An ellipsoid in 3 D is described by 9 independent parameters: 3 for the coordinates of its centre + 6 independent components of a symmetric 3 x 3 matrix. What is the minimum number of points on the surface of an ellipsoid that define it uniquely?
2026-03-29 14:07:22.1774793242
minimum number of points on the surface of a 3D ellipsoid to define it uniquely
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1020784
The minimum IS 9.
The equation for a quadric surface, of which the ellipsoid is a special case, in terms of nine of its points
$[(x_1\mid y_1\mid z_1)\dots (x_9\mid y_9\mid z_9)]$
is $\begin{vmatrix} x^2&y^2&z^2&y·z&z·x&x·y&x&y&z&1\\ x_1^2&y_1^2&z_1^2&y_1·z_1&z_1·x_1&x_1·y_1&x_1&y_1&z_1&1\\ x_2^2&y_2^2&z_2^2&y_2·z_2&z_2·x_2&x_2·y_2&x_2&y_2&z_2&1\\ x_3^2&y_3^2&z_3^2&y_3·z_3&z_3·x_3&x_3·y_3&x_3&y_3&z_3&1\\ x_4^2&y_4^2&z_4^2&y_4·z_4&z_4·x_4&x_4·y_4&x_4&y_4&z_4&1\\ x_5^2&y_5^2&z_5^2&y_5·z_5&z_5·x_5&x_5·y_5&x_5&y_5&z_5&1\\ x_6^2&y_6^2&z_6^2&y_6·z_6&z_6·x_6&x_6·y_6&x_6&y_6&z_6&1\\ x_7^2&y_7^2&z_7^2&y_7·z_7&z_7·x_7&x_7·y_7&x_7&y_7&z_7&1\\ x_8^2&y_8^2&z_8^2&y_8·z_8&z_8·x_8&x_8·y_8&x_8&y_8&z_8&1\\ x_9^2&y_9^2&z_9^2&y_9·z_9&z_9·x_9&x_9·y_9&x_9&y_9&z_9&1\\ \end{vmatrix}=0$.
This gives a quadratic equation in $x,y$ and $z$. Whether the locus of this equation yields an ellipsoid or one of the other quadric surfaces depends on the values of certain functions of the coëfficients of the equation, known as invariants.