A Geometry View
Graphics created in Mathematica using functions from
A Gray, Modern Differential Geometry of Curves and Surfaces, Second Edition,
CRC Press, Boca Raton, 1997.
Animations using LiveGraphics3D software
Geometry is concerned with measurements of distance and angle and
how they change. The most important concept that characterises curves,
surfaces and spaces of higher dimension is curvature.
Curvature measures the departure from flatness
Flatness for a curve means that it is a straight line
Thus, curvature measures the departure from a line and constant curvature will
generate a circle in a plane or a helix in space.
The Fundamental Theorem of Plane Curves states that plane curves are
classified, up to a rotation and translation, by their curvature---which
is the rate of change of angular direction per unit arc length s.
So, given a starting point and the curvature function, the curve can be
found.
Can you guess what a plane curve will look like if its curvature is 1, s, or 1/s ?
These four plane curves have curvatures:
1 (circle), s (clothoid),
sin s, s J0(s)
Observe what happens in the second
curve above, the clothoid, when we have unbounded curvature and in the
last two curves when we have periodic curvature.
An epitrochoid is a curve traced out by a point
attached to a circle rolling outside another circle, so it should
display symmetric periodic curvature.
Here is an epitrochoid and its curvature
Here is the limacon
l(t) = (1 + 2 Cos[t]) {Cos[t], Sin[t]}
and its curvature
The Fundamental Theorem of Space Curves states that space curves are
classified, up to a rotation and translation, by their curvature and torsion.
Torsion measures the departure of a curve from a plane.
So, given a starting point and two functions of arc length representing
curvature and torsion, the curve can be determined. We know that constant
curvature in the plane gives a circle; if we have constant torsion as well
then we obtain a circular helix---shown here as a tube:
The next two curves are again encased in ruled tubes
to make them clearer as two knots: the trefoil and the
figure eight knot. You can see the rulings twisting rapidly where there
is rapid departure from a plane and the torsion is high
We can see this more clearly by rotating the
Trefoil knot
Topology interacts with geometry through curvature of surfaces
and through torsion of curves. We saw that constant curvature of a curve
causes it to close up in a circle and constant curvature of a surface
causes it to close up as a sphere.
A sphere and a torus are both closed surfaces but they are
topologically different---this can be seen graphically by imagining
a map-colouring exercise on each, like on the Earth and on a toroidal
space station shared among different countries. The sphere needs at
most four colours but the torus could need as many as seven, because on
a torus there is the possibility for returning from two different directions,
round the wheel and round the tube:
Here is a micrograph of the DNA of E. Coli,
showing twisting and supercoiling
Schematic of protein (blue/yellow) bound to DNA (green) at a promoter region
inducing expression of a genome (red)
White's Theorem states that the topological link number Lk
for a pair of closed curves is the sum of their twist number Tw
and their writhe number Wr:
Lk = Tw + Wr
This is remarkable because the two terms on the right are geometrical
but their sum is purely a topological count of the linking of the two closed curves.
This has important implications for molecular biology of DNA where the two curves correspond to the edges of the DNA helix. Supercoiling corresponds to Writhe in the theorem and enzymes exist that alter Tw and Lk: compacting the molecule
gives protection from virus attacks and opening up locally allows replication.
Curves illustrating White's Theorem
Surfaces
Gauss put the study of curvature of surfaces on a rigorous basis
and we give his name to the intrinsic function he used
The (Gaussian) curvature of a surface is a scalar
measure of the rate of change of
direction of a unit normal vector round the surface
A plane is flat with zero curvature: it has a constant unit normal direction
A sphere has constant curvature---its unit normal vector
changes at a constant rate everywhere
Conventionally, we say that a sphere has positive curvature
because this surface always lies on the same side of its tangent plane---then
since the curvature is constant it actually closes up. For a saddle,
the surface actually crosses the tangent plane around every point
and then we have negative curvature. Here is a simple saddle given by
z = x^2 - y^2 and its curvature, which we see is -4 at the origin but
tends to zero as we move away from the origin---also shown is a cross-section through the curvature plot
There are more complicated saddle surfaces, here is one
Monkey Saddle: z = x^3 - 3x y^2
Here it is with its Gaussian curvature---note that on this
saddle at the origin the curvature is zero as you see from the graph
of a cross-section of the curvature plot:
We know that a sphere has constant positive curvature and we know that
on a saddle the curvature is negative. So, if we have constant negative curvature
we must expect that around every point the surface looks like a saddle. Then,
unlike a sphere, the surface will not close up, it is unbounded and called
a pseudosphere. Here
is a famous example:
New Surfaces From Old
A Mobius band is made by joining a strip after a twist, resulting in
one edge, one side and `non-orientable'---no continuous normal.
Joining the ends of a tube or cylinder creates a torus, which is orientable:
Joining the ends of a tube with opposing orientations---the
equivalent of a twist for the strip---gives a
Klein bottle, most easily seen by using a double-tube:
Klein Bottle: Non-orientable closed surface
In addition to the Gaussian curvature, which is intrinsic to
the surface and independent of how it is embedded in space,
there is the mean curvature, which measures the average curvature
of perpendicular curves in the surface
Minimal surfaces have minimal area among surfaces with the same boundary
and hence zero mean curvature.
Soap films make minimal surfaces on wire frames: so we see that mean curvature
of the surface is closely related to the elastic energy of the soap film---the
elastic energy is minimized in a stretched film.
The characteristic of a minimal surface is that it must around every point
look positively curved in one principal direction and negatively curved
in the perpendicular direction, then the sum of principal curvatures is zero.
Here are two examples that can be continuously transformed into one another through a family of minimal surfaces:
the helicoid and the catenoid
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