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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.
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.

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