparaboloid[-1,1,-1,1]."

    hyperboloid::usage="{u,v}->hyperboloid[a,c][u,v] is the standard parametrization of a circular hyperboloid of one sheet. {u,v}->hyperboloid[a,b,c][u,v] is the standard parametrization of an elliptical hyperboloid of one sheet.  To plot try hyperboloid[1,2,3][0,3Pi/2,-2,2]."

    hypocycloidinvolutemin::usage="{u,v}->hypocycloidinvolutemin[a,b][t][u,v] is a 1-parameter family of minimal surfaces such that {u,v}->hypocycloidinvolutemin[a,b][0][u,v] is a minimal surface containing the involute of a hypocycloid as a geodesic.  To plot try hypocycloidinvolutemin[4,1][0][-Pi,Pi,-1,1]."

    hy2sheet::usage="{u,v}->hy2sheet[a,c][u,v] is the standard parametrization of a circular hyperboloid of two sheets. {u,v}->hy2sheet[a,b,c][u,v] is the standard parametrization an elliptical hyperboloid of two sheets.  To plot try hy2sheet[1,1,1][-1,1,-1,1]."

    invertedcircularcone::usage="{u,v}->invertedcircularcone[{p1,p2,p3},rho,a,b][u,v] is a parametrization of {u,v}->circularcone[a,b][u,v] inverted with respect to a sphere of radius rho centered at {p1,p2,p3}. {u,v}->invertedcircularcone[rho,a,b][u,v] is a parametrization of {u,v}->{u,v}->circularcone[a,b][u,v] inverted with respect to a sphere of radius rho centered at the origin. An inverted circular cone is a cyclide of Dupin.  To plot try invertedcircularcone[{5,0,0},2,1,1][-Pi,Pi,-12,12]."

    invertedcircularcylinder::usage="{u,v}->invertedcircularcylinder[{p1,p2,p3},rho,a][u,v] is a parametrization of {u,v}->circularcylinder[a][u,v] inverted with respect to a sphere of radius rho centered at {p1,p2,p3}. {u,v}->invertedcircularcylinder[rho,a][u,v] is a parametrization of {u,v}->circularcylinder[a][u,v] inverted with respect to a sphere of radius rho centered at the origin. An inverted circular cylinder is a cyclide of Dupin.  To plot try invertedcircularcylinder[{5,0,0},2,1][-Pi,Pi,-12,12]."

    invertedtorus::usage="{u,v}->invertedtorus[{p1,p2,p3},rho,a,b][u,v] is a parametrization of {u,v}->torus[a,b][u,v] inverted with respect to a sphere of radius rho centered at {p1,p2,p3}. {u,v}->invertedtorus[rho,a,b][u,v] is a parametrization of {u,v}->torus[a,b][u,v] inverted with respect to a sphere of radius rho centered at the origin. An inverted torus is a cyclide of Dupin.  To plot try invertedtorus[{35,0,0},8,8,3][-Pi,Pi,-Pi,Pi]."

    jorgemeekspolar::usage="jorgemeekspolar[n,a][r,theta] is a polar parametrization of a Jorge-Meeks n-noid.  To plot try jorgemeekspolar[3,1][0.001,1.999,-Pi,Pi]."

    kidney::usage="{u,v}->kidney[a][u,v] is a surface of revolution generated by t->nephroid[a][t].  To plot try kidney[1][0,3Pi/2,-Pi/2,Pi/2]."

    kleinbottle::usage="{u,v}->kleinbottle[a][u,v] is a parametrization in R^3 of the surface formed by moving and twisting a figure eight along a circle of radius a. The surface is nonorientable and a neighborhood of the self-intersection curve is nonorientable.  To plot try kleinbottle[2][-Pi/4,3Pi/2,-Pi,Pi]."

    kleinbottlebis::usage="{u,v}->kleinbottlebis[e,f][u,v] is a Klein bottle in R^3. A neighborhood of the self-intersection curve is orientable.  To plot try kleinbottlebis[1,2][-Pi,29.9517,-Pi,Pi]."

    kleinbottlewri::usage="{u,v}->kleinbottlewri[u,v] is WRI's version of a Klein bottle in R^3. A neighborhood of the self-intersection curve is orientable.  To plot try kleinbottlewri[0,2Pi,0,2Pi]."

    klein4::usage="{u,v}->klein4[a,b][u,v] is a parametrization of a Klein bottle as a surface in R^4."

    kuen::usage="{u,v}->kuen[a][u,v] is a parametrization of Kuen's surface. It has constant negative curvature -1/a^2.  To plot try kuen[1][-4,4,0.01,Pi - 0.01]."

    kuentcheb::usage="{u,v}->kuentcheb[a][u,v] is a Tchebyshef principal patch and {p,q}->kuentcheb[a][p + q,p - q] is a Tchebyshef asymptotic patch for Kuen's surface of constant negative curvature -1/a^2.  To plot try kuentcheb[1][-4,4,-4,4]."

    lemniscatemin::usage="{u,v}->lemniscatemin[a][t][u,v] is a 1-parameter family of minimal surfaces such that {u,v}->lemniscatemin[a][0][u,v] is a minimal surface containing a lemniscate as a geodesic.  To plot try lemniscatemin[1][0][-Pi,Pi,-0.2Pi,0.2Pi]."

    logminpolar::usage="{r,theta}->logminpolar[n][t][r,theta] is a 1-parameter family of minimal surfaces whose height function is the identity and whose Gauss map is an iterated logarithm.  To plot try logminpolar[0][0][0.005,2.005,-3Pi,3Pi]."

    menn::usage="{u,v}->menn[a][u,v] is a parametrization of Menn's surface.  To plot try menn[1][-1.5,1.5,-1.5,1.5]."

    mercatorellipsoid::usage="{u,v}->mercatorellipsoid[a,b,c][u,v] is the Mercator injection of an ellipsoid with axes of lengths a, b and c into R^3.  To plot try mercatorellipsoid[1,2,3][0,3Pi/2,-0.49Pi,0.49Pi]."

    mercatorsphere::usage="{u,v}->mercatorsphere[a][u,v] is the Mercator injection of a sphere of radius a into R^3.  To plot try mercatorsphere[1][0,3Pi/2,-0.49Pi,0.49Pi]."

    moebcirc::usage="{u,v}->moebcirc[a][u,v] is a parametrization of a Moebius strip that has a circle as boundary.  To plot try moebcirc[1][0,Pi,0,Pi]."

    moebiusstrip::usage="{u,v}->moebiusstrip[a][u,v] is the standard parametrization of a Moebius strip.  To plot try moebiusstrip[1][0,2Pi,-0.3,0.3]."

    monkeysaddle::usage="{u,v}->monkeysaddle[u,v] is a parametrization of a monkey saddle. {u,v}->monkeysaddle[n][u,v] is a parametrization of a monkey saddle for a monkey with n-2 tails.  To plot try monkeysaddle[-1.1,1.1,-1.1,1.1]."

    monkeysaddlepolar::usage="{r,theta}->monkeysaddlepolar[n][r,theta] is a polar parametrization of a monkey saddle with n - 2 tails.  To plot try monkeysaddlepolar[3][0,1,0,2Pi]."

    nodoid::usage="See nodoidnds.  To plot try nodoid[-9,9][8,3,1,0.6][0,3Pi/2,-9,9]."

    nodoidnds::usage="nodoidnds[vmin,vmax][a,b,c,e][u,v] is an approximation over the interval vmin<v<vmax found by NDSolve to a nodoid. It is a surface of revolution of a curve of Delaunay of eccentricity e and has constant mean curvature."

    parabolamin::usage="{u,v}->parabolamin[a][t][u,v] is a 1-parameter family of minimal surfaces such that {u,v}->parabolamin[a][0][u,v] is a minimal surface containing a parabola as a geodesic.  To plot try parabolamin[1][0][-2,2,-0.3,0.3]."

    paraboliccoor::usage="{u,v,w}->paraboliccoor[d,e,f][a,b,c][w][u,v] is a parabolic coordinate patch on R^3. The parameters d, e and f are plus 1 or minus 1.  To plot try paraboliccoor[1,1,1][2,1,3][4][-9,1,1,2]."

    paraboliccyclide::usage="{u,v}->paraboliccyclide[a,k][u,v] is a parametrization of the cyclide of Dupin of radius k whose focal sets are the parabolas u->{u,0,-u^2/(8*a) + a} and v->{0,v,v^2/(8*a) - a}.  To plot try paraboliccyclide[2,1][-40,40,-40,40]."

    paraboloid::usage="{u,v}->paraboloid[a][u,v] is a circular paraboloid of radius a. {u,v}->paraboloid[a,b][u,v] and {u,v}->paraboloid[a,b,c][u,v] is either an elliptic or a hyperbolic paraboloid.  To plot try paraboloid[1][-1,1,-1,1]."

    paraboloidpolar::usage="{r,theta}->paraboloidpolar[n,a,b][r,theta] is a polar parametrization of a generalized paraboloid.  To plot try paraboloidpolar[2,2,1][0,1,0,3Pi/2]."

    pear::usage="{u,v}->pear[a,b][u,v] is a surface of revolution generated by t->piriform[a,b][t] with the axis of revolution passing through the cusp.  To plot try pear[1,1][0,3Pi/2,-Pi/2,Pi/2]."

    pearbis::usage="{u,v}->pearbis[u,v] is a parametrization of the surface of revolution that resembles an pear.  To plot try pearbis[0,3Pi/2,-Pi,Pi]."

    perturbedms::usage="{u,v}->perturbedms[a][u,v] is a parametrization of a monkey saddle perturbed by a circular paraboloid.  To plot try perturbedms[-1.0][-1,1,-1,1]."

    perturbedmspolar::usage="{r,theta}->perturbedmspolar[n,a][r,theta] is a polar parametrization of a monkey saddle of order n perturbed by a circular paraboloid.  To plot try perturbedmspolar[3,-1][0,1,0,2Pi]."
    
    pillow::usage="{u,v}->pillow[a][u,v] is a pillow-shaped surface.  To plot try pillow[1][-Pi,Pi,-Pi,Pi]."

    plane::usage="{u,v}->plane[a1,a2,b1,b2,c1,c2][u,v] is a parametrization of a plane in R^3."

    planepolar::usage="{r,theta}->planepolar[a][r,theta] is a polar parametrization of the plane."

    planetocat::usage="{u,v}->planetocat[t][u,v] is a 1-parameter family of minimal surfaces containing a plane and a catenoid. The principal curves of each surface are planar.  To plot try planetocat[0.1][0,3Pi/2,-2,2]."

    plucker::usage="{u,v}->plucker[u,v] is the standard parametrization of Plucker's surface. {u,v}->plucker[n][u,v] is a parametrization of Plucker's surface with n folds. {u,v}->plucker[m,n,a][u,v] is a generalization of the monkey saddle and Plucker's surface.  To plot try plucker[-1,1,-1,1]."

    pluckerpolar::usage="{r,theta}->pluckerpolar[n][r,theta] is the polar parametrization of Plucker's surface with n folds. {r,theta}->pluckerpolar[m,n,a][r,theta] is a generalization of the polar parametrizations of the monkey saddle and Plucker's surface.  To plot try pluckerpolar[0,5,1][0,1,-Pi,Pi]."

    pseudocrosscap::usage="{u,v}->pseudocrosscap[u,v] is a parametrization of a pseudocrosscap in R^3.  To plot try pseudocrosscap[-1,0.5,0,2Pi]."

    pseudosphere::usage="{u,v}->pseudosphere[a][u,v] is the standard parametrization of a pseudosphere of constant negative curvature -1/a^2.  To plot try pseudosphere[1][0,3Pi/2,0.05,Pi-0.05]."

    pseudospherebis::usage="{u,v}->pseudospherebis[a][u,v] is a parametrization of a pseudosphere of constant negative curvature -1/a^2.  To plot try pseudospherebis[1][0,3Pi/2,0.01,2]."

    pseudospheretcheb::usage="{u,v}->pseudospheretcheb[a][u,v] is a Tchebyshef principal patch and {p,q}->pseudospheretcheb[a][p + q,p - q] is a Tchebyshef asymptotic patch for a pseudosphere of constant negative curvature -1/a^2.  To plot try pseudospheretcheb[1][-2.8,2.8,0,3Pi/2]."

    rconoid::usage="{u,v}->rconoid[u,v] is a right conoid. It coincides with {u,v}->2*pluckerpolar[1][v/2,u].  To plot try rconoid[-Pi,Pi,-2,2]."

    richmond::usage="{u,v}->richmond[u,v] is a minimal surface with one planar end.  To plot try richmond[-0.5,2,0.5,2]."

    richmondpolar::usage="{r,theta}->richmondpolar[n][t][r,theta] is the polar parametrization of a 1-parameter family of minimal surfaces such that {r,theta}->richmondpolar[n][0][r,theta] is a minimal surface with one planar end of degree n.  To plot try richmondpolar[1][0][0.3,1.3,0,2Pi]."

    roman::usage="{u,v}->roman[a][u,v] is a parametrization of Steiner's Roman surface of radius a in R^3.  To plot try roman[1][0,Pi,-Pi/2,Pi/2]."

    rosemin::usage="{u,v}->rosemin[n,a][t][u,v] is a 1-parameter family of minimal surfaces such that {u,v}->rosemin[n,a][0][u,v] contains the rose t->{a*Cos[t]*Cos[n*t],a*Cos[n*t]*Sin[t]} as a geodesic.  To plot try rosemin[3,1][0][0,Pi,-0.3,0.3]."

    scherk::usage="{u,v}->scherk[a][u,v] is a parametrization of Scherk's minimal surface. {u,v}->scherk[a,theta][u,v] is a parametrization of twisted Scherk's minimal surface of angle theta.  To plot try scherk[1][-Pi/2 + 0.01,3Pi/2 + 0.01,-Pi/2 + .01,3Pi/2 + 0.01]."

    scherk5::usage="{u,v}->scherk5[a,b,c][u,v] is a parametrization of Scherk's fifth minimal surface, where a, b, c are plus 1 or minus 1.  {u,v}->scherk5[u,v] is the same as {u,v}->scherk5[a,b,c][u,v]. To plot try scherk5[-1,1,-1,1]."

    shoe::usage="{u,v}->shoe[a,-a][u,v] resembles the instep of a shoe.  To plot try shoe[1,-1][-1.5,1.5,-1.5,1.5]."

    shoeasym::usage="{u,v}->shoeasym[a,b][u,v] is an asymptotic parametrization of a shoe surface.  To plot try shoeasym[1,-1][-1,1.2,-1,1.2]."

    sievert::usage="{u,v}->sievert[a][u,v] is a parametrization of Sievert's surface of constant positive curvature a^2.  To plot try sievert[1][-Pi/2,Pi/2,0.001,Pi - 0.001]."

