Enter point coordinates (x, y), inversion center (a, b), and a positive radius r. Use the same dimensionless coordinate scale for every input. The calculator returns the inverse point (x′, y′) for inversive-geometry constructions.
The original point and its inverse lie on the same ray from the center, and their distances from the center multiply to r². This definition is given in Fractals and Circle Inversions.
Consequently, points outside the inversion circle move inside it, noncentral points inside move outside, and points on the circle stay fixed. Applying the same inversion twice recovers the original point, apart from numerical rounding. The scale factor depends on the point’s distance from the center, so inversion is nonlinear.
| Point (x, y) | Center (a, b) | Radius r | d² | Inverse point (x′, y′) |
|---|---|---|---|---|
| (4, 0) | (0, 0) | 2 | 16 | (1, 0) |
| (1, 0) | (0, 0) | 2 | 1 | (4, 0) |
| (0, 2) | (0, 0) | 2 | 4 | (0, 2) |
| (4, 6) | (1, 2) | 5 | 25 | (4, 6) |
| (1, 2) | (1, 2) | 5 | 0 | Unavailable |
The center itself has no finite inverse: when x = a and y = b, d² is zero and both inverse coordinates are unavailable. An unavailable result is not zero.
This calculator accepts radii of at least 10⁻¹² coordinate units; this is an input limit, not a mathematical restriction on circle inversion. Rescale your coordinates if a smaller radius is needed. All coordinates and the radius must use the same scale.
Points very close to the center can produce very large inverse coordinates. Extreme inputs can exceed the numerical range, making dependent results unavailable. Results are displayed to eight decimal places, so very small nonzero values may appear as zero.