Course MT3818 Topics in Geometry

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Isometries of the plane

Many of the observations made about the group of the line can also be applied to I(R2): the isometries of the plane R2.

Again we can divide the isometries into two subsets: those for which the orthogonal transformation is in SO(2) (with determinant +1) and those with determinant -1. These are called direct and opposite symmetries respectively.

  1. Direct symmetries

    Here is a complete classification of direct symmetries.

    Theorem
    Any direct symmetry is either a translation or rotation about some point.

    Proof

    1. If f has a fixed point b then T-bcomp f comp Tb(0) = 0 and so this is a length preserving map which fixes the origin and hence is a linear map. Thus it is in SO(2) and is rotation about 0. Hence f is rotation about the point b.

    2. We have f = Tacomp L with L belongs SO(2) implies f(x) = a + L(x). Now we try to solve f(x) = x = a + L(x) to find a fixed point. That is, we look for a solution of (L - I)x = -a. The only case in which we could not find such a solution is if L - I were not invertible. That is, if +1 were an eigenvalue of L. But the only element in SO(2) with +1 as an eigenvalue is the identity. Hence if f fails to have a fixed point, L = I and f is a translation.


  2. Opposite symmetries

    We now classify the opposite symmetries. There are two kinds.

    Theorem
    An opposite symmetry of R2 is either reflection in any line or is a glide reflection.

    Proof
    Note that the reflection may be in a line not necessarily through 0.
    A glide reflection (or glide) is a reflection in a line followed by a translation in a direction parallel to that line.

    Since any element in O(2) - SO(2) is a reflection in a line through 0, the result follows from:

    Lemma
    Let Rb be reflection in a line through 0 containing the vector b. Then Ta comp Rb is a reflection if a and b are perpendicular and a glide-reflection otherwise.

    Proof
    From the diagram, if a and b are perpendicular, then Ta comp Rb is reflection in the line l parallel to b through the point a/2.


    If a and b are not perpendicular, then points of l are mapped to other points of l and so f acts on l by translation. Points on one side of l are mapped to the other side and so we have a glide along l.





Remarks
  1. Note that glide reflections are really the most general form of an opposite symmetry. A reflection could be considered as a glide where the translation happens to be trivial.

  2. Glide reflections have no fixed points and so we get the following summary about isometries of R2.

    DirectOpposite
    Fixed pointRotationReflection
    No fixed pointTranslationGlide

  3. Products of direct symmetries are direct. It is however, possible for the product of rotations to be a translation.

  4. Products of reflections are rotations if the "mirror lines" meet and translations if they are parallel.

  5. Any isometry of R2 can be written as a product of at most three reflections in lines.


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JOC February 2003