    sinovaloid::usage="{u,v}->sinovaloid[n,a][u,v] is the surface of revolution generated by t->sinoval[n,a][t].  To plot try sinovaloid[3,1][0,3Pi/2,-Pi/2,Pi/2]."

    sinsurface::usage="{u,v}->sinsurface[a][u,v] is the {u,v}->a*{Sin[u],Sin[v],Sin[u + v]}. The implicit equation is 4*x^2*y^2*(a^2 - z^2) == a^2*(x^2 + y^2 - z^2)^2.  To plot try sinsurface[1][-Pi,Pi,-Pi,Pi]."

    snail::usage="{u,v}->snail[u,v] is a surface resembling a snail.  To plot try snail[-Pi,Pi,-Pi,Pi]."

    sphere::usage="{u,v}->sphere[a][u,v] is the standard parametrization of a sphere of radius a centered at the origin.  To plot try sphere[1][0,3Pi/2,-Pi/2,Pi/2]."

    spherehelix::usage="{u,v}->spherehelix[a,b][u,v] is ahelical prolongation of a sphere into R^4."

    stereographicellipsoid::usage="{u,v}->stereographicellipsoid[a,b,c][u,v] is a stereographic parametrization of an ellipsoid with axes of lengths a, b and c.  To plot try stereographicellipsoid[5,3,1][-2.8,2.8,-2.8,2.8]."

    stereographicsphere::usage="{u,v}->stereographicsphere[a][u,v] is a stereographic parametrization of a sphere of radius a in R^3.  To plot try stereographicsphere[1][-2.8,2.8,-2.8,2.8]."

    stereographicspherepolar::usage="{r,theta}->stereographicspherepolar[a][r,theta] is a polar stereographic parametrization of a sphere of radius a in R^3.  To plot try stereographicspherepolar[1][0,3,0,2Pi]."

    swallowtail::usage="{u,v}->swallowtail[u,v] is a parametrization of a swallow tail shaped surface.  To plot try swallowtail[-3,2,-0.8,0.8]."

    tetrahedral::usage="{u,v}->tetrahedral[m,n][A,B,C][a,b,c][u,v] is a parametrization of a tetrahedral surface."

    thomsen::usage="{u,v}->thomsen[a,b][u,v] is a parametrization of Thomsen's minimal surface.  To plot try thomsen[1,-2][-Pi/2,Pi/2,0,2Pi]."

    torus::usage="{u,v}->torus[a,b][u,v] is a parametrization of the torus formed by revolving a circle of radius b in the xz-plane about the z-axis along a circle of radius a in the xy-plane. {u,v}->torus[a,b,c][u,v] is a parametrization of the torus formed by revolving an ellipse with axes of lengths b and c in the xz-plane about the z-axis along a circle of radius a in the xy-plane. {u,v}->torus[a,b,b,d,d][u,v] is a surface formed by moving a circle of radius b in the xz-plane along an ellipse in the xy-plane. {u,v}->torus[a,b,c,d,e][u,v] is a surface formed by moving an ellipse in the xz-plane along an ellipse in the xy-plane.  To plot try torus[8,3][0,3Pi/2,-Pi,Pi]."

    torusasym::usage="{p,q}->torusasym[pm,a][p,q] is an asymptotic parametrization of {u,v}->torus[a,a][u,v]. The parameter pm is plus or minus 1.  To plot try torusasym[1,1][-Pi,Pi,-Pi,Pi]."

    torusbis::usage="{u,v}->torusbis[pm,a,b][u,v] is a parametrization of a torus. The parameter pm is plus or minus 1.  To plot try torusbis[1,8,3][5,11,0,2Pi]."

    twiflat::usage="{u,v}->twiflat[pm,a,c][sl][u,v] is a parametrization of a generalized helicoid of zero Gaussian curvature of slant sl. The parameter pm is plus or minus 1.  To plot try twiflat[1,1,1][1][0,7Pi/2,1,5]." 

    twisphere::usage="{u,v}->twisphere[a,b][u,v] is a parametrization of a generalized helicoid formed by rotating and twisting a semicircle. It is also called a corkscrew surface.  To plot try twisphere[1,1][0,2Pi,-Pi,Pi]." 

    veronese::usage="{u,v}->veronese[a][u,v] is a parametrization of a Veronese surface in R^5."

    wallis::usage="{u,v}->wallis[a,b,c][u,v] is a parametrization of a conical edge of Wallis.  To plot try wallis[1,1,3][Pi/4,3Pi/4,0,1]."

    whitneyumbrella::usage="{u,v}->whitneyumbrella[u,v] is a parametrization of a Whitney umbrella.  To plot try whitneyumbrella[-1,1,-1,1]."

    wrinkles::usage="{u,v}->wrinkles[a][u,v] is the Monge patch {u,v}->{u,v,a*(u/v + v/u)}.  To plot try wrinkles[0.05][-1,1,-1,1]."

(* Implicitly Defined Surfaces *)

    cossurfaceimplicit::usage="cossurfaceimplicit[a][x,y,z]==0 is the nonparametric form of the surface {u,v}->{a*Cos[u],a*Cos[v],a*Cos[u + v]}."

    crosscapimplicit::usage="crosscapimplicit[a,b][x,y,z]==0 is the nonparametric form of a crosscap."

    ellipsoidimplicit::usage="ellipsoidimplicit[a,b,c][x,y,z]==0 is the nonparametric form of an ellipsoid."

    ellipticparaboloidimplicit::usage="ellipticparaboloidimplicit[a,b,c][x,y,z]==0 is the nonparametric form of an elliptic paraboloid."

    equihom1implicit::usage="equihom1implicit[x,y,z]==0 is the nonparametric form of the first equiaffinely homogeneous surface."

    equihom2implicit::usage="equihom2implicit[x,y,z]==0 is the nonparametric form of the second equiaffinely homogeneous surface."

    equihom3implicit::usage="equihom3implicit[x,y,z]==0 is the nonparametric form of the third equiaffinely homogeneous surface."

    equihom4implicit::usage="equihom4implicit[x,y,z]==0 is the nonparametric form of the fourth equiaffinely homogeneous surface."

    equihom5implicit::usage="equihom5implicit[x,y,z]==0 is the nonparametric form of the fifth equiaffinely homogeneous surface."

    equihom6implicit::usage="equihom6implicit[x,y,z]==0 is the nonparametric form of the sixth equiaffinely homogeneous surface."

    goursatimplicit::usage="goursatimplicit[a,b,c][x,y,z]==0 is the nonparametric form of Goursat's surface."

    hyperbolicparaboloidimplicit::usage="hyperbolicparaboloidimplicit[a,b,c][x,y,z]==0 is the nonparametric form of a hyperbolic paraboloid."

    hyperboloidimplicit::usage="hyperboloidimplicit[a,b,c][x,y,z]==0 is the nonparametric form of a hyperboloid of one sheet."

    hy2sheetimplicit::usage="hy2sheetimplicit[a,b,c][x,y,z]==0 is the nonparametric form of a hyperboloid of two sheets."

    kazoolaimplicit::usage="kazoolaimplicit[a,b,c,d][x,y,z]==0 is the nonparametric form of a kazoola."

    kummerimplicit::usage="kummerimplicit[x,y,z]==0 is the nonparametric form of Kummer's surface."

    scherkimplicit::usage="scherkimplicit[a][x,y,z]==0 is the nonparametric form of Scherk's minimal surface. scherkimplicit[a,phi][x,y,z]==0 is the nonparametric form of a twisted Scherk's minimal surface."

    scherk5implicit::usage="scherk5implicit[x,y,z]==0 is the nonparametric form of a Scherk's fifth minimal surface."

    sinsurfaceimplicit::usage="sinsurfaceimplicit[a][x,y,z]==0 is the nonparametric form of the surface {u,v}->{a*Sin[u],a*Sin[v],a*Sin[u + v]}."

    sphereimplicit::usage="sphereimplicit[a][x,y,z]==0 is the nonparametric form of a sphere. sphereimplicit[n,a][x,y,z]==0 is the nonparametric form of the surface x^(2*n) + y^(2*n) +z^(2*n) = a^(2*n)."

    torusimplicit::usage="torusimplicit[a,b][x,y,z]==0 is the nonparametric form of a torus."

    twocuspsimplicit::usage="twocuspsimplicit[x,y,z]==0 is the nonparametric form of a surface with two cusps."

(* Drum Plots *)

    circularmembrane::usage="{r,theta}->circularmembrane[m,n,t][r,theta] is a parametrization of a circular membrane of degree {m,n} at time t."

    circularmembraneplot::usage="circularmembraneplot[m,n,t] plots a circular membrane of degree {m,n} at time t."

    circularmembranemovie::usage=" circularmembranemovie[m,n,{t,tmin,tmax,numf}] animates a circular membrane of degree {m,n} from tmin to tmax with numf frames."

    squaremembrane::usage="{u,v}->squaremembrane[m,n,t][u,v] is a parametrization of a square membrane of degree {m,n} at time t."

    squaremembraneplot::usage="squaremembraneplot[m,n,t] plots a square membrane of degree {m,n} at time t."

    squaremembranemovie::usage="squaremembranemovie[m,n,{t,tmin,tmax,numf}] animates a square membrane of degree {m,n} from tmin to tmax with numf frames."

(* Minimal Curves *)

    astroidmincurve::usage="z->astroidmincurve[a][z] is the minimal curve generated by the astroid t->astroid[a][t]."

    bourmincurve::usage="z->bourmincurve[m][z] is the minimal curve corresponding to Bour's minimal surface of order m. z->bourmincurve[m,n][z] is the minimal curve corresponding to Bour's minimal surface of order {m,n}. Note that bourmincurve[m]=bourmincurve[m,1], enneper[n]=bourmincurve[2,n] and richmond[n][z]=bourmincurve[0,n + 1][z]/2."

    cardioidmincurve::usage="z->cardioidmincurve[a][z] is the minimal curve generated by the cardioid t->cardioid[a][t]."

    catalanmincurve::usage="z->catalanmincurve[a][z] is the minimal curve corresponding to Catalan's minimal surface. It contains a cycloid as a geodesic."

    catenoidmincurve::usage="z->catenoidmincurve[a][z] is the minimal curve corresponding to a catenoid. It contains a circle as a geodesic."

    cayleysexticmincurve::usage="z->cayleysexticmincurve[a][z] is the minimal curve generated by the Cayley's sextic t->cayleysextic[a][t]."

    circleinvolutemincurve::usage="z->circleinvolutemincurve[n,a][z] is the minimal curve generated by the nth involute starting at {a,0} of the circle t->{a*Cos[t],a*Sin[t]}."

    cissoidmincurve::usage="z->cissoidmincurve[a][z] is the minimal curve generated by the cissoid t->cissoid[a][t]."

    clothoidmincurve::usage="z->clothoidmincurve[a][z] is the minimal curve generated by the clothoid t->clothoid[1,a][z]."

    cnchyprofilemincurve::usage="z->cnchyprofilemincurve[pm,a,b][z] is the minimal curve generated by the curve t->cnchyprofile[pm,a,b][t]. The parameter pm is plus 1 or -1."

    costamincurve::usage="z->costamincurve[z] is the minimal curve that determines Costa's minimal surface."

    cpcprofilemincurve::usage="z->cpcprofilemincurve[pm,a,b][z] is the minimal curve generated by the curve t->cpcprofile[pm,a,b][t]. The parameter pm is plus 1 or -1."

    deltoidinvolutemincurve::usage="z->deltoidinvolutemincurve[a][z] is the minimal curve generated by the involute starting at t=0 of t->deltoid[a][t]."

    deltoidmincurve::usage="z->deltoidmincurve[a][z] is the minimal curve generated by the deltoid t->deltoid[a][t]."

    ellipsemincurve::usage="z->ellipsemincurve[a,b][z] is the minimal curve generated by the ellipse t->ellipse[a,b][t]."

    ennepercatenoidmincurve::usage="z->ennepercatenoidmincurve[n,a,b][z] is the minimal curve whose Weierstrass representation is given by f[z_]:=2 and g[z_]:=a/z + b*z^n."

    ennepermincurve::usage="z->ennepermincurve[n][z] is the minimal curve corresponding to Enneper's minimal surface of order n."

    epicycloidmincurve::usage="z->epicycloidmincurve[a,b][z] is the minimal curve generated by the epicycloid t->epicycloid[a,b][t]."

    hennebergmincurve::usage="z->hennebergmincurve[z] is the minimal curve corresponding to Henneberg's minimal surface."

    hyperbolamincurve::usage="z->hyperbolamincurve[a,b][z] is the minimal curve generated by the hyperbola t->hyperbola[a,b][t]."

    hypocycloidinvolutemincurve::usage="z->hypocycloidinvolutemincurve[a,b][z] is the minimal curve generated by the involute starting at t=0 of t->hypocycloid[a,b][t]."

    hypocycloidmincurve::usage="z->hypocycloidmincurve[a,b][z] is the minimal curve generated by the hypocycloid t->hypocycloid[a,b][t]."

    jorgemeeksmincurve::usage="z->jorgemeeksmincurve[n,a][z] is the minimal curve used to generate the Jorge-Meeks n-noid."

    jorgemeeksmincurveprime::usage="z->jorgemeeksmincurveprime[n,a][z] is the derivative of z->jorgemeeksmincurve[n,a][z]."

    lemniscatemincurve::usage="z->lemniscatemincurve[a][z] is the minimal curve generated by the lemniscate t->lemniscate[a][t]."

    logmincurve::usage="z->logmincurve[n][z] is a minimal curve whose height function is the identity and whose Gauss map is an iterated logarithm."

    logspiralmincurve::usage="z->logspiralmincurve[a,b][z] is the minimal curve generated by the log spiral t->logspiral[a,b][t]."

    nephroidmincurve::usage="z->nephroidmincurve[a][z] is the minimal curve generated by the nephroid t->nephroid[a][t]."

    nielsenspiralmincurve::usage="z->nielsenspiralmincurve[a][z] is the minimal curve generated by Nielsen's spiral t->nielsenspiral[a][t]."

    parabolamincurve::usage="z->parabolamincurve[a][z] is the minimal curve generated by the parabola t->parabola[a][t]."

    richmondmincurve::usage="z->richmondmincurve[n][z] is the minimal curve corresponding to Richmond's minimal surface of order n."

    rosemincurve::usage="z->rosemincurve[n,a][z] is the minimal curve generated by the rose t->rose[n,a][t]."

    scherktowermincurveprime::usage="z->scherktowermincurveprime[n][z] is the integrand of the Scherk tower minimal curve."

    scmincurve::usage="z->scmincurve[z] is the minimal curve generated by the semicubical parabola t->sc[t]."

    tractrixmincurve::usage="z->tractrixmincurve[a][z] is the minimal curve generated by the tractrix t->tractrix[a][t]."

