### Projection of Cubic Curve

The general 2D cubic curve is:

ax^{3} + bx^{2}y + cxy^{2} + dy^{3}
+ ex^{2} + fxy + gy^{2}
+ 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:

ax^{3} + bx^{2}y + cxy^{2} + dy^{3}
+ ex^{2}z + fxyz + gy^{2}z
+ hxz^{2} + iyz^{2} + jz^{3} = 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)).