Let $x,y,z$ be real numbers and $$F(x,y,z)=ax^3+bx^3+cy^3+dx^2y+ex^2z+fxy^2+gxz^2+hy^2z+iyz^2+jxyz+k$$ $(a,b,c,\cdots$ are nonzero integers)
If I write $z=rx+sy$ where $r,s$ are real numbers and substitute $z$ in $F$, I will obtain a bivariate polynomial of degree 3 with real coefficients. I am curious to know the geometrical implication of such variable change? Is this a projection of the curve F on a 2-dimensional plane? Please let me know. Thanks.
It may be helpful to first consider a lower dimensional example. Suppose $\,F(x,y)\,$ is a real valued function of two real variables. Then the equation $\,z=F(x,y)\,$ is a surface in three space. If we make a substitution such as $\,y=rx+s,\,$ now $\,z=F(x,y)\,$ is a curve in three space which is a slice of the original surface. That is, we find the intersection of the surface with a plane whose result is a curve embedded in three space.
In your example of $\,F(x,y,z)\,$ and substitution $\, z=rx+sy,\,$ a very similar process happens. The equation $\,w=F(x,y,z)\,$ is a hypersurface in four space. After the substitution we get the intersection of the hypersurface with a hyperplane. The result is a two dimensional surface embedded in four space.