(* Surface Metrics *)

    geopolar::usage="geopolar[f] is the geodesic polar coordinate metric relative to the function f."

    liouville::usage="liouville[f,g] is the metric of a Liouville patch corresponding to the functions f and g."

    poincare::usage="poincare[lambda] is the metric on the Poincare upper half-plane with constant curvature -1/lambda^2. poincare[m,n,lambda] is a generalization of the metric poincare[lambda]."

    poindisk::usage="poindisk[lambda] is the metric on the Poincare disk with constant curvature -1/lambda^2."

    schwarzschildplane::usage="schwarzschildplane[m] is the 2-dimensional Schwarzschild metric. schwarzschildplane[m,f] is the 2-dimensional Schwarzschild metric relative to the function f."

Begin["`Private`"]

(* Parametrically Defined Surfaces *)

    agnesirev[a_][u_,v_]:=
    {-2*a*Cos[u]*Cos[v]^2,-2*a*Cos[v]^2*Sin[u],2*a*Tan[v]}

    agnesirev[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[agnesirev[a][u,v],{u,umin,umax},{v,vmin,vmax},opt];

    apple[u_,v_]:= {Cos[u]*(4 + 3.8*Cos[v]),
                    Sin[u]*(4 + 3.8*Cos[v]),
        (Cos[v] + Sin[v]-1)*(1 + Sin[v])*
         Log[1 - Pi*v/10] + 7.5*Sin[v]}

    apple[umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[apple[u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    astell[a_,b_,c_][u_,v_]:= {a*Cos[u]*Cos[v],
                               b*Sin[u]*Cos[v],c*Sin[v]}^3

    astell[a_,b_,c_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[astell[a,b,c][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    bohdom[a_,b_,c_][u_,v_]:= {a*Cos[u],
                               a*Sin[u] + b*Cos[v],c*Sin[v]}

    bohdom[a_,b_,c_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[bohdom[a,b,c][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    bour[u_,v_]:= {(2*u^2 - u^4 - 2*v^2 + 6*u^2*v^2 - v^4)/4,
                    u*v*(-1 - u^2 + v^2),(2/3)*(u^3 - 3*u*v^2)}

    bour[umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[bour[u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    bour[m_][t_][u_,v_]:= Simplify[ComplexExpand[
        analytictominimal[bourmincurve[m]][t][u,v],
        TargetFunctions->{Re,Im}]]

    bourpolar[m_][t_][r_,theta_]:=
      {r^(m - 1)*Cos[t - (m - 1)*theta]/(m - 1) -
       r^(m + 1)*Cos[t - (m + 1)*theta]/(m + 1),
       r^(m - 1)*Sin[t - (m - 1)*theta]/(m - 1) +
       r^(m + 1)*Sin[t - (m + 1)*theta]/(m + 1),
       2*r^m*Cos[t - m*theta]/m}

    bourpolar[m_,n_][t_][r_,theta_]:=
      {r^(m - 1)*Cos[t - (m - 1)*theta]/(m - 1) -
       r^(m + 2*n - 1)*Cos[t - (m + 2*n - 1)*theta]/(m + 2*n - 1),
       r^(m - 1)*Sin[t - (m - 1)*theta]/(m - 1) +
       r^(m + 2*n - 1)*Sin[t - (m + 2*n - 1)*theta]/(m + 2*n - 1),
       2*r^(m + n - 1)*Cos[t - (m + n - 1)*theta]/(m + n - 1)}

    bourpolar[m_,n_][t_:0][rmin_,rmax_,thetamin_,thetamax_,opt___]:=
    pplot3d[bourpolar[m,n][t][r,theta],
	{r,rmin,rmax},{theta,thetamin,thetamax},
	opt,PlotPoints->{20,40}];
    
    boy[a_,b_,c_][u_,v_]:=
      {(a/2)*(-Cos[v]^2 + 2*Cos[u]^2*Sin[v]^2 + 
        Cos[v]*Cos[u]*Sin[v]*(-Cos[v]^2 + Cos[u]^2*Sin[v]^2) - 
        Sin[v]^2*Sin[u]^2 + 2*Cos[v]*Sin[v]*Sin[u]*
        (-Cos[v]^2 + Sin[v]^2*Sin[u]^2) - Sin[v]^4*Sin[4*u]/4),
       (Sqrt[3]/2)*b*(-Cos[v]^2 + 
        Cos[v]*Cos[u]*Sin[v]*(Cos[v]^2 - Cos[u]^2*Sin[v]^2) + 
        Sin[v]^2*Sin[u]^2 - (Sin[v]^4*Sin[4*u])/4),
       c*(Cos[v] + Cos[u]*Sin[v] + Sin[u]*Sin[v])*
         (-4*Sin[v]*(Cos[u] - Sin[u])*(-Cos[v] + Cos[u]*Sin[v])*
         (Cos[v] - Sin[u]*Sin[v]) + 
         (Cos[v] + Cos[u]*Sin[v] + Sin[u]*Sin[v])^3)}

    boy[a_,b_,c_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[boy[a,b,c][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    catalan[a_][u_,v_]:= {a*(u - Sin[u]*Cosh[v]),
                          a*(1 - Cos[u]*Cosh[v]),
                          -4*a*Sin[u/2]*Sinh[v/2]}

    catalan[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[catalan[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    catalandef[a_][t_][u_,v_]:=
        {a*Cos[t]*(u - Cosh[v]*Sin[u]) + 
         a*Sin[t]*(v - Cos[u]*Sinh[v]),
         a*Cos[t]*(1 - Cos[u]*Cosh[v]) + 
         a*Sin[t]*Sin[u]*Sinh[v],
         4*a*(1 - Cos[u/2]*Cosh[v/2])*Sin[t] - 
         4*a*Cos[t]*Sin[u/2]*Sinh[v/2]}

    catalandef[a_][t_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[catalandef[a][t][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    catenoid[a_][u_,v_]:= {a*Cos[u]*Cosh[v/a],
                           a*Sin[u]*Cosh[v/a],v}

    catenoid[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[catenoid[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    catenoid[a_,b_,c_][u_,v_]:= {a*Cos[u]*Cosh[v],
                                 b*Sin[u]*Cosh[v],c*v}

    catenoidasym[a_][p_,q_]:=
        {a*Cos[(p + q)/2]*Cosh[(q - p)/2],
         a*Sin[(p + q)/2]*Cosh[(q - p)/2],
         a*(q - p)/2}

    catenoidasym[a_][pmin_,pmax_,qmin_,qmax_,opt___]:=
    pplot3d[catenoidasym[a][p,q],{p,pmin,pmax},{q,qmin,qmax},
	opt];

    catenoidiso[a_][u_,v_]:= {a*Cos[u/a]*Cosh[v/a],
                              a*Sin[u/a]*Cosh[v/a],v}

    catenoidiso[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[catenoidiso[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    circleinvolutemin[1,a_][t_][u_,v_]:=
       {a*Cos[t]*(Cos[u]*Cosh[v] +
           u*Cosh[v]*Sin[u] - v*Cos[u]*Sinh[v]) + 
        a*Sin[t]*(v*Cosh[v]*Sin[u] +
           u*Cos[u]*Sinh[v] - Sin[u]*Sinh[v]),
        a*Sin[t]*(-v*Cos[u]*Cosh[v] + Cos[u]*Sinh[v] + 
           u*Sin[u]*Sinh[v]) + a*Cos[t]*(-u*Cos[u]*Cosh[v] +
           Cosh[v]*Sin[u] - v*Sin[u]*Sinh[v]),
        -a*u*v*Cos[t] + a*(u^2 - v^2)*Sin[t]/2}

    circleinvolutemin[1,a_][t_:0][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[circleinvolutemin[1,a][t][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    circularcone[a_,b_][u_,v_]:= {a*v*Cos[u],a*v*Sin[u],b*v}

    circularcone[a_,b_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[circularcone[a,b][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    circularcylinder[a_][u_,v_]:= {a*Cos[u],a*Sin[u],v}

    circularcylinder[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[circularcylinder[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    clothoidmin[a_][t_][u_,v_]:=
      {a*Sqrt[Pi]*Cos[t]*Re[FresnelS[(u + I*v)/Sqrt[Pi]]] + 
       a*Sqrt[Pi]*Im[FresnelS[(u + I*v)/Sqrt[Pi]]]*Sin[t],
       a*Sqrt[Pi]*Cos[t]*Re[FresnelC[(u + I*v)/Sqrt[Pi]]] + 
       a*Sqrt[Pi]*Im[FresnelC[(u + I*v)/Sqrt[Pi]]]*Sin[t],
       -(a*v*Cos[t]) + a*u*Sin[t]}

    clothoidmin[a_][t_:0][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[clothoidmin[a][t][u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{60,10}];

    cnccosurfrev[a_,b_][u_,v_]:= {b*Cos[u]*Sinh[v/a],
                                  b*Sin[u]*Sinh[v/a],
                      -I*Sqrt[a^2 - b^2]*EllipticE[I*v/a,
                      -b^2/(a^2 - b^2)]}

    cnccosurfrev[a_,b_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[cnccosurfrev[a,b][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    cnccotcheb[a_,b_][u_,v_]:=
        {(a/b)*JacobiCN[b*u,1/b^2]*Cos[b*v],
         (a/b)*JacobiCN[b*u,1/b^2]*Sin[b*v],
         a*(b*u - EllipticE[JacobiAmplitude[b*u,1/b^2],1/b^2])}
         
    cnccotcheb[a_,b_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[cnccotcheb[a,b][u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{43,25}];

    cnccotcheb[a_,b_,c_][u_,v_]:=
        {(a/b)*JacobiCN[b*u,1/b^2]*Cos[b*v],
         (a/b)*JacobiCN[b*u,1/b^2]*Sin[b*v],
         c*(b*u - EllipticE[JacobiAmplitude[b*u,1/b^2],1/b^2])}

    cnchysurfrev[a_,b_][u_,v_]:= {b*Cos[u]*Cosh[v/a],
                                  b*Sin[u]*Cosh[v/a],
                           - I*a*EllipticE[I*v/a,-b^2/a^2]}

    cnchysurfrev[a_,b_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[cnchysurfrev[a,b][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    cnchytcheb[a_,b_][u_,v_]:=
        {(a/b)*JacobiDN[u,b^2]*Cos[b*v],
         (a/b)*JacobiDN[u,b^2]*Sin[b*v],
         (a/b)*(u - EllipticE[JacobiAmplitude[u,b^2],b^2])}
         

    cnchytcheb[a_,b_,c_][u_,v_]:=
        {(a/b)*JacobiDN[u,b^2]*Cos[b*v],
         (a/b)*JacobiDN[u,b^2]*Sin[b*v],
         (c/b)*(u - EllipticE[JacobiAmplitude[u,b^2],b^2])}

    cnchytcheb[a_,b_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[cnchytcheb[a,b][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    cossurface[a_][u_,v_]:= {a*Cos[u],a*Cos[v],a*Cos[u + v]}

    cossurface[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[cossurface[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    costa[u_,v_]:=
        {(1/2)*Re[-WeierstrassZeta[u + I*v,{189.07272,0}] +
        3.14159*u + 0.35889 +
        0.228473*(WeierstrassZeta[u + I*v - 1/2,{189.07272,0}] -
             WeierstrassZeta[u + I*v - I/2,{189.07272,0}])],
        (1/2)*Re[-I*WeierstrassZeta[u + I*v,{189.07272,0}] +
        3.14159*v + 0.35889 -
        0.228473*(I*WeierstrassZeta[u + I*v - 1/2,{189.07272,0}] -
             I*WeierstrassZeta[u + I*v - I/2,{189.07272,0}])],
        0.626657*Log[Abs[(WeierstrassP[u + I*v,{189.07272,0}] -
             6.87519)/
             (WeierstrassP[u + I*v,{189.07272,0}] + 6.87519)]]}

    costa[umin_,umax_,vmin_,vmax_,opt___]:=
    Module[{costaplot80},
    costaplot80=ParametricPlot3D[costa[u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->80,DisplayFunction->Identity];
	selectgraphics3d[costaplot80[[1]],8,
           Boxed->False,ViewPoint->{2.9,-1,4},DisplayFunction->dip[EdgeForm[]]]]
       

    cpcsurfrev[a_,b_][u_,v_]:= {b*Cos[u]*Cos[v/a],
                                b*Sin[u]*Cos[v/a],
                                a*EllipticE[v/a,b^2/a^2]}

    cpcsurfrev[a_,b_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[cpcsurfrev[a,b][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    crosscap[a_][u_,v_]:= {a^2*Sin[u]*Sin[2*v]/2,
                 a^2*Sin[2*u]*Cos[v]^2,a^2*Cos[2*u]*Cos[v]^2}

    crosscap[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[crosscap[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    deltoidinvolutemin[a_][t_][u_,v_]:=
          {(a/3)*Cos[t]*(8*Cos[u/2]*Cosh[v/2] + 2*Cos[u]*Cosh[v] - 
            Cos[2*u]*Cosh[2*v]) + 
          (a/3)*Sin[t]*(-8*Sin[u/2]*Sinh[v/2] - 2*Sin[u]*Sinh[v] + 
            Sin[2*u]*Sinh[2*v]),
           (Cos[t]*(-8*a*Cosh[v/2]*Sin[u/2] + 2*a*Cosh[v]*Sin[u] + 
            a*Cosh[2*v]*Sin[2*u]))/3 + 
           (Sin[t]*(-8*a*Cos[u/2]*Sinh[v/2] + 2*a*Cos[u]*Sinh[v] + 
            a*Cos[2*u]*Sinh[2*v]))/3,
           (4*a*Sin[t]*(3*u - 2*Cosh[3*v/2]*Sin[3*u/2]))/9 - 
           (4*a*Cos[t]*(3*v - 2*Cos[3*u/2]*Sinh[3*v/2]))/9}

    deltoidinvolutemin[a_][t_:0][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[deltoidinvolutemin[a][t][u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{40,10}];

    deltoidmin[a_][t_][u_,v_]:=
          {a*Cos[t]*(2*Cos[u]*Cosh[v] + Cos[2*u]*Cosh[2*v]) + 
           2*a*Sin[t]*Sin[u]*(-Sinh[v] - Cos[u]*Sinh[2*v]),
           -2*a*(Cos[u]*Sinh[v]*((-1 + Cos[u]*Cosh[v])*Sin[t] + 
            2*a*Cos[t]*Sin[u]*Sinh[v])) + 
            Cosh[v]*Sin[u]*(2*a*Cos[t]*(1 - Cos[u]*Cosh[v]) + 
            2*a*Sin[t]*Sin[u]*Sinh[v]),
            -(8/3)*a*Cos[3*u/2]*Cosh[3*v/2]*Sin[t] - 
           (8/3)*a*Cos[t]*Sin[3*u/2]*Sinh[3*v/2]}

