Stereographic Projection Calculator

Calculate two independent conversions: sphere coordinates (x, y, z) to plane coordinates, and plane coordinates (u, v) to sphere coordinates. All coordinates are dimensionless. Forward inputs must satisfy x² + y² + z² = 1. This calculator does not check sphere membership or normalize your inputs.

Use the unit sphere centered at the origin, the north pole N = (0, 0, 1), and the equatorial plane z = 0. The line through N and a sphere point meets that plane at (u, v, 0). The inverse uses the second intersection of the line through N and (u, v, 0) with the sphere. This construction is described in the University of Washington stereographic projection reference.

For forward projection, enter x, y and z for a point you have verified lies on the unit sphere. Individual coordinate bounds do not establish sphere membership. For inverse projection, enter u and v independently. Forward outputs are not automatically copied into the inverse inputs; copy them manually to check a round trip. Leave the fixed formula constant at 1.

At the north pole (0, 0, 1), the forward denominator is zero and both forward results are unavailable. They are not zero or finite plane coordinates. The independent inverse conversion can still return results. Near the north pole, a small change in z can greatly change the forward result. Very large plane inputs can exceed numeric range during squaring; unavailable intermediate values propagate to dependent outputs. Display rounding can also affect round-trip comparisons.

This tool maps sphere and plane coordinates. It does not perform latitude/longitude conversion, automatic sphere-membership classification, orthogonal vector projection, affine transformation or planar circle inversion.

Reference

1. University of Washington — Stereographic Projection: geometric construction and inverse mapping.