Enter four 3D points using the same coordinate unit. The results give the center and coefficients in x² + y² + z² + Dx + Ey + Fz + G = 0. Coordinates are bounded to ±10¹²; outputs use the units implied by your coordinates.
The calculation subtracts the first point from the others, solves the resulting three equations using cross products and their determinant, and translates the center back. Shared intermediate results are reused. It does not depend on an arbitrarily chosen nonzero pivot.
Four noncoplanar points determine one sphere. Coplanar or repeated points do not determine a unique sphere and return unavailable when the determinant is zero. Nearly coplanar points, or points extremely close together relative to their coordinate magnitudes, can produce inaccurate finite results due to floating-point precision. A finite result is not a guarantee of numerical accuracy; verify sensitive inputs with higher-precision methods.
The default points lie on the unit sphere: center (0, 0, 0), D = E = F = 0 and G = −1.