    deltoidmin[a_][t_:0][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[deltoidmin[a][t][u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{40,10}];

    dini[a_,b_][u_,v_]:= {a*Cos[u]*Sin[v],
                          a*Sin[u]*Sin[v],
                          a*(Cos[v] + Log[Tan[v/2]]) + b*u}

    dini[a_,b_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[dini[a,b][u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{55,10}];

    dinihelicoid[pm_,a_,b_,c_][u_,v_]:=
          {v*Cos[u],v*Sin[u],c*u +
           pm*(Integrate[Sqrt[(b*c^2 + a^2*vv^2 + b*vv^2 - c^2*vv^2 - vv^4)/
                          (vv^4 -b*vv^2)],vv])} /. vv->v

    dinitcheb[a_,b_][u_,v_]:=
          {a*Cos[b]*Cos[v]/Cosh[(u - v*Sin[b])/Cos[b]],
           a*Cos[b]*Sin[v]/Cosh[(u - v*Sin[b])/Cos[b]],
           a*(u - Cos[b]*Tanh[(u - v*Sin[b])/Cos[b]])}

    dinitcheb[a_,b_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[dinitcheb[a,b][u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{25,55}];

    dinitcheb[a_,b_,c_][u_,v_]:=
          {a*Cos[c]*Cos[v]/Cosh[(u - v*Sin[c])/Cos[c]],
           a*Cos[c]*Sin[v]/Cosh[(u - v*Sin[c])/Cos[c]],
           b*(u - Cos[c]*Tanh[(u - v*Sin[c])/Cos[c]])}

    earth[u_,v_]:= {6378.2*Cos[u]*Cos[v],6378.2*Sin[u]*Cos[v],
                    6356.6*Sin[v]}

    earth[umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[earth[u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    eighteight[u_,v_]:= {Sin[u]*Sin[v],
                         Cos[u]*Sin[u]*Sin[v],Cos[v]*Sin[v]}

    eighteight[umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[eighteight[u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    eightsurface[u_,v_]:=
        {Cos[u]*Sin[2*v],Sin[u]*Sin[2*v],Sin[v]}

    eightsurface[umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[eightsurface[u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    ellhypcyclide[a_,c_,k_][u_,v_]:=
        {(c*(k - c*Cos[u]) + a*Cos[u]*(a - k*Cos[v]))/
                 (a - c*Cos[u]*Cos[v]),
         (Sqrt[a^2 - c^2]*(a - k*Cos[v])*Sin[u])/
                 (a - c*Cos[u]*Cos[v]),
         (Sqrt[a^2 - c^2]*(k - c*Cos[u])*Sin[v])/
                 (a - c*Cos[u]*Cos[v])}

    ellhypcyclide[a_,c_,k_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[ellhypcyclide[a,c,k][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    ellipsemin[a_,b_][t_][u_,v_]:=
        {a*Cos[t]*Cos[u]*Cosh[v] - a*Sin[t]*Sin[u]*Sinh[v],
         b*Cos[t]*Sin[u]*Cosh[v] + b*Sin[t]*Cos[u]*Sinh[v],
        -b*(Cos[t]*Im[EllipticE[u + I*v,1 - a^2/b^2]] - 
            Sin[t]*Re[EllipticE[u + I*v,1 - a^2/b^2]])}

    ellipsemin[a_,b_][t:0][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[ellipsemin[a,b][t][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    ellipsoid[a_,c_][u_,v_]:= {a*Cos[v]*Cos[u],
                               a*Cos[v]*Sin[u],c*Sin[v]}
                              
    ellipsoid[a_,b_,c_][u_,v_]:= {a*Cos[v]*Cos[u],
                                  b*Cos[v]*Sin[u],c*Sin[v]}

    ellipsoid[a_,b_,c_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[ellipsoid[a,b,c][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    ellipticalhyperboloidminus[a_,b_,c_][u_,v_]:=
        {a*Cos[u] + a*v*Sin[u],-b*v*Cos[u] + b*Sin[u],c*v}

    ellipticalhyperboloidminus[a_,b_,c_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[ellipticalhyperboloidminus[a,b,c][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    ellipticalhyperboloidplus[a_,b_,c_][u_,v_]:=
        {a*Cos[u] - a*v*Sin[u],b*v*Cos[u] + b*Sin[u],c*v}

    ellipticalhyperboloidplus[a_,b_,c_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[ellipticalhyperboloidplus[a,b,c][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    ellipticcoor[d_,e_,f_][a_,b_,c_][w_][u_,v_]:=
         {d*Sqrt[(a - u)*(a - v)*(a - w)/((a - b)*(a - c))],
          e*Sqrt[(b - u)*(b - v)*(b - w)/((b - a)*(b - c))],
          f*Sqrt[(c - u)*(c - v)*(c - w)/((c - a)*(c - b))]}

    ellipticcoor[d_,e_,f_][a_,b_,c_][w_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[
    ellipticcoor[d,e,f][a,b,c][w][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    ellipticparaboloid[a_,b_][u_,v_]:=
                       {u,v,u^2/a^2 + v^2/b^2}

    ellipticparaboloid[a_,b_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[ellipticparaboloid[a,b][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    enneper[u_,v_]:= {u - u^3/3 + u*v^2,
                     -v + v^3/3 - v*u^2,u^2 - v^2}

    enneper[umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[enneper[u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
	
    enneper[n_][t_][u_,v_]:= Simplify[ComplexExpand[
        analytictominimal[ennepermincurve[n]][t][u,v],
        TargetFunctions->{Re,Im}]]

    ennepercatenoidpolar[n_][t_][r_,theta_]:=
        {(n*(2*n + 1)*r^2*Cos[t - theta] +
          n*(2*n + 1)*Cos[t + theta] - 
          r^(n + 1)*(n*r^(n + 1)*Cos[t - theta - 2*n*theta] + 
          2*Cos[t - n*theta] + 4*n*Cos[t - n*theta]))/
         (n*(2*n + 1)*r),
         (n*(2*n + 1)*r^2*Sin[t - theta] -
          n*(2*n + 1)*Sin[t + theta] +
          r^(n + 1)*(2*Sin[t - n*theta] + 4*n*Sin[t - n*theta] + 
          n*r^(n + 1)*Sin[t + theta - 2*(n + 1)*theta]))/
         (n*(2*n + 1)*r),
          2*(r^(n + 1)*Cos[t - (n + 1)*theta] + 
         (n + 1)*(Cos[t]*Log[r] + theta*Sin[t]))/(n + 1)}

    ennepercatenoidpolar[n_,a_,b_][t_][r_,theta_]:=
        {(n*(2*n + 1)*r^2*Cos[t - theta] + 
          a^2*n*(2*n + 1)*Cos[t + theta] - 
          b*r^(n + 1)*(2*a*Cos[t - n*theta] + 
          4*a*n*Cos[t - n*theta] + 
          b*n*r^(n + 1)*Cos[t - (2*n + 1)*theta]))/
         (n*(2*n + 1)*r),
         (n*(2*n + 1)*r^2*Sin[t - theta] - 
          a^2*n*(2*n + 1)*Sin[t + theta] + 
          b*r^n*(b*n*r^(n + 2)*Sin[t - theta - 2*n*theta] + 
          2*a*r*Sin[t + theta - (n + 1)*theta] + 
          4*a*n*r*Sin[t + theta - (n + 1)*theta]))/
         (n*(2*n + 1)*r),
         (2*b*r^(n + 1)*Cos[t - (n + 1)*theta] + 
          2*a*(n + 1)*Cos[t]*Log[r] + 2*a*(n + 1)*theta*Sin[t])
          /(n + 1)}

    ennepercatenoidpolar[n_,a_,b_][t_:0][rmin_,rmax_,thetamin_,thetamax_,opt___]:=
    pplot3d[ennepercatenoidpolar[n,a,b][t][r,theta],
	{r,rmin,rmax},{theta,thetamin,thetamax},opt];
    
    enneperpolar[n_][t_][r_,theta_]:=
        {r*Cos[t - theta] - r^(2*n + 1)*
           Cos[t - (2*n + 1)*theta]/(2*n + 1),
         r*Sin[t - theta] + r^(2*n + 1)*
           Sin[t - (2*n + 1)*theta]/(2*n + 1),
         2*r^(n + 1)*Cos[t - (n + 1)*theta]/(n + 1)}

    enneperpolar[n_][t_:0][rmin_,rmax_,thetamin_,thetamax_,opt___]:=
    pplot3d[enneperpolar[n][t][r,theta],
	{r,rmin,rmax},{theta,thetamin,thetamax},
	opt,PlotPoints->{15,69}];
    
    exptwist[a_,c_][u_,v_]:=
             {a*v*Cos[u],a*v*Sin[u],a*E^(c*u)}

    exptwist[a_,c_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[exptwist[a,c][u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{60,10}];

    exptwistasym[a_,c_][p_,q_]:=
            {a*E^(c*(p - q)/2)*Cos[p],
             a*E^(c*(p - q)/2)*Sin[p],
             a*E^(c*p)}

    exptwistasym[a_,c_][pmin_,pmax_,qmin_,qmax_,opt___]:=
    pplot3d[exptwistasym[a,c][p,q],{p,pmin,pmax},{q,qmin,qmax},
	opt,PlotPoints->{48,15}];

    flattorus[a_,b_][u_,v_]:= {a*Cos[u],a*Sin[u],
                               b*Cos[v],b*Sin[v]}

    flattorus[a_,b_,c_,d_][u_,v_]:= {a*Cos[u],b*Sin[u],
                                     c*Cos[v],d*Sin[v]}

    funnel[a_,c_][u_,v_]:= {a*v*Cos[u],
                               c*v*Sin[u],c*Log[v]}

    funnel[a_,b_,c_][u_,v_]:= {a*v*Cos[u],
                               b*v*Sin[u],c*Log[v]}

    funnel[a_,b_,c_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[funnel[a,b,c][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    funnelasym[c_][p_,q_]:= {c*E^(p - q)*Cos[p + q],
                     -c*E^(p - q)*Sin[p + q],c*(p - q)}

    funnelasym[c_][pmin_,pmax_,qmin_,qmax_,opt___]:=
    pplot3d[funnelasym[c][p,q],{p,pmin,pmax},{q,qmin,qmax},
	opt];

    gencube[n_,a_,b_,c_][u_,v_]:=
        {a*(Cos[u]*Sqrt[Cos[u]^2]^(n - 1) +
            Sin[u]*Sqrt[Sin[u]^2]^(n - 1))*
           (Cos[v]*Sqrt[Cos[v]^2]^(n - 1) +
            Sin[v]*Sqrt[Sin[v]^2]^(n - 1)),
         b*(-Cos[u]*Sqrt[Cos[u]^2]^(n - 1) +
            Sin[u]*Sqrt[Sin[u]^2]^(n - 1))*
           (Cos[v]*Sqrt[Cos[v]^2]^(n - 1) +
            Sin[v]*Sqrt[Sin[v]^2]^(n - 1)),
         c*(-Cos[v]*Sqrt[Cos[v]^2]^(n - 1) +
            Sin[v]*Sqrt[Sin[v]^2]^(n - 1))}

    gencube[n_,a_,b_,c_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[gencube[n,a,b,c][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    genhypparab[a_,b_][u_,v_]:= {a*(u + v),b*v,u^2 + 2*u*v}

    genhypparab[a_,b_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[genhypparab[a,b][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
	
    genoctahedron[n_,a_,b_,c_][u_,v_]:=
        {a*Cos[u]*Cos[v]*Sqrt[Cos[u]^2*Cos[v]^2]^(n - 1),
         a*Sin[u]*Cos[v]*Sqrt[Sin[u]^2*Cos[v]^2]^(n - 1),
         a*Sin[v]*Sqrt[Sin[v]^2]^(n - 1)}

    genoctahedron[n_,a_,b_,c_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[genoctahedron[n,a,b,c][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    handkerchief[a_][u_,v_]:= {u,v,
                    (1/3)*u^3 + u*v^2 +a*(u^2 - v^2)}

    handkerchief[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[handkerchief[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
	
    heart[a_][u_,v_]:= {2*a*Cos[u]*(1 + Cos[v])*Sin[v],
                        2*a*Sin[u]*(1 + Cos[v])*Sin[v],
                        -2*a*Cos[v]*(1 + Cos[v])}
 
    heart[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[heart[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
	
    helicoid[a_,c_][u_,v_]:= {a*v*Cos[u],a*v*Sin[u],c*u}

    helicoid[a_,b_,c_][u_,v_]:=
         {a*v*Cos[u],b*v*Sin[u],c*u}

    helicoid[a_,b_,c_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[helicoid[a,b,c][u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{75,15}];
	
    helicoidprin[b_][p_,q_]:=  {b*Cos[p + q]*Sinh[-p + q],
         b*Sin[p + q]*Sinh[-p + q],b*(p + q)}

    helicoidprin[b_][pmin_,pmax_,qmin_,qmax_,opt___]:=
    pplot3d[helicoidprin[b][p,q],{p,pmin,pmax},{q,qmin,qmax},
	opt];

    heltocat[t_][u_,v_]:=
        {Cos[u]*Cosh[v]*Sin[t] + Cos[t]*Sin[u]*Sinh[v],
         Cosh[v]*Sin[t]*Sin[u] - Cos[t]*Cos[u]*Sinh[v],
         u*Cos[t] + v*Sin[t]}

    heltocat[t_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[heltocat[t][u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{50,15}];

    henneberg[u_,v_]:=
        {2*Sinh[u]*Cos[v] - (2/3)*Sinh[3*u]*Cos[3*v],
         2*Sinh[u]*Sin[v] + (2/3)*Sinh[3*u]*Sin[3*v],
         2*Cosh[2*u]*Cos[2*v]}

    henneberg[umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[henneberg[u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{15,50}];

    hyperbolicparaboloid[u_,v_]:= {u,v,u*v}

    hyperbolicparaboloid[a_][u_,v_]:= {u,v,a*u*v}
    
    hyperbolicparaboloid[umin_,umax_,vmin_,vmax_,opt___]:=
	pplot3d[hyperbolicparaboloid[u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    hyperboloid[a_,c_][u_,v_]:= {a*Cosh[v]*Cos[u],
                                 a*Cosh[v]*Sin[u],
                                 c*Sinh[v]}

    hyperboloid[a_,b_,c_][u_,v_]:= {a*Cosh[v]*Cos[u],
                                    b*Cosh[v]*Sin[u],
                                    c*Sinh[v]}

    hyperboloid[a_,b_,c_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[hyperboloid[a,b,c][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    hyperboloidbis[a_,c_][u_,v_]:= {a*Sec[v]*Cos[u],
                                    a*Sec[v]*Sin[u],
                                    c*Tan[v]}

