Projection of Cubic Curve

The general 2D cubic curve is:
ax3 + bx2y + cxy2 + dy3 + ex2 + fxy + gy2 + hx + iy + j = 0.

The real parameters are a, b, … i. If we multiply each parameter by the same non-zero real we get the same curve. That is the only freedom in the parameters.

For projections homogeneous coordinates are superior and our equation becomes:

ax3 + bx2y + cxy2 + dy3 + ex2z + fxyz + gy2z + hxz2 + iyz2 + jz3 = 0.

This defines a 3D set and our original curve is found in the plane where z=1. The parameters are preserved.

We may perform linear transformations on the 3D space and the result produces all of the projective transformations on our spacial plane z=1. We write the 3D transformation (x', y', z') = (x, y, z)A. The 3 by 3 real non-singular matrix A completely defines the projective transformation and conversely except that for any non-zero real q, qA and A define the same projection. We express A as
((Axx, Axy, Axz),
(Ayx, Ayy, Ayz),
(Azx, Azy, Azz)).

(x', y', z') =
((Axx x + Ayx y + Azx z), (Axy x + Ayy y + Azy z), (Axz x + Ayz y + Azz z)).