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The general form of quadric surfaces is

$$Ax^2 + By^2 + Cz^2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0$$

I want to classify all of the possibilities including degenerate cases with the help of general form . I searched on the internet but got really confused ! In the case of conic sections this website gives a complete procedure for classifying : https://brilliant.org/wiki/conics-discriminant/

S.H.W
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  • What do you mean by "all of the possibilities"? Are you looking for a geometric classification for such surfaces over the real numbers? – Servaes Feb 15 '19 at 11:07
  • Yes , that is my goal . – S.H.W Feb 15 '19 at 11:09
  • http://mathworld.wolfram.com/QuadraticSurface.html – Intelligenti pauca Feb 15 '19 at 11:35
  • @Aretino Thanks but it doesn't give the calculations . – S.H.W Feb 15 '19 at 11:43
  • You'll find the details in any book on the subject. See also here: http://math.ucdenver.edu/~tvis/Teaching/4220spring09/Assignments/Brook_Le.pdf – Intelligenti pauca Feb 15 '19 at 11:44
  • @Aretino Thanks again . According to the Stewart Calculus the general form by using the translation and rotation can be brought into $Ax^2 + By^2 + Cz^2 + J = 0$ or $Ax^2 + By^2 + Iz = 0$ . Do know how it is done ? – S.H.W Feb 15 '19 at 11:53
  • It's a coordinate change. You must diagonalise the associated matrix: https://en.wikipedia.org/wiki/Diagonalizable_matrix – Intelligenti pauca Feb 15 '19 at 11:54
  • @Aretino What's the associated matrix ? – S.H.W Feb 15 '19 at 11:57
  • https://cs.nyu.edu/yap/bks/egc/09/21Surfaces.pdf – Intelligenti pauca Feb 15 '19 at 14:31
  • It’s much easier to diagonalize the associated matrix using the method described here, which is at heart a mechanical way of completing squares and doesn’t require computing eigenvalues. The latter can get quite messy. You really only need to know the spectrum of the quadric—the number of positive, negative and zero entries in the diagonalized form—in order to classify it. That’s going to be the same whether you diagonalize via eigenvalues or some other way. – amd Feb 15 '19 at 20:31
  • @amd Why is that true ? I mean why the number of positive, negative and zero entries in the diagonalized form classify the quadric surfaces ? – S.H.W Feb 15 '19 at 22:12
  • That’s the essence of the principal axis theorem in combination with Sylvester’s law of inertia. See https://en.wikipedia.org/wiki/Quadric#Euclidean_space, for instance. – amd Feb 15 '19 at 22:15
  • @amd Very interesting , Is there a reference that completely explains this idea and includes a table for different surfaces ? – S.H.W Feb 15 '19 at 22:26
  • That Wikipedia article I lined has a complete table. – amd Feb 15 '19 at 22:58

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