    hyperboloidbis[a_,b_,c_][u_,v_]:= {a*Sec[v]*Cos[u],
                                       b*Sec[v]*Sin[u],
                                       c*Tan[v]}

    hyperboloidbis[a_,b_,c_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[hyperboloidbis[a,b,c][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    hypocycloidinvolutemin[a_,b_][t_][u_,v_]:=
        {Cos[t]*((a - b)*Cos[u]*Cosh[v] + 
         b*Cos[(a - b)*u/b]*Cosh[(a - b)*v/b]) + 
         Sin[t]*((-a + b)*Sin[u]*Sinh[v] - 
         b*Sin[(a - b)*u/b]*Sinh[(a - b)*v/b]),
         Cos[t]*((a - b)*Cosh[v]*Sin[u] - 
         b*Cosh[(a - b)*v/b]*Sin[(a - b)*u/b]) + 
         Sin[t]*((a - b)*Cos[u]*Sinh[v] - 
         b*Cos[((a - b)*u)/b]*Sinh[(a - b)*v/b]),
         (-4*(a - b)*b*Cos[(a*u)/(2*b)]*Cosh[(a*v)/(2*b)]*Sin[t])/a - 
        (4*(a - b)*b*Cos[t]*Sin[(a*u)/(2*b)]*Sinh[(a*v)/(2*b)])/a}

    hypocycloidinvolutemin[a_,b_][t_:0][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[hypocycloidinvolutemin[a,b][t][u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{99,15}];

    hy2sheet[a_,c_][u_,v_]:= {a*Cosh[u]*Cosh[v],
                              a*Sinh[u]*Cosh[v],
                              c*Sinh[v]}

    hy2sheet[a_,b_,c_][u_,v_]:= {a*Cosh[u]*Cosh[v],
                                 b*Sinh[u]*Cosh[v],
                                 c*Sinh[v]}

    hy2sheet[a_,b_,c_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[hy2sheet[a,b,c][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    invertedcircularcone[{p1_,p2_,p3_},rho_,a_,b_][u_,v_]:=
        {p1 + rho^2*(-p1 + a*v*Cos[u])/
         ((-p3 + b*v)^2 + (-p1 + a*v*Cos[u])^2 + 
          (-p2 + a*v*Sin[u])^2),
         p2 + rho^2*(-p2 + a*v*Sin[u])/
         ((-p3 + b*v)^2 + (-p1 + a*v*Cos[u])^2 + 
          (-p2 + a*v*Sin[u])^2),
         p3 + rho^2*(-p3 + b*v)/
         ((-p3 + b*v)^2 + (-p1 + a*v*Cos[u])^2 + 
          (-p2 + a*v*Sin[u])^2)}

    invertedcircularcone[rho_,a_,b_][u_,v_]:=
        {a*rho^2*Cos[u]/(a^2*v + b^2*v),
         a*rho^2*Sin[u]/(a^2*v + b^2*v),
         b*rho^2/(a^2*v + b^2*v)}

    invertedcircularcone[{p1_,p2_,p3_},rho_,a_,b_][umin_,umax_,
                                                   vmin_,vmax_,opt___]:=
    pplot3d[invertedcircularcone[{p1,p2,p3},rho,a,b][u,v],{u,umin,umax},
                {v,vmin,vmax},opt];
     
    invertedcircularcylinder[{p1_,p2_,p3_},rho_,a_][u_,v_]:=
        {p1 + rho^2*(-p1 + a*Cos[u])/
         ((-p3 + v)^2 + (-p1 + a*Cos[u])^2 + 
          (-p2 + a*Sin[u])^2),
          p2 + rho^2*(-p2 + a*Sin[u])/
          ((-p3 + v)^2 + (-p1 + a*Cos[u])^2 + 
           (-p2 + a*Sin[u])^2),
          p3 + rho^2*(-p3 + v)/
          ((-p3 + v)^2 + (-p1 + a*Cos[u])^2 + 
           (-p2 + a*Sin[u])^2)}

    invertedcircularcylinder[rho_,a_][u_,v_]:=
        {a*rho^2*Cos[u]/(a^2 + v^2),
         a*rho^2*Sin[u]/(a^2 + v^2),
         rho^2*v/(a^2 + v^2)}

    invertedcircularcylinder[{p1_,p2_,p3_},rho_,a_][umin_,umax_,
                                                    vmin_,vmax_,opt___]:=
    pplot3d[
    invertedcircularcylinder[{p1,p2,p3},rho,a][u,v],{u,umin,umax},
                {v,vmin,vmax},opt];
     
    invertedtorus[{p1_,p2_,p3_},rho_,a_,b_][u_,v_]:=
        {p1 + (rho^2*(-p1 + Cos[u]*(a + b*Cos[v])))/
         ((-p1 + Cos[u]*(a + b*Cos[v]))^2 + 
          (-p2 + (a + b*Cos[v])*Sin[u])^2 + (-p3 + b*Sin[v])^2),
         p2 + (rho^2*(-p2 + (a + b*Cos[v])Sin[u]))/
         ((-p1 + Cos[u]*(a + b*Cos[v]))^2 + 
          (-p2 + (a + b*Cos[v])*Sin[u])^2 + (-p3 + b*Sin[v])^2),
         p3 + (rho^2*(-p3 + b*Sin[v]))/
         ((-p1 + Cos[u]*(a + b*Cos[v]))^2 + 
          (-p2 + (a + b*Cos[v])*Sin[u])^2 + (-p3 + b*Sin[v])^2)}

    invertedtorus[rho_,a_,b_][u_,v_]:=
        {rho^2*Cos[u]*(a + b*Cos[v])/(a^2 + b^2 + 2*a*b*Cos[v]),
         rho^2*(a + b*Cos[v])*Sin[u]/(a^2 + b^2 + 2*a*b*Cos[v]),
         b*rho^2*Sin[v]/(a^2 + b^2 + 2*a*b*Cos[v])}

    invertedtorus[{p1_,p2_,p3_},rho_,a_,b_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[
    invertedtorus[{p1,p2,p3},rho,a,b][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    jorgemeekspolar[n_,a_][r_,theta_]:=
        Re[jorgemeeksmincurve[n,a][r*E^(I*theta)]]

    jorgemeekspolar[2,a_][r_,theta_]:=
        {(a/8)*Log[(1 + r^2 + 2*r*Cos[theta])/(1 + r^2 - 2*r*Cos[theta])],
          -a*r*(1 + r^2)*Sin[theta]/(2*(1 + r^4 - 2*r^2*Cos[2*theta])),
          (a - a*r^4)/(4*(1 + r^4 - 2*r^2*Cos[2*theta]))}

    jorgemeekspolar[2,a_][rmin_,rmax_,thetamin_,thetamax_,opt___]:=
    pplot3d[jorgemeekspolar[2,a][r,theta],
	{r,rmin,rmax},{theta,thetamin,thetamax},
	opt];
    
    jorgemeekspolar[3,a_][r_,theta_]:=
    {(a/18)*(3*r^2/(1 + r^2 + r^4 + 2*(r + r^3)*Cos[theta] + 
        2*r^2*Cos[2*theta]) + (3*(r + r^3)*Cos[theta])/
       (1 + r^2 + r^4 + 2*(r + r^3)*Cos[theta] + 2*r^2*Cos[2*theta]) - 
       2*Log[1 + r^2 - 2*r*Cos[theta]] + 
       Log[1 + r^2 + r^4 + 2*(r + r^3)*Cos[theta] + 2*r^2*Cos[2*theta]]),
     -(a*(-(Sqrt[3]*Log[1 + r^2 + r*Cos[theta] - Sqrt[3]*r*Sin[theta]]) - 
       Sqrt[3]*r^6*Log[1 + r^2 + r*Cos[theta] - 
       Sqrt[3]*r*Sin[theta]] + 2*Sqrt[3]*r^3*Cos[3*theta]*
       (Log[1 + r^2 + r*Cos[theta] - Sqrt[3]*r*Sin[theta]] - 
        Log[1 + r^2 + r*Cos[theta] + Sqrt[3]*r*Sin[theta]]) + 
        Sqrt[3]*Log[1 + r^2 + r*Cos[theta] + Sqrt[3]*r*Sin[theta]] + 
        Sqrt[3]*r^6*Log[1 + r^2 + r*Cos[theta] + Sqrt[3]*r*Sin[theta]] + 
        3*(r + r^5)*Sin[theta] + 3*r^2*Sin[2*theta] + 
        3*r^4*Sin[2*theta]))/(18*(1 + r^6 - 2*r^3*Cos[3*theta])),
     (a - a*r^6)/(6*(1 + r^6 - 2*r^3*Cos[3*theta]))}

    jorgemeekspolar[3,a_][rmin_,rmax_,thetamin_,thetamax_,opt___]:=
    pplot3d[jorgemeekspolar[3,a][r,theta],
	{r,rmin,rmax},{theta,thetamin,thetamax},
	opt];
    
    kidney[a_][u_,v_]:=
        {a*Cos[u]*(3*Cos[v] - Cos[3*v]),
         a*Sin[u]*(3*Cos[v] - Cos[3*v]),
         a*(3*Sin[v] - Sin[3*v])}

    kidney[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[
    kidney[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    kleinbottle[a_][u_,v_]:=
        {(a + Cos[u/2]*Sin[v] - Sin[u/2]*Sin[2*v])*Cos[u],
         (a + Cos[u/2]*Sin[v] - Sin[u/2]*Sin[2*v])*Sin[u],
          Sin[u/2]*Sin[v] + Cos[u/2]*Sin[2*v]}

    kleinbottle[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[kleinbottle[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    kleinbottlebis[e_,f_][u_,v_]:=If[u<0,
        {f*Sin[u]/2,(-e - f/2 + f*Cos[u]/2)*Cos[v],
         (e + f/2 - f*Cos[u]/2)*Sin[v]},
        {10*Sin[0.105*u] + 2*(e + 0.033*f*u)*
         (Cos[0.105*u] + 4*Cos[0.21*u])*Cos[v]*
         (Sin[0.105*u] + 2*Sin[0.21*u])/
         Sqrt[100*Cos[0.105*u]^2 + 4*(Cos[0.105*u] + 
          4*Cos[0.21*u])^2*(Sin[0.105*u] + 2*Sin[0.21*u])^2],
         (Sin[0.105*u] + 2*Sin[0.21*u])^2 - 
         (10*(e + 0.033*f*u)*Cos[0.105*u]*Cos[v])/
         Sqrt[100*Cos[0.105*u]^2 + 4*(Cos[0.105*u] + 
          4*Cos[0.21*u])^2*(Sin[0.105*u] + 2*Sin[0.21*u])^2],
         (e + 0.033*f*u)*Sin[v]}]

    kleinbottlebis[e_,f_][umin_,umax_,vmin_,vmax_,opt___]:=
    ParametricPlot3D[
    kleinbottlebis[e,f][u,v],{u,umin,umax},{v,vmin,vmax},
    opt,ViewPoint->{-2,0,-3},PlotPoints->{49,19},Axes->None];
     
    kleinbottlewri[u_,v_]:=If[Pi < u <= 2 Pi,
        {6*Cos[u]*(1 + Sin[u]) + 4*(1 - Cos[u]/2)*Cos[v + Pi],
        16*Sin[u],4*(1 - Cos[u]/2)*Sin[v]},
        {6*Cos[u]*(1 + Sin[u]) + 4*(1 - Cos[u]/2)*Cos[u]*Cos[v],
 	 16*Sin[u] + 4*(1 - Cos[u]/2)*Sin[u]*Cos[v],
 	 4*(1 - Cos[u]/2)*Sin[v]}]

    kleinbottlewri[umin_,umax_,vmin_,vmax_,opt___]:=
    ParametricPlot3D[
    kleinbottlewri[u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    klein4[a_,b_][u_,v_]:=
        {(a + b*Cos[v])*Cos[u],(a + b*Cos[v])Sin[u],
          b*Sin[v]*Cos[u/2],b*Sin[v]*Sin[u/2]}

    kuen[a_][u_,v_]:=
         {2*a*(Cos[u] + u*Sin[u])*Sin[v]/(1 + u^2*Sin[v]^2),
          2*a*(Sin[u] - u*Cos[u])Sin[v]/(1 + u^2*Sin[v]^2),
          a*(Log[Tan[v/2]] + 2*Cos[v]/(1 + u^2*Sin[v]^2))}

    kuen[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[kuen[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    kuentcheb[a_][u_,v_]:=
        {2*a*Cosh[u]*(Cos[v] + v*Sin[v])/(v^2 + Cosh[u]^2),
         2*a*Cosh[u]*(Sin[v] - v*Cos[v])/(v^2 + Cosh[u]^2),
          a*(u - Sinh[2*u]/(v^2 + Cosh[u]^2))}

    kuentcheb[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[kuentcheb[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    kuentcheb[a_,b_][u_,v_]:=
        {2*a*Cosh[u]*(Cos[v] + v*Sin[v])/(v^2 + Cosh[u]^2),
         2*a*Cosh[u]*(Sin[v] - v*Cos[v])/(v^2 + Cosh[u]^2),
          b*(u - Sinh[2*u]/(v^2 + Cosh[u]^2))}

    kuentcheb[a_,b_,c_][u_,v_]:=
        {2*a*Cosh[u]*(Cos[v] + v*Sin[v])/(v^2 + Cosh[u]^2),
         2*b*Cosh[u]*(Sin[v] - v*Cos[v])/(v^2 + Cosh[u]^2),
          c*(u - Sinh[2*u]/(v^2 + Cosh[u]^2))}

    lemniscatemin[a_][t_][u_,v_]:=
        {(-16*a*Cos[u]*Cosh[v]*Sin[u]*Sinh[v]*
         (Cos[u]*Cosh[v]*Sin[t] + Cos[t]*Sin[u]*Sinh[v]))/
         (18 + Cos[4*u] - 12*Cos[2*u]*Cosh[2*v] + Cosh[4*v]) +
         (4*a*(3 - Cos[2*u]*Cosh[2*v])*
         (Cos[t]*Cos[u]*Cosh[v] - Sin[t]*Sin[u]*Sinh[v]))/
         (18 + Cos[4*u] - 12*Cos[2*u]*Cosh[2*v] + Cosh[4*v]),
         (a*(-Sin[t - 4*u] + 6*Cosh[2*v]*Sin[t - 2*u] - 
          6*Cosh[2*v]*Sin[t + 2*u] + Sin[t + 4*u] - 
          6*Sin[t - 2*u]*Sinh[2*v] -  6*Sin[t + 2*u]*Sinh[2*v] +
          2*Sin[t]*Sinh[4*v]))/(2*(-18 - Cos[4*u] +
          12*Cos[2*u]*Cosh[2*v] - Cosh[4*v])),
          a*(-Im[EllipticF[u + I*v,-1]]Cos[t] + 
          Re[EllipticF[u + I*v,-1]]*Sin[t])}

