Circle inversion

Parametric equations

Take the circle centered at the origin with radius $R$ as the circle of inversion. A point $P$ other than the origin goes to $P^{\prime }$ on the ray $OP$ so that the two distances multiply to $R^2$. The origin has no image. Inverting twice returns the original point. A point on the circle of inversion stays where it is.

The image of a line or a circle is still a line or a circle. A line that misses the center becomes a circle through the center. A line through the center stays that same line. A circle through the center becomes a line that misses the center. A circle that misses the center stays a circle. When the two points that determine the line coincide, that line is undefined.

Interaction

• Drag the two points on the gold line. The white image switches between a circle through the center and the same line.
• Drag the center of the gold circle, or the point on its circumference, to change the radius, and see whether the white image is a circle or a line.
• Inversion radius R: move the slider to change $R$; both images change together. The paler white circle is the circle of inversion
• Reset: put the line, the circle, and $R$ back at the start

What to notice

• A point on the circle of inversion stays where it is, so if the original circle meets the circle of inversion, those intersection points lie on the image too.
• When the line misses the center, the image is a circle through the center. When the line passes through the center, the image is still that line.
• When the circle passes through the center, the image is a line. When the circle misses the center, the image is still a circle.

Related

• Radians and arc length
• A triangle on the disk

Further reading

• Inversive geometry (Wikipedia)