** 3.3.2 Collinearity for the Analytic Euclidean
Plane Model
***Printout*

*The
human brain is the best pattern recognizer in history.
—Heinz-Karl Winkler (1955–?)*

** ** Corollary 3.2
stated that collinearity is invariant under an isometry of a neutral plane;
therefore, collinearity is invariant under an isometry of a Euclidean plane.
Further, in Exercise 3.25 of the Isometry section, you have shown that
collinearity is not necessarily an invariant property for a
transformation of a
Euclidean plane. But, a stronger result is possible with an
affine
transformation of the Euclidean plane; that is, an affine transformation of the
Euclidean plane does not need
to be an isometry for collinearity to be preserved.

** Exercise 3.35.** Find an affine transformation of the Euclidean plane that
is not an isometry.

** Proposition 3.5. Collinearity is invariant under an affine
transformation of the Euclidean plane. (Video lecture
at end of **
Isometry - Invariant Properties

*Proof.* Let *A* be a matrix of an affine
transformation of the Euclidean plane. Assume *X, Y, *and *Z* are distinct points. Let *X'* = *AX*,
*Y'* = *AY*, and *Z'* = *AZ*. Then Take the determinant of both sides of the
equation, to obtain Since* A* is the matrix of an affine
transformation of the Euclidean plane, det(*A*) is nonzero. Hence, by
Proposition 3.1,
the distinct points *X, Y, *and *Z* are collinear if and only if the points *X', Y', *and *Z'*
are collinear . Therefore, collinearity is
invariant under an affine transformation of the Euclidean plane.//

This result allows us to determine a matrix equation for determining lines.

*Propostion 3.6. **If A is the matrix of an affine transformation of the
Euclidean plane, then the image of a line l under this transformation is given
by kl' *=* lA*^{ –1 } *for some
nonzero real number k where l' is the image of l.*

*Proof.* Let *A* be the matrix of an affine
transformation of the Euclidean plane. Assume *l* is a line. By Proposition 3.5, the image of *l* is a line, denote it by *l'.*
For any point *X* in the plane, *X* satisfies the matrix
equation *lX* = 0 if and only if *X* is on *l*. Further, since *X'* = *AX,* *X'*
satisfies the matrix equation *l'X'* =
0 if and only if *X'* is on *l'*. Hence, *l'AX* = 0 if and only if *X*
is on *l. *Thus, *l'AX* = 0 if and only if *lX*
= 0. Since this is true for all points *X*,
there is a nonzero real number *k* such
that *kl'A* = *l*. Multiply both sides of this equation on the right by the inverse
of *A,* to obtain* kl' *=* lA*^{ –1 } for some nonzero real number* k.//*

** Important Notes.** The value of

** Example.** Let .
The lines

** Exercise 3.36.** (a) Verify the above example. (b) Find the matrix of an
affine transformation that maps

3.3.1 Isometry3.3.3 Model - Isometry for the Analytic Euclidean Plane |

© Copyright 2005, 2006 - Timothy Peil |