![]() $$\det T(A\circ B)=\det(T(A)\circ T(B))=\det T(A)\cdot\det T(B).$$įrom this it's very easy to deduce an odd number of orientation-reversing maps compose to an orientation-reversing map, and now we are done once we check that glide reflections are orientation-reversing (straightforward) and vice versa (less obvious, requires classification of isometries). It's not hard to check that $T(A\circ B)=T(A)\circ T(B)$, therefore A less ad-hoc definition is as follows: we call $A$ orientation-preserving if $\det T(A)>0$ and orientation-reversing if $\det T(A)<0$. Let me denote the latter by $T(A)$ for an isometry $A$. Every isometry is an affine transformation, which means that it can be written as a composition of a translation and some invertible linear transformation. Now, let me just say that if you are willing to accept some linear algebra, then you can make all of the above less ad-hoc. Since the vector of translation and the axis of reflection are parallel, it does not matter which motion is done first in the glide reflection. The vector of translation v v and the axis of reflection m m must be parallel to each other. Therefore, a composition of 1981 glide reflections is still a glide reflection (or a pure reflection). A glide reflection is a combination of a translation and a reflection. It is easy to deduce from those facts that a composition of an odd number of orientation-reversing is orientation-reversing (and of an even number is orientation-preserving). A school wishes to create a track in the shape of a loop. Rotational symmetry is the quality a design has if it maintains all characteristics when it is rotated about a.
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