Euclidean similarities

 

You may recall that a pair of similar triangles are two triangles and with sides , , , i=1,2, respectively, for which after choosing the appropriate correspondence between the sides, the ratios of the corresponding lengths , , and are all the same. A similarity is a transformation of the plane that stretches all distances by exactly the same scale factor. A congruence or isometry (for ``equal distance'') is a transformation that leaves all distances exactly the same (i.e. the scale factor is 1). In this section, we would like to describe exactly what all the possible similarities are. Here we shall make reference to Barnsley [Bar93], Chapter III, section 2, although we shall adopt a less general approach.

We begin with the distance formula between two points and , given by Pythagoras' theorem:

We think of transformations T of the plane as functions , and we are particularly interested in how transformations change distances in the plane. Properties related to distance are often called metrical. As it turns out, the key aspects of distance are shared by many formulas other than that provided by Pythagoras. There is a more general concept called a metric defined as follows:

1. A metric is a function on pairs of points that takes only nonnegative real values.
2. If , then .
3. .
4. For any three points , , , we have

   

   This is called the triangle inequality

For just about all arguments involving distance, these are the only properties we need to use. As it turns out, the concept of metric is useful even in subjects where ``distance'' doesn't seem to make sense, for instance, in symbolic dynamics we often introduce metrics on sets of strings in the symbols. For now, we will just work with Pythagoras' metric.

A similarity is a transformation that changes all distances by the same factor, that is:

for all points , . The constant c>0 is the scaling factor of the similarity T. The basic example of a similarity is the dilation

for positive c. This transformation stretches or shrinks all points by the same factor; the origin (0,0) is fixed by . Suppose T is any other similarity with scaling factor c. Consider the composition . Then . To ease our notation a little, we shall write

so that we may write

Thus, for two points , , we obtain

A similarity with scaling factor 1 is called an isometry (for ``equal distance'') or congruence. Therefore, we conclude that any similarity may be composed with a dilation to obtain an isometry. We are left with determining the isometries.

The simplest example of an isometry is a translation

where is a fixed point in . All points P are then shifted by the fixed amount . We may check that is an isometry by the computation

after cancelling the terms.

If and are any two isometries, the composition is also an isometry, due to the calculation

Let's now assume that T is an arbitrary isometry. Then T maps the origin 0 to some point . Consider the composition . Then . This shows that we can use a translation to modify any isometry into one that fixes the origin.

We may now work with an isometry T that fixes 0. Consider the point T(1,0). Since (1,0) is at distance 1 from the origin, the same should be true for T(1,0). Using a little trigonometry, this means for some angle . We now pull another isometry out of the hat

This clearly has the property that . Also, we can check that . That it's an isometry as well takes a bit of stamina to prove

after observing that . Now the most important part is that . Thus, when we compose , we obtain an isometry that not only fixes 0 but also (1,0). Our trigonometry has guided us to the transformation that rotates the plane clockwise through an angle of about the origin.

Finally, we are left to consider isometries T that fix both the points (0,0) and (1,0). To finish the job, we must resurrect the old Side-Side-Side criterion of Euclidean geometry that says that any two triangles with the same sides are congruent. For our purposes this says that given distances d((0,0), P) and d((1,0),P) there is exactly zero, one or two solution points P=(x,y). If the triangle inequality is violated, there are no solutions. If there is a solution P=(x,0) on the horizontal axis, there is only one solution. If there is a solution P=(x,y) with , there is exactly one other solution (x,-y), the reflection through the horizontal axis. The relevance of this fact to our isometry is that

since T fixes both (0,0) and (1,0). It follows that, for P=(x,y), either T(P)=(x,y) or T(P)=(x,-y). We would hope that this choice of T(P) be consistent for all (x,y). To prove this requires an investigation into the continuity of isometries. We will leave that task for courses on real analysis, and simply say that isometries are perforce continuous maps from to itself. Moreover, this fact implies that either T(x,y)=(x,y) for all (x,y) or T(x,y)=(x,-y) for all (x,y). Let's define W(x,y)=(x,-y).

We have at last completed a stage-by-stage dissection of similarities of . Working backwards, we see that we have shown that all similarities can be represented as a composition of a dilation, a translation, a rotation, and possibly the reflection W.

THEOREM: Every similarity may be expressed in the form

for some c>0, , and angle .

We caution the reader that in the process of rewriting our previous analysis into the statement of the theorem we have taken advantage of the fact that the inverse maps to and are and , respectively. Thus, the constants and in the theorem are actually the opposites of the ones that arose in the analysis. We should also point out that is the inverse map to . The similarities of the first form in the theorem are called orientation-preserving (because they preserve the notions of left and right), while those of the second form are called orientation-reversing.

Coincidentally, all the similarities have the form

for some constants a,b,c,d,e,f. These are known as affine linear transformations, or affine maps for short. Not all affine maps are similarities. Commonly, they stretch distances by one factor in one direction and by a different factor in a different direction. In matrix form (we beg the reader's indulgence in introducing a few concepts from linear algebra without a thorough explanation), an affine linear map may be written

Writing vectors in boldface , we may write affine maps as , where A is a matrix and is a constant vector. The mapping is a translation, and so is an isometry. Therefore, the question of whether or not is a similarity depends entirely on the matrix A. It's a similarity if and only if there is a constant c>0 such that

 

for all vectors . In the exercises, we pose the problem of converting this condition into an equation involving solely the matrix A.

While we shall limit ourselves to two dimensions, some of the most interesting considerations are higher-dimensional. However, the classification of similarities of for is not terribly much different from what we have just done. Translations and dilations are exactly analogous; there are a few more degrees of freedom in the rotations.