In this section, our factorization of isometries into osometries and translations pays off handsomely. It "factors" the difficulty of proving that isometries preserve displacements, angles, barycentric coordinates, lines and triangles, etc. into separate (and easier) proofs for osometries and translations.

1. Isometries Preserve Angles

(The next theorem is a correction of the one given in Tondeur’s textbook, Theorem 4.1. There it is tacitly assumed that the origin is at the vertex of the angle.)

Proof. Before beginning the proof, let’s review a formula from vector calculus. For any vectors X and Y, the dot product is related to the angle between them by

 X Y = |X||Y|\ cos\ /_XY. (1)

Substituting, we have

 $(P^\alpha - Q^\alpha)(R^\alpha - Q^\alpha)\ =$ $| P^\alpha - Q^\alpha | | R^\alpha - Q^\alpha |\ cos\angle P^\alpha Q^\alpha R^\alpha$ $=_{bb{1}}$ $| P - Q | | R - Q |\ cos \angle P^\alpha Q^\alpha R^\alpha$ (2)

where equality $=_{bb{1}}$ follows from the definition of isometry. So we have reduced the problem of showing that isometries preserve dot products, to showing that they preserve angles.

The idea now is to take advantage of the factorization of alpha by proving the theorem for translations and osometries, and then proving it for compositions of isometries.

Step 1: Proof for $\alpha=\tau_D$.

Assuming $\alpha$ is a translation by $D$, we have

 $P^\alpha-Q^\alpha\$ = $P^\tau-Q^\tau$ = $(P+D)-(Q+D)$ = $P-Q$.

Thus,

 $(P^tau-Q^tau)(R^tau-Q^tau)\ =_{bb{1}}$ $(P-Q)(R-Q)$ $=_{bb{2}}$ $|P-Q| |R-Q |\ cos \angle PQR$

where $=_{bb{1}}$ is by substitution, and $=_{bb{2}}$ follows from equation (1) above.

Combining this with equation (2) yields

 $|P-Q||R-Q|\ cos \angle P^\tau Q^\tau R^\tau\ =\ |P-Q||R-Q|\ cos \anglePQR$.

And so,

 $cos\ \angle P^\tau Q^\tau R^\tau\ =\ cos\ \angle PQR$.

This completes Step 1. The next step is to consider osometries:

Step 2: Proof for $\alpha=\beta$, where $\beta$ is an osometry.

 $(P^\beta-Q^\beta) (R^\beta-Q^\beta)\$ = $(P-Q)^\beta (R-Q)^\beta$ (by O3, O4) = $(P-Q) (R-Q)$ (by O2) = $|P-Q| |R-Q| \ cos \angle PQR$. (by (1))

Then combining this with (2), we get our result

 $cos\ \angle P^\beta Q^\beta R^\beta\ =\ cos\ \angle PQR$.

We now have enough to prove the theorem for any isometry:

Step 3: Proof for general $\alpha$.

5. Conclusion

Since isometries are such versatile transformations, wouldn’t it be nice to know all the different kinds of isometries? In one sense we already do. Every isometry is the composition of two unique isometries, a translation and an osometry (linear isometry). As we will see later, osometries are composed of only two kinds, rigid rotations about the origin, and reflections in lines through the origin. So, why isn’t that enough? And it is for many purposes.

As "scientists" of geometry we want two things further, discover the decomposition of an isometry into utterly simple factors, and partition the set of non-trivial isometries into mutually exclusive sets. The former goal is to identify the "atomic" isometries that generate all isometries under composition. The second is the classification of isometries by their composition which then determines what they do to points.

Comment.

Our exposition could diverge here from the geometrical approach taken by Tondeur. We could discover that every linear isometry can be represented by a matrix, which takes every point to the product of the matrix by the coordinates of the point. In fact, we could, by going one dimension higher, concoct a scheme whereby translation becomes a linear transformation, but in the projective plane. This is the approach taken by computer graphics.

We could diverge yet another way, namely by imposing the complex numbers and their algebra as the numerical description of the Euclidean plane. That is the approach taken by Michael Hvidsten, the author of the textbook for MA 402, and also a PhD student of Prof. Tondeur. This approach lets one study the Euclidean and non-Euclidean plane using identical mathematical methods.

Recall that the concept of isometry is the correct physical interpretation of Euclid’s concept of congruence. Each different way of studying the isometry group lends new insight in what profound contribution to human knowledge Euclid and his Greek colleagues made some 2300 years ago. Geometry, along with the theory of numbers, astronomy and music (the quadrivium or graduate curriculum of the Graeco-Roman world) is truly foundational, and well worth our study of it.