TABLE OF CONTENTS Affine maps preserve the structure of the mapped affine spaces. Of particular interest are the properties which remain invariant when figures undergo affine mappings. These properties depend on the affine rules used to construct a figure not on the position of the figure in space. Often a special position allows for a simple proof of a general theorem. A pair of points and their midpoint form an example of a simple affine figure. The B zier and B-spline representation of curves have affine properties which are rather intriguing and most crucial for geometric design.
Literature: Blaschke, Coxeter, Pedoe
A number of classical and useful theorems can be verified by the relationships illustrated in Figure 10.3:
Ceva's Theorem
The dashed lines of the figure intersect at a common point p if and only if
For a proof observe that each ratio ? 1 : ? 2 corresponds to a unique point on bc. Hence, Figure 10.3 asserts that the product above equals 1 if and only if ? 1 : ? 2 corresponds to the black point on bc.
Menelaus' Theorem
The point q of the figure to the left is collinear with c and p if and only if
This theorem can also be derived from Figure 10.3. Each ratio ? 1 : ? 2 corresponds to a unique point on ab. Hence, the product above equals -1 if and only if ? 1
