# Similarity in the plane: similarity

### 13.15.6 Similarity in the plane: similarity

See Section 14.14.6 for similarities in space.

The similarity command creates a command to rotate and scale about a given point.

- similarity takes three mandatory arguments and one optional argument:
  - P, a point (the center of the rotation).
  - r, a real number (the scaling ratio).
  - θ, a real number (the angle of rotation).
  - Optionally, G, a geometric object.
- similarity(P,r,θ) returns a new command which rotates about P through an angle of θ and scales about P by a factor of r.
- similarity(P,r,θ,G) returns and draws the transformation of G.

#### Examples.

- Input:

  s:= similarity(i, 2, -pi/2)
  s(1+i)

  Output:
  then:
  s(circle(1+i,1))
  Output:
- Input:
  similarity(i, 2, -pi/2, 1 + i)
  Output:
- Input:
  similarity(i, 2, -pi/2, circle(1+i,1))
  Output:

Note that for a point P and real numbers r and θ, the command similarity(P,r,θ) is the same as homothety(P,k*exp(i*a)).