    lemniscatemin[a_][t_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[lemniscatemin[a][t][u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{100,20}];
     
    logminpolar[n_][t_][r_,theta_]:=
        analytictominimalpolar[logmincurve[n]][t][r,theta]

    logminpolar[n_][t_:0][rmin_,rmax_,thetamin_,thetamax_,opt___]:=
    pplot3d[logminpolar[n][t][r,theta],
	{r,rmin,rmax},{theta,thetamin,thetamax},
	opt,PlotPoints->{10,60},ViewPoint->{-2.5,-1.5,1.6}];
    
    menn[a_][u_,v_]:= {u,v,a*u^4 + u^2*v - v^2}

    menn[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[menn[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    mercatorellipsoid[a_,b_,c_][u_,v_]:=
        {a*Sech[v]*Cos[u],b*Sech[v]*Sin[u],c*Tanh[v]}

    mercatorellipsoid[a_,b_,c_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[mercatorellipsoid[a,b,c][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    mercatorsphere[a_][u_,v_]:=
        {a*Sech[v]*Cos[u],a*Sech[v]*Sin[u],a*Tanh[v]}

    mercatorsphere[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[mercatorsphere[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    mercatorsphere[a_,b_][u_,v_]:=
        {a*(Cos[u] + b*Cosh[v])/(Cosh[v] + b*Cos[u]),
         a*Sqrt[1 - b^2]*Sin[u]/(Cosh[v] + b*Cos[u]),
         a*Sqrt[1 - b^2]*Sinh[v]/(Cosh[v] + b*Cos[u])}

    mercatorsphere[a_,b_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[mercatorsphere[a,b][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    moebcirc[a_][u_,v_]:={a*(-2*Cos[2*v]*Sin[u])/
     (-2 + Sqrt[2]*Cos[u]*Sin[v] + Sqrt[2]*Sin[u]*Sin[2*v]),
      a*(Sqrt[2]*(Cos[u]*Sin[v] - Sin[u]*Sin[2*v]))/
     (-2 + Sqrt[2]*Cos[u]*Sin[v] + Sqrt[2]*Sin[u]*Sin[2*v]),
       a*(-2*Cos[u]*Cos[v])/
     (-2 + Sqrt[2]*Cos[u]*Sin[v] + Sqrt[2]*Sin[u]*Sin[2*v])}

    moebcirc[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[moebcirc[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    moebcirc[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[moebcirc[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    moebiusstrip[a_][u_,v_]:= {a*(Cos[u] + v*Cos[u/2]*Cos[u]),
        a*(Sin[u] + v*Cos[u/2]*Sin[u]),a*v*Sin[u/2]}

    moebiusstrip[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[moebiusstrip[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{40,5}];
     
    monkeysaddle[u_,v_]:= {u,v,u^3 - 3*u*v^2}
    
    monkeysaddle[umin_,umax_,vmin_,vmax_,opt___]:=
	pplot3d[monkeysaddle[u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    monkeysaddle[n_][u_,v_]:={u,v,
        Simplify[ComplexExpand[Re[(u + I*v)^n],
           TargetFunctions->{Re,Im}]]}

    monkeysaddlepolar[n_][r_,theta_]:=
        {r*Cos[theta],r*Sin[theta],r^n*Cos[n*theta]}

    monkeysaddlepolar[n_][rmin_,rmax_,thetamin_,thetamax_,opt___]:=
    pplot3d[monkeysaddlepolar[n][r,theta],
	{r,rmin,rmax},{theta,thetamin,thetamax},
	opt,PlotPoints->{10,33}];
    
    nodoidnds[vmin_,vmax_][a_,b_,c_,e_][u_,v_]:=
         Module[{tmpintr},
         tmpintr=
             intrinsic[delaunaykappa2[c,e],0,{0,0,-Pi/2},
             MaxSteps->2000,{vmin,vmax}][v];
         {Cos[u]*(a + b*tmpintr[[1]]),Sin[u]*(a + b*tmpintr[[1]]),
            b*tmpintr[[2]]}]
            
    nodoid[vmin_,vmax_][a_,b_,c_,e_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[nodoidnds[vmin,vmax][a,b,c,e][u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{21,51}];
     
     parabolamin[a_][t_][u_,v_]:=
        {2*a*(u*Cos[t] + v*Sin[t]),
         a*((u^2 - v^2)*Cos[t] + 2*u*v*Sin[t]),
         (a*(-2*(u*v + ArcTan[u + Sqrt[1 + u^2],2*v])*Cos[t] + 
         (-2*v^2 + Log[(u + Sqrt[1 + u^2])^2 + 4*v^2])*Sin[t] + 
          2*Sqrt[1 + u^2]*(-(v*Cos[t]) + u*Sin[t])))/2}

    parabolamin[a_][t_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[parabolamin[a][t][u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{40,10}];
     
    paraboliccoor[d_,e_,f_][a_,b_,c_][w_][u_,v_]:=
         {d*Sqrt[(a - u)*(a - v)*(a - w)/(b - a)],
          e*Sqrt[(b - u)*(b - v)*(b - w)/(a - b)],
          f*(c - a - b + u + v + w)/2}

    paraboliccoor[d_,e_,f_][a_,b_,c_][w_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[paraboliccoor[d,e,f][a,b,c][w][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    paraboliccyclide[a_,k_][u_,v_]:=
        {u*(8*a^2 + k + v^2)/(16*a^2 + u^2 + v^2),
         v*(8*a^2 - k + u^2)/(16*a^2 + u^2 + v^2),
        (16*a^2*(k - u^2 + v^2) - k*(u^2 + v^2))/
        (8*a*(16*a^2 + u^2 + v^2))}

    paraboliccyclide[a_,k_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[paraboliccyclide[a,k][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    paraboloid[a_][u_,v_]:= {u,v,a*(u^2 + v^2)}

    paraboloid[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[paraboloid[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    paraboloid[a_,b_][u_,v_]:= {u,v,a*u^2 + b*v^2}

    paraboloid[a_,b_,c_][u_,v_]:=
    {u,v,a*u^2 + b*v^2 + c*u*v}

    paraboloidpolar[n_,a_,b_][r_,theta_]:=
    {a*r*Cos[theta],b*r*Sin[theta],b*r^n}

    paraboloidpolar[n_,a_,b_][rmin_,rmax_,thetamin_,thetamax_,opt___]:=
    pplot3d[paraboloidpolar[n,a,b][r,theta],
	{r,rmin,rmax},{theta,thetamin,thetamax},
	opt];
    
    pear[a_,b_][u_,v_]:= {b*Cos[u]*Cos[v]*(1 + Sin[v]),
                          b*Sin[u]*Cos[v]*(1 + Sin[v]),
                            -a*(1 + Sin[v])}
                            
    pear[a_,b_][umin_,umax_,vmin_,vmax_,opt___]:=
	pplot3d[pear[a,b][u,v],
	{u,umin,umax},{v,vmin,vmax},
	opt];

    pearbis[u_,v_]:={Cos[u]*(4 + 3.8*Cos[v]),
                     Sin[u]*(4 + 3.8*Cos[v]),
        ((Cos[v] - 3.5)*(1 + Sin[v])*Log[1 - Pi*v/10] +
          10*Sin[v])*(1 - 0.4*Cos[v/2]*(1 + 0.2*Sin[v^2]))}
          
    pearbis[umin_,umax_,vmin_,vmax_,opt___]:=
	pplot3d[pearbis[u,v],
	{u,umin,umax},{v,vmin,vmax},
	opt];

    perturbedms[a_][u_,v_]:=
            {u,v,u^3 - 3*u*v^2 + a*(u^2 + v^2)}

    perturbedms[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[perturbedms[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    perturbedms[a_,b_,c_][u_,v_]:=
            {u,v,c*(u^3 - 3*u*v^2) + a*u^2 + b*v^2}

    perturbedms[a_,b_,c_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[perturbedms[a,b,c][u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{10,40}];
     
    perturbedmspolar[n_,a_][r_,theta_]:=
        {r*Cos[theta],r*Sin[theta],r^n*Cos[n*theta] + a*r^2}

    perturbedmspolar[n_,a_][rmin_,rmax_,thetamin_,thetamax_,opt___]:=
    pplot3d[perturbedmspolar[n,a][r,theta],
	{r,rmin,rmax},{theta,thetamin,thetamax},
	opt];
    
    pillow[a_][u_,v_]:= {a*Cos[u],a*Cos[v],a*Sin[u]*Sin[v]}

    pillow[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[pillow[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    plane[a1_,a2_,b1_,b2_,c1_,c2_][u_,v_]:=
        {a1*u + a2*v,b1*u + b2*v,c1*u + c2*v}

    planepolar[a_][r_,theta_]:= {a*r*Cos[theta],a*r*Sin[theta],0}

    planepolar[a_,b_][r_,theta_]:= {a*r*E^(b*theta)*Cos[theta],
                                    a*r*E^(b*theta)*Sin[theta],0}

    planepolar[a_,b_][rmin_,rmax_,thetamin_,thetamax_,opt___]:=
    pplot3d[planepolar[a,b][r,theta],
	{r,rmin,rmax},{theta,thetamin,thetamax},
	opt];
    
    planetocat[t_][u_,v_]:=
        {Cos[t]*u + Sin[u]*Cosh[v],Sin[t]*Cos[u]*Cosh[v],
         v + Cos[t]*Cos[u]*Sinh[v]}
            
    planetocat[t_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[planetocat[t][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    plucker[u_,v_]:= {u,v,2*u*v/(u^2 + v^2)}
    
    plucker[umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[plucker[u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    plucker[n_][u_,v_]:= {u,v,Sin[n*ArcTan[u,v]]}
        
    plucker[m_,n_,a_][u_,v_]:= {a*u,a*v,
                    a*(u + v)^(m/2)*Sin[n*ArcTan[u,v]]}

    pluckerpolar[n_][r_,theta_]:=
                    {r*Cos[theta],r*Sin[theta],Sin[n*theta]}

    pluckerpolar[m_,n_,a_][r_,theta_]:=
        {a*r*Cos[theta],a*r*Sin[theta],a*r^m*Sin[n*theta]}

    pluckerpolar[m_,n_,a_][rmin_,rmax_,thetamin_,thetamax_,opt___]:=
    pplot3d[pluckerpolar[m,n,a][r,theta],
	{r,rmin,rmax},{theta,thetamin,thetamax},
	opt,PlotPoints->{10,75}];
    
    pseudocrosscap[u_,v_]:=  
                 {(1 - u^2)*Sin[v],(1 - u^2)*Sin[2*v],u}

    pseudocrosscap[umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[pseudocrosscap[u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{24,48}];
     
    pseudosphere[a_][u_,v_]:= {a*Cos[u]*Sin[v],
                               a*Sin[u]*Sin[v],
                               a*(Cos[v] + Log[Tan[v/2]])}

    pseudosphere[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[pseudosphere[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    pseudosphere[a_,b_][u_,v_]:= {a*Cos[u]*Sin[v],
                                  a*Sin[u]*Sin[v],
                                  b*(Cos[v] + Log[Tan[v/2]])}

    pseudosphere[a_,b_,c_][u_,v_]:= {a*Cos[u]*Sin[v],
                                     b*Sin[u]*Sin[v],
                                     c*(Cos[v] + Log[Tan[v/2]])}

    pseudospherebis[a_][u_,v_]:= {a*Cos[u]*Tanh[v],
                                  a*Sin[u]*Tanh[v],
                                  a*(Sech[v] + Log[Tanh[v/2]])}

    pseudospherebis[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[pseudospherebis[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    pseudospheretcheb[a_][u_,v_]:= {a*Cos[v]/Cosh[u],
                                    a*Sin[v]/Cosh[u],
                                    a*(u - Tanh[u])}

    pseudospheretcheb[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[pseudospheretcheb[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    pseudospheretcheb[a_,b_][u_,v_]:= {a*Cos[v]/Cosh[u],
                                       a*Sin[v]/Cosh[u],
                                       b*(u - Tanh[u])}

    pseudospheretcheb[a_,b_,c_][u_,v_]:= {a*Cos[v]/Cosh[u],
                                          b*Sin[v]/Cosh[u],
                                          c*(u - Tanh[u])}

    pseudospheretcheb[a_,b_,c_,d_][u_,v_]:= {a*Cos[v]/Cosh[u],
                                             b*Sin[v]/Cosh[u],
                                             c*u - d*Tanh[u]}

    rconoid[u_,v_]:= {v*Cos[u],v*Sin[u],2*Sin[u]}

    rconoid[umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[rconoid[u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    richmond[u_,v_]:=
        {(-3*u - u^5 + 2*u^3*v^2 + 3*u*v^4)/(6*(u^2 + v^2)),
         (-3*v - 3*u^4*v - 2*u^2*v^3 + v^5)/(6*(u^2 + v^2)),u}

    richmond[umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[richmond[u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    richmond[n_][t_][u_,v_]:= Simplify[ComplexExpand[
        analytictominimal[richmondmincurve[n]][t][u,v],
        TargetFunctions->{Re,Im}]]

    richmondpolar[n_][t_][r_,theta_]:=
        {-Cos[t + theta]/(2*r) - 
          r^(2*n + 1)*Cos[t - (2*n + 1)*theta]/(4*n + 2),
         -Sin[t + theta]/(2*r) + 
          r^(2*n + 1)*Sin[t - (2*n + 1)*theta]/(4*n + 2),
          r^n*Cos[t - n*theta]/n}

    richmondpolar[n_][t_:0][rmin_,rmax_,thetamin_,thetamax_,opt___]:=
    pplot3d[richmondpolar[n][t][r,theta],
	{r,rmin,rmax},{theta,thetamin,thetamax},
	opt];
    
    roman[a_][u_,v_]:= {(a^2/2)*Sin[2*u]Cos[v]^2,
                        (a^2/2)*Sin[u]*Sin[2*v],
                        (a^2/2)*Cos[u]*Sin[2*v]}

    roman[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[roman[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    roman[n_,a_][u_,v_]:= {(a^2/2)*(Cos[u]*Cos[v]^2*Sin[u])^n,
                           (a^2/2)*(Cos[v]*Sin[u]*Sin[v])^n,
                           (a^2/2)*(Cos[u]*Cos[v]*Sin[v])^n}

