zahlenteufel

The Appolonius Cone: In this geometrical construction we can see the 4 types of Conic Section. This demo is made in Processing. (source code available here)

To do this, we need to intersect lines and planes in 3 dimensions. As in this case we can put the coordinates wherever we want, we can choose them to make our code simpler. So we choose \((0,0,0)\) as the cone’s vertex and \({y=1}\) as plane where the base is, and we also choose 1 as the base radius.
The planes we will use will be parallel to the z-axis (the depth one), so they have a trivial representation as a linear function in 2d (except the vertical plane, but we will deal with that later) : \(y=mx+b\)

class SpecialPlane {
    float m, b; // Plane: {(x,y,z) / m*x+b = y}

As all the lines that we will use to intersect pass through the origin (i.e. the cone’s vertex), we will only need to know which other point is coming through. In this case that will be \((\cos \theta,1,\sin \theta)\) where \(\theta\) sweeps al the angles between \(0\) and \(2\pi\). Finally with a little of Linear Algebra we deduce the formula:

PVector intersect(float lx, float ly) {
    float f = -b / (m*lx-1);
    return new PVector(f * lx, f, f * ly);
}