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Sunday, July 29, 2007

Conic sections

The circle, ellips, parabola and hyperbola are examples of conic sections:



A conic section is an affine variety like V(x^2 + y^2 - z^2, ax + by + cz). Where x^2 + y^2 - z^2 = 0 is the equation of a cone and ax + by + cz = 0 is the equation of a plane in three dimensional affine space. Depending on the values of a, b and c the affine variety takes the form of a circle, ellips, parabola or hyperbola.

Interesting, not? Let's experiment in Mathematica.
Show[
  ContourPlot3D[x^2 + y^2 - z^2, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}],
  ContourPlot3D[-2x + y - z, {x, -2, 2}, {y, -1, 1}, {z, -2, 2}]
  ]


This images almost asks to be rotated in 3D for further visual inspection. This is where the superb JavaView comes in but that is something for another time.
Posted by at 9:38:00 PM
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