    rosemin[n_,a_][t_][u_,v_]:=
    {Cos[n*u]*Cosh[n*v]*(a*Cos[t]*Cos[u]*Cosh[v] - 
       a*Sin[t]*Sin[u]*Sinh[v]) + 
         Sin[n*u]*(-a*Cos[u]*Cosh[v]*Sin[t] - 
         a*Cos[t]*Sin[u]*Sinh[v])*Sinh[n*v],
       -Cos[u]*Sinh[v]*(-a*Cos[n*u]*Cosh[n*v]*Sin[t] - 
         a*Cos[t]*Sin[n*u]*Sinh[n*v]) + 
         Cosh[v]*Sin[u]*(a*Cos[t]*Cos[n*u]*Cosh[n*v] - 
         a*Sin[t]*Sin[n*u]*Sinh[n*v]),
       -a*Cos[t]*Im[EllipticE[n*(u + I*v),1 - n^2]]/n + 
        a*Re[EllipticE[n*(u + I*v),1 - n^2]]*Sin[t]/n}

    rosemin[n_,a_][t_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[rosemin[n,a][t][u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{45,15}];
     
    scherk[a_][u_,v_]:= {u,v,(1/a)*Log[Cos[a*v]/Cos[a*u]]}

    scherk[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[scherk[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    scherk[a_,theta_][u_,v_]:=
        {Sec[2*theta]*(Cos[theta]*u + Sin[theta]*v),
         Sec[2*theta]*(Sin[theta]*u + Cos[theta]*v),
         (1/a)*Log[
         Cos[a*Sec[2*theta]*(Sin[theta]*u + Cos[theta]*v)]/
         Cos[a*Sec[2*theta]*(Cos[theta]*u + Sin[theta]*v)]]}
   
    scherk5[u_,v_]:= {ArcSinh[u],ArcSinh[v],ArcSin[u*v]}

    scherk5[umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[scherk5[u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    scherk5[a_,b_,c_][u_,v_]:= {a*ArcSinh[u],
                                b*ArcSinh[v],
                                c*ArcSin[u*v]}

    shoe[a_,b_][u_,v_]:= {u,v,a*u^3 + b*v^2} 

    shoe[a_,b_][umin_,umax_,vmin_,vmax_,opt___]:=
	pplot3d[shoe[a,b][u,v],
	{u,umin,umax},{v,vmin,vmax},opt];
    
    shoeasym[a_,b_][p_,q_]:=
    {(-3*b/(4*a))^(1/3)*(p - q)^(2/3),
                      -p - q,(b/4)*(p^2 + 14*p*q + q^2)}

    shoeasym[a_,b_][pmin_,pmax_,qmin_,qmax_,opt___]:=
    pplot3d[Re[
	 shoeasym[a,b][p,q]],{p,pmin,pmax},{q,qmin,qmax},
	opt];

    sievert[a_][u_,v_]:=
       {(2/(a + 1 - a*Sin[v]^2*Cos[u]^2))*
        (Sqrt[(a + 1)*(1 + a*Sin[u]^2)]*Sin[v]/Sqrt[a])*
         Cos[-u/Sqrt[a + 1] + ArcTan[Sqrt[a + 1]*Tan[u]]],
         (2/(a + 1 - a*Sin[v]^2*Cos[u]^2))*
         (Sqrt[(a + 1)*(1 + a*Sin[u]^2)]*Sin[v]/Sqrt[a])*
         Sin[-u/Sqrt[a + 1] + ArcTan[Sqrt[a + 1]*Tan[u]]],
         Log[Tan[v/2]]/Sqrt[a] + 2*(a + 1)*Cos[v]/
                   ((a + 1 - a*Sin[v]^2*Cos[u]^2)*Sqrt[a])}

    sievert[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[sievert[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    sinovaloid[n_,a_][u_,v_]:= {a*Cos[u]*Cos[v],
                                a*Sin[u]*Cos[v],
                                a*Nest[Sin,v,n]}

    sinovaloid[n_,a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[sinovaloid[n,a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    sinsurface[a_][u_,v_]:=
        {a*Sin[u],a*Sin[v],a*Sin[u + v]}

    sinsurface[a_][umin_,umax_,vmin_,vmax_,opt___]:=
	pplot3d[sinsurface[a][u,v],
	{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{40,40}];
    
    snail[u_,v_]:=
        {u*Cos[v]*Sin[u],u*Cos[u]*Cos[v],-u*Sin[v]}

    snail[umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[snail[u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    sphere[a_][u_,v_]:=
        {a*Cos[v]*Cos[u],a*Cos[v]*Sin[u],a*Sin[v]}
        
    sphere[a_][umin_,umax_,vmin_,vmax_,opt___]:=
	pplot3d[sphere[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    spherehelix[a_,b_][u_,v_]:=
    {a*Cos[u]*Cos[v],a*Cos[v]*Sin[u],a*Sin[v],b*v}

    stereographicellipsoid[a_,b_,c_][u_,v_]:=
        {2*a*u/(u^2 + v^2 + 1),2*b*v/(u^2 + v^2 + 1),
         c*(u^2 + v^2 - 1)/(u^2 + v^2 + 1)}

    stereographicellipsoid[a_,b_,c_][umin_,umax_,vmin_,vmax_,opt___]:=
	pplot3d[stereographicellipsoid[a,b,c][u,v],
	{u,umin,umax},{v,vmin,vmax},
	opt];
    
    stereographicsphere[a_][u_,v_]:=
        {2*a*u/(u^2 + v^2 + 1),2*a*v/(u^2 + v^2 + 1),
         a*(u^2 + v^2 - 1)/(u^2 + v^2 + 1)}

    stereographicsphere[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[stereographicsphere[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    stereographicspherepolar[a_][r_,theta_]:=
        {2*a*r*Cos[theta]/(1 + r^2),
         2*a*r*Sin[theta]/(1 + r^2),
         a*(-1 + r^2)/(1 + r^2)}

    stereographicspherepolar[a_][rmin_,rmax_,thetamin_,thetamax_,opt___]:=
    pplot3d[stereographicspherepolar[a][r,theta],
	{r,rmin,rmax},{theta,thetamin,thetamax},
	opt];
    
    stereographicspherepolar[a_,b_][r_,theta_]:=
        {(2*a*E^(b*theta)*r*Cos[theta])/(1 + E^(2*b*theta)*r^2),
         (2*a*E^(b*theta)*r*Sin[theta])/(1 + E^(2*b*theta)*r^2),
         (a*(-1 + E^(2*b*theta)*r^2))/(1 + E^(2*b*theta)*r^2)}

    stereographicspherepolar[a_,b_][rmin_,rmax_,thetamin_,thetamax_,opt___]:=
    pplot3d[stereographicspherepolar[a,b][r,theta],
	{r,rmin,rmax},{theta,thetamin,thetamax},
	opt];
    
    swallowtail[u_,v_]:={3*v^4 + u*v^2,-4*v^3 - 2*u*v,u}

    swallowtail[umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[swallowtail[u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    tetrahedral[m_,n_][A_,B_,C_][a_,b_,c_][u_,v_]:=
              {A*(u - a)^m*(v - a)^n,
               B*(u - b)^m*(v - b)^n,
               C*(u - c)^m*(v - c)^n}

    thomsen[a_,b_][u_,v_]:=
        {b*u/a + Sqrt[1+b^2]*Sinh[a*u]*Cos[a*v]/a^2,
        Sqrt[1 + b^2]*v/a + b*Cosh[a*u]*Sin[a*v]/a^2,
        Sinh[a*u]*Sin[a*v]/a^2}

    thomsen[a_,b_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[thomsen[a,b][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    torus[a_,b_][u_,v_]:= {(a + b*Cos[v])*Cos[u],
                           (a + b*Cos[v])*Sin[u],
                            b*Sin[v]}

    torus[a_,b_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[torus[a,b][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    torus[a_,b_,c_][u_,v_]:= {(a + b*Cos[v])*Cos[u],
                              (a + b*Cos[v])*Sin[u],
                               c*Sin[v]}

    torus[a_,b_,c_,d_][u_,v_]:= 
        {(a + b*Cos[v])*Cos[u],(d + b*Cos[v])*Sin[u],c*Sin[v]}

    torus[a_,b_,c_,d_,e_][u_,v_]:=
        {(a  +b*Cos[v])*Cos[u],(d + e*Cos[v])*Sin[u],c*Sin[v]}

    torusasym[pm_,a_][p_,q_]:=
        {(2*a*Cos[(p + q)/2]/(1 + Cosh[(p - q)/Sqrt[8]]^2)),
         (2*a*Sin[(p + q)/2]/(1 + Cosh[(p - q)/Sqrt[8]]^2)),
         pm*(2*a*Cosh[(p - q)/Sqrt[8]]/(1 + Cosh[(p - q)/Sqrt[8]]^2))}

    torusasym[pm_,a_][pmin_,pmax_,qmin_,qmax_,opt___]:=
    pplot3d[torusasym[pm,a][p,q],{p,pmin,pmax},{q,qmin,qmax},
	opt];

    torusbis[pm_,a_,b_][r_,theta_]:={r*Cos[theta],r*Sin[theta],
                                     pm*Sqrt[b^2 - (r - a)^2]}

    torusbis[pm_,a_,b_][rmin_,rmax_,thetamin_,thetamax_,opt___]:=
    pplot3d[torusbis[pm,a,b][r,theta],{r,rmin,rmax},{theta,thetamin,thetamax},
	opt,PlotPoints->{10,33}];

    twiflat[pm_,a_,c_][sl_][u_,v_]:= {v*Cos[u],v*Sin[u],
        sl*u + ArcSin[c/(v*a)]*c + Sqrt[v^2*a^2 - c^2]*pm}

    twiflat[pm_,a_,c_][sl_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[twiflat[pm,a,c][sl][u,v],{u,umin,umax},{v,vmin,vmax},
	opt,PlotPoints->{50,15}];
     
    twisphere[a_,b_][u_,v_]:= {a*Cos[u]*Cos[v],
                               a*Sin[u]*Cos[v],
                               a*Sin[v] + b*u}

    twisphere[a_,b_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[twisphere[a,b][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];

    veronese[a_][u_,v_]:= {u,v,u^2,u v,v^2}

    wallis[a_,b_,c_][u_,v_]:= {v*Cos[u],v*Sin[u],
                               c*Sqrt[a^2 - b^2*Cos[u]^2]}
 
    wallis[a_,b_,c_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[wallis[a,b,c][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    whitneyumbrella[u_,v_]:= {u*v,u,v^2}

    whitneyumbrella[umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[whitneyumbrella[u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
    wrinkles[a_][u_,v_]:= {u,v,a*(u/v + v/u)}

    wrinkles[a_][umin_,umax_,vmin_,vmax_,opt___]:=
    pplot3d[wrinkles[a][u,v],{u,umin,umax},{v,vmin,vmax},
	opt];
     
(* Implicitly Defined Surfaces *)

    cossurfaceimplicit[a_][x_,y_,z_]:=
               (z - a*x*y)^2 - a^2(1 - x^2)*(1 - y^2)

    crosscapimplicit[a_,b_][x_,y_,z_]:=
        (a*x^2 + b*y^2)*(x^2 + y^2 + z^2) - 2*z*(x^2 + y^2)

    ellipsoidimplicit[a_,b_,c_][x_,y_,z_]:=
               (x/a)^2 + (y/b)^2 + (z/c)^2 - 1

    ellipticparaboloidimplicit[a_,b_,c_][x_,y_,z_]:=
               (x/a)^2 + (y/b)^2 - c*z
    
    equihom1implicit[x_,y_,z_]:= x*y*z - 1

    equihom2implicit[x_,y_,z_]:= (x^2 + y^2)*z - 1

    equihom3implicit[x_,y_,z_]:= x^2*(z - y^2)^3 - 1

    equihom4implicit[x_,y_,z_]:= x^2*(z - y^2)^3 + 1

    equihom5implicit[x_,y_,z_]:= z - x*y - x^3/3

    equihom6implicit[x_,y_,z_]:= z - x*y + Log[x]

    goursatimplicit[a_,b_,c_][x_,y_,z_]:= x^4 + y^4 + z^4 +
                   a*(x^2 + y^2 + z^2)^2 + b*(x^2 + y^2 + z^2) + c

    hyperboloidimplicit[a_,b_,c_][x_,y_,z_]:=
               (x/a)^2 + (y/b)^2 - (z/c)^2 - 1

    hy2sheetimplicit[a_,b_,c_][x_,y_,z_]:=
               (x/a)^2 - (y/b)^2 - (z/c)^2 - 1
    
    hyperbolicparaboloidimplicit[a_,b_,c_][x_,y_,z_]:=
               (x/a)^2 - (y/b)^2 - c*z

    kazoolaimplicit[a_,b_,c_,d_][x_,y_,z_]:=
        d + c*x^2y^2z^2 - a*(x^2 + y^2 + z^2) - b*(x^4 + y^4 + z^4)

    kummerimplicit[x_,y_,z_]:= x^4 + y^4 + z^4 -
             (y^2*z^2 + z^2*x^2 + x^2*y^2) - (x^2 + y^2 + z^2) + 1

    scherkimplicit[a_][x_,y_,z_]:=
               E^(a*z)*Cos[a*x] -  Cos[a*y]

    scherkimplicit[a_,phi_][x_,y_,z_]:=
               Tan[x/(a*Cos[phi])]*Tan[y/(a*Sin[phi])] - Tanh[z/a]

    scherk5implicit[x_,y_,z_]:=
               Sinh[x]*Sinh[y] - Sin[z]

    sinsurfaceimplicit[a_][x_,y_,z_]:=
               4*x^2*y^2*(a^2 - z^2) - a^2*(x^2 + y^2 - z^2)^2

    sphereimplicit[a_][x_,y_,z_]:=
               x^2 + y^2 + z^2 - a^2

    sphereimplicit[n_,a_][x_,y_,z_]:=
               x^(2*n) + y^(2*n) + z^(2*n) - a^(2*n)

    torusimplicit[a_,b_][x_,y_,z_]:=
              z^2 + (Sqrt[x^2 + y^2] - a)^2 - b^2

    twocuspsimplicit[x_,y_,z_]:= (z - 1)^2*(x^2 - z^2) -
           (x^2 - z)^2 - y^4 - y^2*(2*x^2 + z^2 + 2*z - 1)

(* Drum Plots *)

    circularmembrane[m_,n_,t_][r_,theta_]:=
        Flatten[{r*Cos[theta],r*Sin[theta],
            BesselJ[m,r*BesselJZeros[m,{n,n}]]*
            Cos[m*theta]*
            Cos[t*BesselJZeros[m,{n,n}]]}]

