Euclidean Space $\mathbb{E} ^3$

This is the space of our normal experience and we distinguish between $\mathbb{R} ^3,$ the vector space or linear space of triples of real numbers, and Euclidean 3-space $\mathbb{E} ^3,$ the point space of triples of real numbers. Intuitively, we can think of a vector in $\mathbb{R} ^3$ as an arrow corresponding to the directed line in $\mathbb{E} ^3$ from one point (the blunt end of the vector arrow) to another point (the sharp end of the vector arrow).

In this course we shall be concerned only with three dimensional $\mathbb{E} ^3$ but the basic definitions of points, difference vectors and distances are the same for all $\mathbb{E} ^n$ with $n=1,2,3,\ldots;$ of course, in dimensions higher than 3, the extra directions will arise from other features than ordinary space--such as time, temperature, pressure etc. The important fact to hang onto is that $\mathbb{E} ^3$ consists of points represented by coordinates p=(p₁,p₂,p₃) while the directed difference between a pair of such points p,q is a vector $\overline{q-p}$ with components (q₁-p₁,q₂-p₂,q₃-p₃). In modern mathematics, it is customary to omit the overbar when writing vectors and this will be our usual practice; we identify vectors with their sets of components and points with their sets of coordinates.

The space $\mathbb{E} ^3$ has one particularly important feature: the availability of the vector cross product on $\mathbb{R} ^3,$which simplifies many geometrical proofs.

Our main interest in this course is to develop the geometry of curves and surfaces in $\mathbb{E} ^3.$ The basic ideas are very simple: a curve is a continuous image of an interval and a surface is a continuous image of a product of intervals; in each case the intervals may be open or closed or neither.

Difference vectors and distances

The difference map gives the vector arrow from one point to another and is defined by

$${\rm difference}:\mathbb{E} ^3\times\mathbb{E} ^3\rightarrow \mathbb{R} ^3: (p,q)\mapsto v=q-p.$$

The distance map takes non-negative real values and is defined by

$${\rm distance}:\mathbb{E} ^3\times \mathbb{E} ^3 \rightarrow [0,\infty) :(p,q)\mapsto \vert\vert q-p\vert\vert$$

here, $\vert\vert \ \vert\vert$ denotes the operation of taking the norm or absolute value of the vector, defined by

$$\vert\vert(q_1-p_1,q_2-p_2,q_3-p_3)\vert\vert=+\sqrt{(q_1-p_1)^2+(q_2-p_2)^2+(q_3-p_3)^2}$$

Then we can view $\mathbb{E} ^3$ as the set of points representing ordinary space, together with the standard Euclidean angles and Pythagorean distances and $\mathbb{R} ^3$ provides the vectors of directed differences between points. Not all books make this distinction so you need to be prepared to encounter the unstated identification $\mathbb{E} ^3=\mathbb{R} ^3.$ Often, we use the coordinates (x,y,z) for points in $\mathbb{E} ^3$ and denote by $\mathbb{E} ^2$ the set of points in $\mathbb{E} ^3$ with z=0 and then we abbreviate (x,y,0) to (x,y).

The standard unit sphere $\mathbb{S} ^n$ in a Euclidean n-space is the set of points unit distance from the origin; we shall often use $\mathbb{S} ^1$ in $\mathbb{E} ^2$ and $\mathbb{S} ^2$ in $\mathbb{E} ^3.$A parametric equation for the unit 2-sphere $\mathbb{S} ^2$ in $\mathbb{E} ^3$ is given by

$$g:[0,2\pi]\times [-\pi/2,\pi/2]: \rightarrow \mathbb{E} ^3 : (u,v) \mapsto (\cos v \cos u, \cos v \sin u, \sin v).$$