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Figure 2.19.
Proving Theorem 2.4.6.
L B
L
D
L C
C
E = A + 2AB
B
A
form A + r AB + s AC for all rational numbers r and s. These points are a dense set of
points in the plane. The final step handles the points where r or s are irrational. See
[Gans69].
2.4.7. Corollary. An affine transformation of the plane is completely determined by
what it does to three noncollinear points.
Proof. Showing that the corollary follows from Theorem 2.4.6 uses an, by now stan-
dard, argument that is left as an exercise for the reader.
We are ready to state and prove a fundamental theorem about affine maps.
2.4.8. Theorem. Every affine transformation of the plane can be described uniquely
by equations of the form (2.25). The determinant in (2.25b) is called the determinant
of the affine transformation . Conversely, every such pair of equations defines an affine
transformation.
Proof. We start with the converse. A transformation T defined by equations (2.25)
has an inverse that is again defined by linear equations of the same form. Let f(x,y)
= 0 be the equation of a line L . Then the set L ¢=T( L ) is defined by the equation
f(T -1 (x,y)) = 0. This shows that L ¢ is a line and that T is an affine map.
Next, let T be an affine map and choose three noncollinear points. By Theorem
2.4.5 there is a map M defined by equations (2.25) that agrees with T on those points.
Since we just showed that T is an affine map, we have two affine maps that act the
same on three noncollinear points. By Corollary 2.4.7, T = M and the theorem is
proved.
Because of Theorem 2.4.8 everything proved for the maps defined by equations
(2.25) holds for affine maps. We restate these properties to emphasize their validity
for affine maps.
(1) Every affine map in the plane is a composition of translations, rotations,
shears, and/or scaling transformations. Conversely, every composition of such
maps is an affine map.
(2) There is a unique affine transformation in the plane that maps three non-
collinear points into any other three noncollinear points.
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