Spiarl similarity is a geometric transformation which is a combination of a homothety and a rotation with the same center. Both are thus particular cases of spiral similarity. For a rotation, the coefficient of homothety is 1. For a homothety, the angle of rotation is 0.
(In the applet below, various spiral similarities are controlled by a hollow blue point - the center of rotation, and a dial with a draggable point which determines the angle of rotation and the coefficient of homothety (relative to the circle shown.) 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. You'll see the applet in action after you Add a shape to apply transformations to.)
The following observations are noteworthy:
Spiral similarity preserves the orientation. For example, if a polygon is traversed clockwise, its image under a spiral similarity is likewise traversed clockwise.
Spiral similarity preserves angles.
Spiral similarity maps parallel lines onto parallel lines.
Except for the trivial case of rotation through a zero angle which is identical, spiral similarities have a single fixed point - the common center of the homothety and rotation.
In the most general case, successive spiral similarities result in a spiral similarity, but may also give a translation.
The product of spiral similarities is not in general commutative. Two spiral similarities with a common center commute as a matter of course.
Copyright © 1996-2018 Alexander Bogomolny