Circle Inversion Fractals

Inversion Formula for a Circle

To invert the circle S with center (c, d) and radius s across the circle with center (a, b) and radius r,

first find the points p and q on S lying on the line determined by (a, b) and (c, d).

The unit vector from (a, b) to (c, d) is

So to find the point q, start at the center (c, d) of the circle S and move the distance s in the direction from (a, b) to (c, d).

That is, start at (c, d) and add s times the unit vector from (a, b) to (c, d):

To find the point p, start at (c, d) and move the distance s in the direction opposite that from (a, b) to (c, d). That is,

Using the formula for the inverse of a point, we see that inverting p and q in C gives

and

Then, the center of the inverted circle is the midpoint of p' and q'. That is, the center is

The diameter of the inverted circle is the distance between p' and q', so the radius is

Note the center of the inverted circle is NOT the inverse of the center of S. Click the animation to stop.