Dilative Rotations or Spiral Similarities are particular cases of transformations that we call *similarities*.

Similarities are transformations that change each figure into a similar figure, preserving angles, "though it may alter distances.
However, all distances are increased (or decreased) in the same ratio, called the ratio of magnification" (Coxeter and Greitzer, p.94)

In applets and images we are using one classical illustration from Alice's Adventures in Wonderland, drawn by John Tenniel.

If similarities preserve distances they are *isometries* or *congruent transformations*. Isometries transform each figure
into a congruent figure. Similarities include, as special cases, isometries.

The simplest isometry is a translation.

A rotation is a similarity that is also an isometry:

"The simplest kind of similarity is a *dilatation* [also called *dilation*], which transform each line into a parallel line.
Any dilatation that is not merely a translation is called a *central dilatation*, because all the lines joining corresponding
points of the figure and its image are concurrent." (Coxeter and Greitzer, p.94)

Half-turns are isometries that can be seen as rotations or central dilatations:

Central dilatations have an invariant point and it is true that any dilatation that is not a translation has an unique invariant point
(see Coxeter, p. 69).

The combination of a central dilatation and a rotation about the same center is called a *dilatative rotation* or
*spiral similarity*.

In the next applet you can see a dilative rotation as a continuous transformation: first, rotation, then, dilatation.

REFERENCES

Coxeter H. S. M. - Introduction to Geometry (John Whiley and Sons, Second Edition, 1969)

Coxeter, H. S. M. and Greitzer, S. L. - Geometry Revisited (Mathematical Association of America, 1967)