Plane Isometries

The first component of the word isometry is from the Greek isos "equal," of unknown origin. The second is from the Greek metron "a measure" [Schwartzman]. In mathematics, an isometry is a transformation (same as transform, function, operator) that preserves measurements, and more specifically distances between points. If f is such a transformation, then the definition means that

 (1) dist(f(P), f(Q)) = dist(P, Q),

where "dist" is a metric function on the space at hand. (We shall be mainly concerned with the isometries of the plane.) In the context of plane transformation "dist" is the common Euclidean metric.

A transformation that preserves distances is (by SSS) also bound to preserve angles.

A plane isometry is a function that is defined for any point of the plane. If we consider plane figures as collections of points, then every such collection S has an image f(S) under the isometry f. The definition of isometry assures that relative positions of points in S are preserved in f(S), such that the two sets of points - S and f(S) - are equal, which is what they are sometimes called. This mostly happens when an isometry is regarded as a rigid motion of the plane. (The term "rigid motion," with its intuitive appeal, may be confusing. Indeed, if both S and f(S) lie in the same plane which rigidly moves somehow, then surely both S and f(S) have been moving along. What the terminology intends to imply is that, as the result of such motion of the plane as a whole, the "new position" of S is the same as the "old position" of f(S).)

It is more common nowadays to call S and f(S) congruent. By definition, two sets S and T are said to be congruent iff there is an isometry f such that T = f(S).

Usually the terms come in pairs. Two triangles are equal (congruent) if there is a rigid motion (isometry) of the plane that maps one onto the other. (As an aside, I was taught and am used to thinking of isometries and equal triangles.) I prefer the "isometry/congruent" pair, although do think that at times the "rigid motion" appellation may be handy. For example, if we really could move a plane, there would be two ways of doing that: one is to let it glide over itself, the other requires lifting the plane into the 3D space and putting it back to its original location. If an isometry is such that, in terms of rigid motions, it can't be performed without lifting the plane into the space, it is called improper. It is proper, otherwise.

Two figures congruent under a proper isometry are said to be directly congruent. An improper isometry induces indirect congruence; two congruent figures under an improper isometry are called oppositely congruent.

The difference between the two kinds of isometries is that the proper ones preserve the orientation while the improper isometries change it. For example, translations and rotations are proper isometries. Reflections and glide reflections are improper.

Any isometry f is a 1-1 correspondence and, as such has an inverse f -1, which is also an isometry. Indeed, (1) is equivalent to

 (2) dist(P', Q') = dist(f -1(P'), f -1(Q')),

where P' = f(P) and Q' = Q(P).

The product of two isometries is naturally an isometry. However, in general the product is not commutative. Thus, the collection of all plane isometries is a noncommutative group, where the identical transformation plays the role of the unit element. The identity transformation may be looked at as either a trivial rotation or a trivial translation.

All isometries are of one of the four classes I mentioned above as examples. Any proper isometry is either a translation or a rotation. An improper isometry is either a reflection or a glide reflection [Coxeter, Yaglom].

An isometry may or may not have invariant, or fixed, sets, i.e., sets S that satisfy S = f(S). For example, a non-trivial rotation has a unique fixed point - its rotocenter. A non-trivial translation does not have fixed points; nonetheless it leaves invariant all the lines parallel to the vector for translation. The points on each of the lines are not fixed, they move, but they move along the line they are on.

A point P not on the axis of reflection f and its image f(P) determine the axis and, therefore, the reflection itself uniquely. The axis of reflection is simply the perpendicular bisector of the segment joining P and f(P). A translation also is uniquely determined by a point and its image. In general, to determine a rotation or a glide reflection one needs two point/image pairs.

The product of two reflections is either a reflection or a rotation; for each of these two reflections can be found (and not even uniquely) whose successive applications result in a given transformation. A glide reflection is the product of three reflections.

The following table summarizes the relevant facts about isometries

  Proper Specified by Invariant sets # point/image pairs No Axis of symmetry Each point on the axis.Each line perpendicular to the axis 1 1 Yes Vector of translation Each line parallel to the vector of translation 1 2 Yes Center and angle of rotation Center of rotation.Each circle around the center (*). 2 2 No Parallel vector and axis Axis of reflection 2 3

Note

(*) A half-turn fixes all the lines through the center of rotation.

Here is a multiplication table template for the group operation of plane isometries:

  Reflection Translation Rotation Translation or Rotation Glide Reflection Glide Reflection Translation or Rotation Glide Reflection Translation Rotation Reflection or Glide reflection Glide Reflection Rotation Rotation or Translation Glide Reflection Translation or Rotation Reflection or Glide Reflection Glide reflection Translation or Rotation

(If you wonder how this table has been filled, ask yourself, given the properties of the four transformations and knowing the fact that any product/entry is bound to be one of them, what might it be? Start with determining whether it is proper or not.)

References

1. H. S. M. Coxeter, Introduction to Geometry, John Wiley & Sons, 1961
2. C. Dodge, Euclidean Geometry and Transformations, Dover, 2004
3. H. Eves, A Survey of Geometry, Allyn and Bacon, 1972
4. H. R. Jacobs, Geometry, 3rd edition, W. H. Freeman and Company, 2003
5. D. Pedoe, Geometry: A Comprehensive Course, Dover, 1988
6. S. Schwartzman, The Words of Mathematics, MAA, 1994
7. I. M. Yaglom, Geometric Transformations I, MAA, 1962