Find coefficients of quadric equation

matricesquadricssystems of equations

I try to find the ten coefficients that describe the equation of a quadric in a 3D space by knowing ten points belonging to that quadric.
I have a system of equations with ten equations that are homogeneous. Because they are homogeneous, the only and trivial solution using matrix method is {0,0,0,0,0,0,0,0,0,0}. So how can I really solve my system ?

Thanks

EDIT the equation of a quadric in a 3D space is :

Ax² + By² + Cz² + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0, where the unknowns are A B C D E F G H I and J.

Knowing 10 points belonging to the quadric, we have a system of equation:

Ax² + By² + Cz² + Dxy + Eyz + Fxz + Gx + Hy + Iz + 1J = 0
Ax² + By² + Cz² + Dxy + Eyz + Fxz + Gx + Hy + Iz + 1J = 0
Ax² + By² + Cz² + Dxy + Eyz + Fxz + Gx + Hy + Iz + 1J = 0
Ax² + By² + Cz² + Dxy + Eyz + Fxz + Gx + Hy + Iz + 1J = 0
Ax² + By² + Cz² + Dxy + Eyz + Fxz + Gx + Hy + Iz + 1J = 0
Ax² + By² + Cz² + Dxy + Eyz + Fxz + Gx + Hy + Iz + 1J = 0
Ax² + By² + Cz² + Dxy + Eyz + Fxz + Gx + Hy + Iz + 1J = 0
Ax² + By² + Cz² + Dxy + Eyz + Fxz + Gx + Hy + Iz + 1J = 0
Ax² + By² + Cz² + Dxy + Eyz + Fxz + Gx + Hy + Iz + 1J = 0
Ax² + By² + Cz² + Dxy + Eyz + Fxz + Gx + Hy + Iz + 1J = 0

(Of course, x y and z of each line are those of one point).

With the matrix method, we want to solve aX=b => X=a-1b
with "a" being the matrix of the known coef of the system (x², y², y², xy, yz, xz, x, y, z, 1) for each point, X being the vector to be solved (A, B, C, D, E, F, I, J,) and "b" being the second member vector (0,0,0,0,0,0,0,0,0,0) because the equations of the system are homogeneous.

So the result is X=a-1b => X = a-1*0 => X=0 in all cases, and that's my issue.

Best Answer

You are not counting correctly. Because the system is homogeneous only the ratios of the coefficients are important, and with ten coefficients, you can define only nine ratios independently. That's what sets your degrees of freedom. You have to draw your quadric surface through nine points rather than ten.

Having properly selected nine points, you may use approaches by amd (this answer) to solve the system. Note that this method is superior to assuming a specific coefficient such as $J$ is nonzero because in practice, no individual coefficient is guaranteed nonzero for every quadric surface.

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