(*
    circularmembrane[m_,n_,t_][r_,theta_]:=
        {r*Cos[theta],r*Sin[theta],
            BesselJ[m,r*If[ListQ[BesselJZeros[m,{n,n}]]==True,
                  First[BesselJZeros[m,{n,n}]],BesselJZeros[m,{n,n}]]]*
            Cos[m*theta]*
            Cos[t*If[ListQ[BesselJZeros[m,{n,n}]==True],
                  First[BesselJZeros[m,{n,n}]],BesselJZeros[m,{n,n}]]]}
*)


      circularmembraneplot[m_,n_,t_,opts___]:=
        Module[{zzz},
             zzz=circularmembrane[m,n,t][r,theta];
             ParametricPlot3D[Evaluate[zzz],
                 {r,0,1},{theta,0,2*Pi},opts]]

      circularmembranemovie[m_,n_,{t_,tmin_,tmax_,numf_},
          opts___]:=
          Do[Module[{zzz},
             zzz=circularmembrane[m,n,t][r,theta];
             ParametricPlot3D[Evaluate[zzz],
                 {r,0,1},{theta,0,2*Pi},opts]],
                 {t,tmin,tmax,numf - 1}]

      squaremembrane[m_,n_,t_][u_,v_]:=
             {u,v,Sin[m*u]*Sin[n*v]*Cos[Sqrt[m^2 + n^2]*t]}

      squaremembraneplot[m_,n_,t_,opts___]:=
             ParametricPlot3D[Evaluate[
                squaremembrane[m,n,t][u,v]],
                 {u,0,Pi},{v,0,Pi},opts]

      squaremembranemovie[m_,n_,{t_,tmin_,tmax_,numf_},
          opts___]:=
             Do[ParametricPlot3D[Evaluate[
                squaremembrane[m,n,t][u,v]],
                 {u,0,Pi},{v,0,Pi},opts],
                 {t,tmin,tmax,numf - 1}]

(* Minimal Curves *)

    astroidmincurve[a_][z_]:= {a*Cos[z]^3,a*Sin[z]^3,
                               -3*I/4*a*Cos[2*z]}

    bourmincurve[m_][z_]:=
        {z^(m - 1)/(m - 1) - z^(m + 1)/(m + 1),
         I*(z^(m - 1)/(m - 1) + z^(m + 1)/(m + 1)),2*z^m/m}

    bourmincurve[m_,n_][z_]:=
        {z^(m - 1)/(m - 1) - z^(m + 2*n - 1)/(m + 2*n - 1),
         I*(z^(m - 1)/(m - 1) + z^(m + 2*n - 1)/(m + 2*n - 1)),
         2*z^(m + n - 1)/(m + n - 1)}

    cardioidmincurve[a_][z_]:= {2*a*Cos[z]*(1 + Cos[z]),
                  2*a*(1 + Cos[z])*Sin[z],8*I*a*Sin[z/2]}

    catalanmincurve[a_][z_]:= {a*z - a*Sin[z],a - a*Cos[z],
                                    -4*I*a*Cos[z/2] + 4*I*a}

    catenoidmincurve[a_][z_]:= {a*Cosh[z/a],z,-I*a*Sinh[z/a]}

(*
different: harmonictoanalytic[catenoid[a]][z]
*)

    cayleysexticmincurve[a_][z_]:=
         {4*a*Cos[z/2]^4*(-1 + 2*Cos[z]),
          4*a*Cos[z/2]^3*Sin[3*z/2],3*I*a*(z + Sin[z])}

    circleinvolutemincurve[n_,a_][z_]:=
        Module[{tmp1,tmp2,tmp3},
            tmp1=E^(I*tt)Normal[Series[E^(-I*tt),{tt,0,n}]];
            tmp2=a*ComplexExpand[{Re[tmp1],Im[tmp1]}];
            tmp3=tmp2 /. tt->z;
            Append[tmp3,I*a*z^(n + 1)/(n + 1)!]]

    cissoidmincurve[a_][z_]:= {2*a*z^2/(1 + z^2),
         2*a*z^3/(1 + z^2),2*I*a*(Sqrt[4 + z^2] - 
          Sqrt[3]*ArcTanh[Sqrt[4 + z^2]/Sqrt[3]])}

    clothoidmincurve[a_][z_]:=
            {a*Sqrt[Pi]*FresnelS[z/Sqrt[Pi]],
             a*Sqrt[Pi]*FresnelC[z/Sqrt[Pi]],I*a*z}

    cnchyprofilemincurve[pm_,a_,b_][z_]:=
        {pm*b*Cosh[z/a],-I*a*EllipticE[I*z/a,-b^2/a^2],I*z}

    costamincurve[z_]:=
        {(1/2)*(-WeierstrassZeta[z,{189.07272,0}] + Pi*(z - I) +
        (1 + I)*0.35889 +
        0.228473*(WeierstrassZeta[z - 1/2,{189.07272,0}] -
             WeierstrassZeta[z - I/2,{189.07272,0}])),
        (I/2)*(-WeierstrassZeta[z,{189.07272,0}] + Pi*(-z + 1) -
        (1 + I)*0.35889 -
        0.228473*(WeierstrassZeta[z - 1/2,{189.07272,0}] -
             WeierstrassZeta[z - I/2,{189.07272,0}])),
        Sqrt[2*Pi]/4*(Log[(WeierstrassP[z,{189.07272,0}] - 6.87519)/
             (WeierstrassP[z,{189.07272,0}] + 6.87519)] - Pi*I)}

    cpcprofilemincurve[pm_,a_,b_][z_]:= {b*pm*Cos[z/a],
                          a*EllipticE[z/a,b^2/a^2],I*z}

    deltoidinvolutemincurve[a_][z_]:=
        {(a/3)*(8*Cos[z/2] + 2*Cos[z] - Cos[2*z]),
         (a/3)*(-8*Sin[z/2] + 2*Sin[z] + Sin[2*z]),
          (4*a*I/9)*(3*z - 2*Sin[3*z/2])}

    deltoidmincurve[a_][z_]:= {-a + 2*a*Cos[z]*(1 + Cos[z]),
             2*a*(1 - Cos[z])*Sin[z],-8*(I/3)*a*Cos[3*z/2]}
    
    ellipsemincurve[a_,b_][z_]:= {a*Cos[z],b*Sin[z],
                                  I*b*EllipticE[z,-a^2/b^2]}

    ennepercatenoidmincurve[n_][z_]:=
        {1/z + z - 2*z^n/n - z^(2*n + 1)/(2*n + 1),
         I*(2*z^(n + 1) + 2*n^2*(z^2 - 1) + n*(-1 + z^2 +
         4*z^(n + 1) + z^(2*(n + 1))))/(n*(2*n + 1)*z),
         2*z^(n + 1)/(n + 1) + 2*Log[z]}

    ennepercatenoidmincurve[n_,a_,b_][z_]:=
        {a^2/z + z - 2*a*b*z^n/n - b^2*z^(2*n + 1)/(2*n + 1),
         I*(-a^2*n*(2*n + 1) + 2*a*b*(2*n + 1)*z^(n + 1) + 
         n*z^2*(1 + 2*n + b^2*z^(2*n)))/(n*(2*n + 1)*z),
         2*b*z^(n + 1)/(n + 1) + 2*a*Log[z]}    

    ennepermincurve[n_][z_]:= {z - z^(2*n + 1)/(2*n + 1),
        I*(z + z^(2*n + 1)/(2*n + 1)),2*z^(n + 1)/(n + 1)}

    epicycloidmincurve[a_,b_][z_]:=
              {(a + b)*Cos[z] - b*Cos[(a + b)*z/b],
               (a + b)*Sin[z] - b*Sin[(a + b)*z/b],
               -4*I*b*(a + b)*Cos[a*z/(2*b)]/a}

    hennebergmincurve[z_]:= {-8*Sinh[z]^3/3,
          8*I/3 - 2*I*Cosh[z] - 2*I/3*Cosh[3*z],2*Cosh[2*z]}

    hyperbolamincurve[a_,b_][z_]:= {a*Cos[z],b*Sin[z],
                                    b*EllipticE[I*z + a^2/b^2]}

    hypocycloidinvolutemincurve[a_,b_][z_]:=
       {(a^2*Cos[z] - 3*a*b*Cos[z] + 2*b^2*Cos[z] - 
         a*b*Cos[z - a*z/b] + 2*b^2*Cos[z - a*z/b] + 
         4*a*b*Cos[z - a*z/(2*b)] - 4*b^2*Cos[z - a*z/(2*b)])/a,
        (a^2*Sin[z] - 3*a*b*Sin[z] + 2*b^2*Sin[z] - 
         a*b*Sin[z - a*z/b] + 2*b^2*Sin[z - a*z/b] + 
         4*a*b*Sin[z - a*z/(2*b)] -  4*b^2*Sin[z - a*z/(2*b)])/a,
        2*I*(a^2 - 3*a*b + 2*b^2)*(a*z - 2*b*Sin[a*z/(2*b)])/a^2}

    hypocycloidmincurve[a_,b_][z_]:=
           {(a - b)*Cos[z] + b*Cos[(a - b)*z/b],
            (a - b)*Sin[z] - b*Sin[(a - b)*z/b],
            -4*I*(a - b)*b*Cos[a*z/(2*b)]/a}

    jorgemeeksmincurve[n_,a_][z_]:=
        {Integrate[jorgemeeksmincurveprime[n,a][zz][[1]],{zz,0,z}],
         Integrate[jorgemeeksmincurveprime[n,a][zz][[2]],{zz,0,z}],
         a*(1 + z^n)/(2*n*(1 - z^n))}

    jorgemeeksmincurve[2,a_][z_]:=
    {-I/4*a*Pi  + (a/4)*Log[(z + 1)/(z - 1)],
     -I*a*z/(2*(z^2 - 1)),a*(1 + z^2)/(4*(1 - z^2))}

    jorgemeeksmincurve[3,a_][z_]:=
    {(a/18)*(4*I*Pi + 3*z/(1 + z + z^2) - 4*Log[z - 1] + 2*Log[1 + z + z^2]),
      -(I*a/54)*(2*Sqrt[3]*Pi + 3*(3*z*(z + 1) - 
       4*Sqrt[3]*(z^3 - 1)*ArcTan[(2*z + 1)/Sqrt[3]])/(z^3 - 1)),
     a*(z^3 + 1)/(6*(1 - z^3))}

    jorgemeeksmincurve[4,a_][z_]:=
    {3*I/16*a*Pi + (a*(2*z - 3*(1 + z^2)*Log[-1 + z] + 
        3*(1 + z^2)*Log[1 + z]))/(16*(1 + z^2)),
     -I*a*z/(8*(z^2 - 1)) + 3*I*a*ArcTan[z]/8,
      a*(1 + z^4)/(8*(1 - z^4))}

    jorgemeeksmincurveprime[n_,a_][z_]:=
        {(a*(1 - z^(2*n - 2)))/(2*(z^n - 1)^2),
         (a*I*(1 + z^(2*n - 2)))/(2*(z^n - 1)^2),
          a*z^(n - 1)/(z^n - 1)^2}

    lemniscatemincurve[a_][z_]:=
        {(a*Cos[z])/(1 + Sin[z]^2),
         (a*Cos[z]*Sin[z])/(1 + Sin[z]^2),I*a*EllipticF[z,- 1]}

    logmincurve[n_][z_]:=
        Module[{ii,nn,zz},
            ii[nn_][zz_]:=Nest[Integrate[#,zz]&,1/zz,nn];
            wei[#&,ii[n]][z]]

    logspiralmincurve[a_,b_][z_]:= {a*E^(b*z)*Cos[z],
          a*E^(b*z)*Sin[z],(I*a*Sqrt[1 + b^2]*E^(b*z))/b}

    nephroidmincurve[a_][z_]:= {a*(3*Cos[z] - Cos[3*z]),
                   a*(3*Sin[z] - Sin[3*z]),-6*I*a*Cos[z]}

    nielsenspiralmincurve[a_][z_]:= 
        {a*CosIntegral[z],a*SinIntegral[z],I*a*Log[z]}

    parabolamincurve[a_][z_]:= {2*a*z,a*z^2,
        I*a*(z*Sqrt[1 + z^2] + Log[z + Sqrt[1 + z^2]])}

    richmondmincurve[n_][z_]:=
            {-1/(2*z) - z^(2*n + 1)/(4*n + 2),
             -I/(2*z) + I*z^(2*n + 1)/(4*n + 2),z^n/n}

    rosemincurve[n_,a_][z_]:= {a*Cos[z]*Cos[n*z],a*Cos[n*z]*Sin[z],
                               I*a*EllipticE[n*z,1 - n^2]/n}

    scherktowermincurveprime[n_][z_]:=
        {(1 - z^(2*n - 2))/(2*(1 + z^(2*n))),
         (I*(1 + z^(2*n - 2)))/(2*(1 + z^(2*n))),
          z^(n - 1)/(1 + z^(2*n))}

    scmincurve[z_]:= {z^2,z^3,
         I*(4/(27*z) + z/3)*Sqrt[4*z^2 + 9*z^4]}

    tractrixmincurve[a_][z_]:= {a*Sin[z],
                                a*(Cos[z] + Log[Tan[z/2]]),
                        I*a*(Log[Cos[z/2]] + Log[Sin[z/2]])}

(* Surface Metrics *)

    ee[geopolar[f_]][u_,v_]:= 1

    ff[geopolar[f_]][u_,v_]:= 0

    gg[geopolar[f_]][u_,v_]:= f[u,v]^2

    ee[liouville[f_,g_]][u_,v_]:= f[u] + g[v]

    ff[liouville[f_,g_]][u_,v_]:= 0

    gg[liouville[f_,g_]][u_,v_]:= f[u] + g[v]

    ee[poincare[lambda_]][u_,v_]:= 1/(lambda*v)^2

    ff[poincare[lambda_]][u_,v_]:= 0

    gg[poincare[lambda_]][u_,v_]:= 1/(lambda*v)^2

    ee[poincare[m_,n_,lambda_]][u_,v_]:= 1/(lambda*v)^m

    ff[poincare[m_,n_,lambda_]][u_,v_]:= 0

    gg[poincare[m_,n_,lambda_]][u_,v_]:= 1/(lambda*v)^n

    ee[poindisk[lambda_]][u_,v_]:=
                  1/(1 - lambda^2*(u^2 + v^2)/4)^2

    ff[poindisk[lambda_]][u_,v_]:= 0

    gg[poindisk[lambda_]][u_,v_]:=
                  1/(1 - lambda^2*(u^2 + v^2)/4)^2

    ee[schwarzschildplane[m_]][u_,v_]:=  -(1 - 2*m/v)

    ff[schwarzschildplane[m_]][u_,v_]:= 0

    gg[schwarzschildplane[m_]][u_,v_]:= 1/(1 - 2*m/v)

    ee[schwarzschildplane[m_,f_]][u_,v_]:= -(1 - 2*m/v + f[v])

    ff[schwarzschildplane[m_,f_]][u_,v_]:= 0

    gg[schwarzschildplane[m_,f_]][u_,v_]:= 1/(1 - 2*m/v + f[v])

End[]

EndPackage[]