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I am writing a piece of software involving conic sections. I need to write a method that returns a different number (0, 1, 2...) for each type of conic section.

I have determined the following conic sections:

  • 0 - Circle
  • 1 - Ellipse
  • 2 - Parabola
  • 3 - Hyperbola
  • 4 - Rectangular Hyperbola

and these degenerate conics:

  • 5 - Line
  • 6 - Point
  • 7 - Intersecting Lines
  • 8 - Parallel Lines
  • 9 - No Graph

My Question: How to determine which of these ten types a given conic section is using only the values of A, B, C, D, E, and F in the equation below:

$$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $$

Thanks in advance for any help, Zach Hilman

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  • $\begingroup$ I am removing the "algebraic-geometry" tag here. "algebraic-geometry" is the study of zeros of multivariate polynomials, and is different than analytic geometry, which this problem does concern. $\endgroup$ – Mark Fischler Oct 28 '16 at 20:58
  • $\begingroup$ Sorry, I was a little confused about that one. $\endgroup$ – Zach Hilman Oct 28 '16 at 20:58
  • $\begingroup$ I mean, this is fundamentally asking about classifying affine varieties given by the general form above. I see no reason why it couldn't be considered part of algebraic geometry (although another tag may indeed be more appropriate). $\endgroup$ – Michael Lee Oct 28 '16 at 21:02
  • $\begingroup$ Have a look at (math.stackexchange.com/q/1539317) $\endgroup$ – Jean Marie Oct 28 '16 at 21:19
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    $\begingroup$ This en.m.wikipedia.org/wiki/Conic_section#Discriminant has a formulation in terms of only the coefficients and the determinant $\delta=B^2-4AC$ $\endgroup$ – Sophie Dec 14 '16 at 3:53

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