An equation of a sphere calculator, given four points on the sphere, is presented.
The equation of a sphere whose center is at the point \( (h,k,l) \) and has radius \( r \) is given by
\[ (x-h)^2 + (y - k)^2 + (z - l)^2 = r^2 \]
Expanding the above equation and grouping terms, we obtain
\[ x^2 + y^2 + z^2 - 2xh - 2 y k - 2 z l + h^2 + k^2 + l^2 = r^2 \]
or
\[ x^2 + y^2 + z^2 = 2 x h + 2 y k + 2 z l + r^2 - ( h^2 + k^2 + l^2) \]
Let , \( A = 2 h \) , \( B = 2 k \) , \( C = 2 l \) and \( D = r^2 - (h^2 + k^2 + l^2) \) and rewrite the above as follows
\[ A x + B y + C z + D = x^2 + y^2 + z^2 \]
For the 4 points with coordinates \( (x_1 , y_1 , z_1 ) \), \( (x_2 , y_2 , z_2 ) \), \( (x_3 , y_3 , z_3 ) \) and \( (x_4 , y_4 , z_4 ) \) to be on the same sphere, the following system of four equations with the unknowns \( A, B , C \) and \( D \) must have a solution:
\[ \begin{align*}
A x_1 + B y_1 + C z_1 + D & = x_1^2 + y_1^2 + z_1^2 \\
A x_2 + B y_2 + C z_2 + D & = x_2^2 + y_2^2 + z_2^2 \\
A x_3 + B y_3 + C z_3 + D & = x_3^2 + y_3^2 + z_3^2 \\
A x_4 + B y_4 + C z_4 + D & = x_4^2 + y_4^2 + z_4^2
\end{align*} \]
Using Cramer's rule, we solve for \( A, B , C \) and \( D \) as follows:
\[ M = \begin{vmatrix}
&x_1&&y_1&&z_1&&1&\\
\\
&x_2&&y_2&&z_2&&1&\\
\\
&x_3&&y_3&&z_3&&1&\\
\\
&x_4&&y_4&&z_4&&1&\\
\end{vmatrix} \]
and
\[ A = \dfrac{\begin{vmatrix}
&x_1^2 + y_1^2 + z_1^2&&y_1&&z_1&&1&\\
\\
&x_2^2 + y_2^2 + z_2^2 &&y_2&&z_2&&1&\\
\\
&x_3^2 + y_3^2 + z_3^2&&y_3&&z_3&&1&\\
\\
&x_4^2 + y_4^2 + z_4^2&&y_4&&z_4&&1&\\
\end{vmatrix}}{M} \]
\[ B = \dfrac{\begin{vmatrix}
&x_1&&x_1^2 + y_1^2 + z_1^2&&z_1&&1&\\
\\
&x_2&&x_2^2 + y_2^2 + z_2^2&&z_2&&1&\\
\\
&x_3&&x_3^2 + y_3^2 + z_3^2&&z_3&&1&\\
\\
&x_4&&x_4^2 + y_4^2 + z_4^2&&z_4&&1&\\
\end{vmatrix}}{M} \]
\[ C = \dfrac{\begin{vmatrix}
&x_1&&y_1&&x_1^2 + y_1^2 + z_1^2 &&1&\\
\\
&x_2&&y_2&&x_2^2 + y_2^2 + z_2^2&&1&\\
\\
&x_3&&y_3&&x_3^2 + y_3^2 + z_3^2&&1&\\
\\
&x_4&&y_4&&x_4^2 + y_4^2 + z_4^2&&1&\\
\end{vmatrix}}{M} \]
\[ D = \dfrac{\begin{vmatrix}
&x_1&&y_1&&z_1&&x_1^2 + y_1^2 + z_1^2&\\
\\
&x_2&&y_2&&z_2&&x_2^2 + y_2^2 + z_2^2&\\
\\
&x_3&&y_3&&z_3&&x_3^2 + y_3^2 + z_3^2&\\
\\
&x_4&&y_4&&z_4&&x_4^2 + y_4^2 + z_4^2&\\
\end{vmatrix}}{M} \]
Enter the coordinates of four 3D points:
