The order of reflections matters: two reflections do not commute. Successive reflections in two axes that meet in a point O is equivalent to a rotation around O through double the angle between them. Successive reflections in two parallel axes result in a translation in the direction perpendicular to the axes to twice the distance between them. Īll points on the axis of refleclation are fixed as are all the lines perpendicular to the axis.Reflection maps parallel lines onto parallel lines. Reflection is isometry: a reflection preserves distances.
Reflection changes the orientation: if a polygon is traversed clockwise, its image is traversed counterclockwise, and vice versa. The following observations are noteworthy:
TRANSLATION GEOMETRY REFLECTION INSTALL
If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet. This applet requires Sun's Java VM 2 which your browser may perceive as a popup. They rotate if dragged near the applet's border, or translate if dragged nearer their midpoint. In the applet, you can create polygons with a desired number of vertices, drag the vertices one at a time, or drag the polygon as a whole. On the other hand, if S' is known to be a mirror image of S, then any pair of points P and P' not fixed by the reflection (P ≠ P'), the axes of reflection is uniquely determined as the perpendicular bisector of PP'. This is exactly what has been done in the applet below. To determine S L(S) when S is a polygon, suffice it to reflect its vertices. Each point of a given shape S is reflected in L, and the collection of these reflections is the symmetric image of S: S L(S). The reflection transform S L applies to arbitrary shapes point-by-point. So that repeated reflection does noting: it does not move a point. If P' = S L(P), then P is the reflection in L of P': P = S L(P'). The line L is called the axis of symmetry or axis of reflection. P' is said to be a mirror or symmetric image of P in L. In other words, P' is located on the other side of L, but at the same distance from L as P. Reflection P' of P in L is the point such that PP' is perpendicular to L, and PM = MP', where M is the point of intersection of PP' and L.
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