Sphere Equation Calculator Given 4 Points

Enter four points in 3D space (decimal values allowed). The calculator finds the unique sphere passing through them using Cramer's Rule as long as the four points are not coplanar.

Sphere Calculator

Enter four non coplanar points in 3D space (decimal values allowed).

🔹 Point 1 (x₁, y₁, z₁)
🔸 Point 2 (x₂, y₂, z₂)
🔹 Point 3 (x₃, y₃, z₃)
🔸 Point 4 (x₄, y₄, z₄)

Results

Method: Determinants (Cramer's Rule)

The general sphere equation is: \[ x^2 + y^2 + z^2 + Ax + By + Cz + D = 0 \] For each point \((x_i,y_i,z_i)\) we have: \[ A x_i + B y_i + C z_i + D = -(x_i^2+y_i^2+z_i^2) \]

Using Cramer's rule, we solve the 4×4 linear system for A, B, C, D.

\[ \begin{vmatrix} x_1 & y_1 & z_1 & 1 \\ x_2 & y_2 & z_2 & 1 \\ x_3 & y_3 & z_3 & 1 \\ x_4 & y_4 & z_4 & 1 \end{vmatrix} \cdot \begin{pmatrix} A \\ B \\ C \\ D \end{pmatrix} = \begin{pmatrix} -S_1 \\ -S_2 \\ -S_3 \\ -S_4 \end{pmatrix} \] where \(S_i = x_i^2+y_i^2+z_i^2\).