Conic sections · scipython.com

P6.5.6Conic sections

The implicit formula for a conic section may be written as the second-degree polynomial,

\[Q = Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,\]

or in matrix form using the homogeneous coordinate vector,:

\mathbf{x} = \begin{pmatrix}x \\ y \\ 1\end{pmatrix},

as $\mathbf{x}^T\mathbf{Q}\mathbf{x} = 0$ , where

\mathbf{Q} = \begin{pmatrix} A & B/2 & D/2 \\ B/2 & C & E/2\\ D/2 & E/2 & F \end{pmatrix}.

Conic sections may be classified according to the following properties of $\mathbf{Q}$ , where the submatrix $\mathbf{Q}_{33}$ is

\mathbf{Q}_{33} = \begin{pmatrix}A & B/2\\B/2 & C\end{pmatrix}.

If $\mathrm{det} \mathbf{Q} = 0$, the conic is degenerate in one of the following forms:
- if $\mathrm{det} \mathbf{Q}_{33} < 0$ , the equation represents two intersecting lines,
- if $\mathrm{det} \mathbf{Q}_{33} = 0$ , the equation represents two parallel lines,
- if $\mathrm{det} \mathbf{Q}_{33} > 0$ , the equation represents a single point.
If $\mathrm{det} \mathbf{Q} \ne 0$:
- if $\mathrm{det} \mathbf{Q_{33}} < 0$ the conic is a hyperbola,
- if $\mathrm{det} \mathbf{Q_{33}} = 0$ the conic is a parabola,
- if $\mathrm{det} \mathbf{Q_{33}} > 0$, the conic is an ellipse:
  - if $$A=C$$ and $$B=0$$ , the ellipse is a circle.

Write a program to classify the conic section represented by the 6 coefficients $$A, B, C, D, E$$ and $$F$$ .

Some test-cases (coefficients not given are zero):

Hyperbola: $$B = 1, F = -9$$ .
Parabola: $A = \frac{1}{2}, D = 2, E = -\frac{1}{2}$ .
Circle: $A = \frac{1}{2}, C = \frac{1}{2}, D = -2, E = -3, F = 2$ .
Ellipse: $$A = 9, C = 4, F = -36$$ .
Two parallel lines: $$A = 1, F = -1$$ .
A single point: $$A = 1, C = 1$$ .

Show solution

The code below classifies conic sections, given the coefficients $$A, B, C, D, E$$ and $$F$$ on the command line.

conicsections

import sys
import numpy as np


def classify_conic(A, B, C, D, E, F):
    Q = np.array(((A, B / 2, D / 2), (B / 2, C, E / 2), (D / 2, E / 2, F)))
    Q = np.matrix(Q)
    Q33 = Q[:2, :2]

    detQ = np.linalg.det(Q)
    detQ33 = np.linalg.det(Q33)
    if detQ == 0:
        if detQ33 < 0:
            return "two intersecting lines"
        if detQ33 == 0:
            return "two parallel lines"
        return "a single point"

    if detQ33 < 0:
        return "a hyperbola"
    if detQ33 == 0:
        return "a parabola"
    if A == C and B == 0:
        return "a circle"
    return "an ellipse"


if __name__ == "__main__":
    A, B, C, D, E, F = [float(arg) for arg in sys.argv[1:]]
    print(classify_conic(A, B, C, D, E, F))

For example,

$ python classify.py 9 0 4 0 0 -36
an ellipse