** 3.2.3 Affine Transformation of the Euclidean
Plane
***Printout*

*A mathematician,
like a painter or a poet, is a maker of patterns. If his patterns are more
permanent than theirs, it is because they are made with ideas.
—*

** **

What is the form of a transformation matrix for the analytic
model of the Euclidean plane? We investigate this question. Let *A *= [*a _{ij}*]
be a transformation matrix for the Euclidean plane and (

.

Since the last matrix must be the matrix of a point in the
Euclidean plane, we must have *a*_{31}*x* + *a*_{32}*y* + *a*_{33}
= 1 for every point (*x*, *y*, 1) in the Euclidean plane. In
particular, the point (0, 0, 1) must satisfy the equation. Hence, *a*_{33} = 1. Further, the points
(0, 1, 1) and (1, 0, 1) satisfy the equation and imply *
a*_{32}
= 0 and *a*_{31} = 0,
respectively.
Therefore, the transformation matrix must have the form

,

which motivates the following definition.

* *

** Definition.** An

where each *a _{ij}
*is a real number.

* *

** Exercise 3.19.** Prove that every affine transformation of the Euclidean
plane has an inverse that is an affine transformation of the
Euclidean plane.

*Proposition 3.3. An affine transformation of the Euclidean plane is a
transformation of the Euclidean plane.*

**
Exercise 3.20.** Prove Proposition 3.3.

*Click here to see
an animation of a sequence of affine
transformations.
*

*Proposition 3.4.*
*The set of
affine transformations of the Euclidean plane form a group under matrix
multiplication.*

* *

*Proof.* Since the
identity matrix is clearly a matrix of an affine transformation of the
Euclidean plane and the product of matrices is associative, we need only show
closure and that every transformation has an inverse.

Let *A* and *B* be the matrices
of affine transformations of the Euclidean plane. Since det(*A*) and det(*B*)
are both nonzero, we have that det(*AB*) = det(*A*)* *
* ·** *det(*B*)
is not zero. Also,

is a matrix of an affine transformation of the Euclidean
plane. *(The last row of the matrix is 0 ,
0, 1.)* Hence closure holds.

Complete the proof by showing the
inverse property.//

** Exercise 3.21.** Given three points

** Exercise 3.22.** Find the matrix
of an affine transformation that maps

** Exercise 3.23.** Show the group of
affine transformations of the Euclidean plane is not commutative.

3.2.2 An Analytic Model for the Euclidean Plane3.3.1 Isometry |

© Copyright 2005, 2006 - Timothy Peil |