** 3.3.3 Isometry for the Analytic Euclidean
Plane Model
***Printout*

*In
mathematics the art of asking questions is more valuable than solving problems.
—*

** ** A natural
question arises, "What is the form of the matrix of an
isometry that is an
affine transformation of the Euclidean plane?" We investigate that question.
The matrix of an affine
transformation of the Euclidean plane has the form .
What restrictions need to be placed on the values *a _{ij}* for

and

If we assume *A* is
an isometry, then *d*(*X, Y*) = *d*(*X', Y'*). Hence

For the first and last expressions to be equal, we must have

Assume *a*_{11} = 0. Then by (1), *a*_{21} = ±1. And by (3), *a*_{22} = 0. Finally, by (2), *a*_{12} = ±1.

Assume *a*_{11} is nonzero. Then by (3), Substitute into (2), Hence, by (1), *a*_{22} = ±* a*_{11}.
If *a*_{22} = *a*_{11}, then (3) implies *a*_{12} = *–a*_{21}. If *a*_{22}
= *–a*_{11}, then (3) implies *a*_{12} = *a*_{21}.

Since **,
**there is a real number such that and Further, note that there are no restrictions on

Thus, we summarize the results in the following proposition, which only has the converse left to be proven.

*Proposition 3.7. An affine transformation of the Euclidean plane is an
isometry if and only if the matrix representation is*

* (direct isometry)
or (indirect isometry).*

** Corollary to Proposition 3.7. The determinant of a direct isometry is 1 and the determinant of an indirect isometry
is –1.**

* *

** Examples. **Which is a direct isometry? Which is an indirect
isometry? Note the positions of the triangles. What happens with the measures
of the angles between the sides? Use the definition of the measure of the angle
between two lines to check your conjectures. Investigate further by looking at
the animation video clips. (See below between the two examples for the links.)

* Click here for an animation of the
graphic examples:* example above
or example below.

*Proposition 3.8. The product of the matrices of two affine direct or
two affine indirect isometries of the Euclidean plane is the matrix of an affine direct
isometry. Further, the product of an affine direct and an affine indirect
isometry of the Euclidean plane is an affine indirect isometry of the Euclidean
plane.*

*Proposition 3.9. The set of affine direct isometries of the
Euclidean plane is a group.*

*Proposition 3.10. The set of affine isometries of the
Euclidean plane is a group.*

We partially
examine the questions asked before the illustrations above. Notice that the
first diagram illustrates a direct isometry; whereas, the second diagram is an
indirect isometry.
If we label the vertices in the original triangle clockwise
as *A, B*, and *C,* what happens with the vertices in the image figure for each diagram?
In the first diagram, the vertices of the image remain in the same clockwise
order but reverse to a counter-clockwise order in the second diagram. It
appears that a direct isometry keeps the orientation the same, and an indirect
isometry reverses the orientation.

Examine this further by computing
the measures of the angles between the lines determined by the sides for both
diagrams. The angle between the lines *l*[1, –1, 0] and *m*[1,
–3, 2]
measures approximately –0.464, where *(Check
the computations determining the two lines and the measure of the angle.)* The
measure of the angle between the two image lines *l'*[1, –3.085, –4.322] and *m'*[1, 6.655, 8.210], in the first diagram,
is approximately –0.464. The angles between the lines *l* and *m *and the lines *l'* and *m'* measure the same. The measure of the angle between the two image lines *l'*[1, 0.325, –1.424] and *m'*[1, 0.985, –1.751], in the second
diagram, is approximately 0.464. The measure of the angle between the image
lines *l'* and *m'* has the opposite sign of the measure of the angle between the
lines *l* and *m.* Compute the values for the other two angles, and .

The observations in the above
examples lead us to conjecture the next two propositions.

*Proposition 3.11. For an affine direct isometry of the Euclidean plane,
the measure of the angle between two lines equals the measure of the angle
between the two image lines.*

*Proof.* Let the
lines *p'* and *q'* be the images of lines *p*
and *q* under a direct isometry with
matrix *A.* Let *B*
be the inverse of the matrix *A*. By
Proposition 3.9, *B* is the matrix of a
direct isometry. By Proposition 3.6, there are nonzero real numbers *k*_{1} and *k*_{2} such that *k*_{1}*p'* = *pB*
and *k*_{2}*q'* = *qB*. We use the
results of the previous two sentences together with
Proposition 3.7 to compute .
The line *q' *may be expressed in a similar form. Compute the measure of the
angle between *p'* and *q' *where the angle is not a right angle. *(The case for a right angle is
left for you to verify.)*

* *

*Proposition 3.12. For an affine indirect isometry of the Euclidean
plane, the measure of the angle between the two image lines has the opposite
sign of the measure of the angle between the two lines. *

** Exercise 3.37.** Let

** Exercise 3.38.** An affine transformation maps

** Exercise 3.39.** Complete the proof of Proposition 3.7.

** Exercise 3.40.** Prove Proposition 3.8.

** Exercise 3.41.** Prove Proposition 3.9.

** Exercise 3.42.** Prove Proposition 3.10.

** Exercise 3.43.** Fill in the missing steps of the two computations in the
proof of Proposition 3.11.

** Exercise 3.44.** Prove the inverse of an affine
indirect isometry of the Euclidean plane is an
affine indirect isometry of the Euclidean plane.

** Exercise 3.45.** Prove Proposition 3.12.

3.3.2 Model - Collinearity for the Analytic Euclidean
Plane |

© Copyright 2005, 2006 - Timothy Peil |