文档介绍：Thomas Banchoff, Shiing-Shen Chern, and
William Pohl
Differential Geometry of Curves
and Surfaces, 1st Edition
SPIN C346 Differential Geometry Banchoff/Chern/Pohl
Monograph – Mathematics –
January 29, 2003
Springer
Berlin Heidelberg NewYork
Barcelona Hong Kong
London Milan Paris
Tokyo
Part I
Introduction
1 Review of Euclidean Geometry
Motions
Three-dimensional Euclidean space E consists of points which have as coor-
dinates ordered triples of real numbers x1, x2, x3. In vector notation, we write
x = (x1, x2, x3). The distance between two points is given by the formula
v
u 3
uX 2
d(x, y) = t (xi − yi) . ()
i=1
Note that d(x, y) ≥ 0 and d(x, y) = 0 if and only if x = y.
An affine transformation T from R3 to R3 is defined by T (x) = Ax + b,
where A is a 3 × 3 matrix and b is a vector in R3. An affine transformation
that preserves distance between points, such that d(T (x), T (y)) = d(x, y) for
all x and y is called a motion of R3.
Proposition 1. An affine transformation is a motion if and only if A is an
orthogonal matrix, . a matrix with columns that are mutually perpendicular
unit vectors.
Proof. Let the points (x1, x2, x3) and (y1, y2, y3) be mapped by an affine
0 0 0 0 0 0
transformation T onto the points (x1, x2, x3) and (y1, y2, y3) respectively, so
that
3
0 X
xi = aijxj + bi
j=1
3
0 X
yi = aijyj + bi ,
j=1
where aij denotes the entries of the matrix A and the bi denotes po-
nents of b. If we subtract these two equations, we get
3
0 0 X
xi − yi = aij (xj − yj) .
j=1
4 1 Review of Euclidean Geometry
Taking the sum of the squares gives us
X 0 0 2 X
(xi − yi) = aijaik (xj − yj) (xk − yk) ,
i i,j,k
where all the indices run from 1 to 3. This equality will only hold true if
3
X
aijaik = δjk , ()
i=1
where δjk = 1 if j = k and 0 otherwise.
0 P3
Given an affine transformation xi = j=1 aijxj + bi, we can solve ex-
0 0 P3
plicitly for the xi in terms of the xi. We first set xi − bi =