The reduced_quadric command finds the reduced equation of a quadric.
reduced_quadric(7*x^2+4*y^2+4*z^2+ 4*x*y-4*x*z-2*y*z-4*x+5*y+4*z-18) |
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The output is a list containing:
⎡ ⎢ ⎢ ⎣ |
| ,− |
| ,− |
| ⎤ ⎥ ⎥ ⎦ | . |
| . |
9 x2+3 y2+3 z2− |
| . |
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Hence the quadric is an ellipsoid 9x2+3y2+3z2+(−602)/27=0. After the change of origin to [11/27,(−26)/27,(−29)/54], the matrix of basis change is:
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Its parametric equation is:
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Note that if the quadric is degenerate and made of one or two plane(s), each plane is not given by its parametric equation but by the list of a point in the plane and a normal vector.
reduced_quadric(x^2-y^2+3*x+y+2) |
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