- A necessary definition
$F(x,y,z)$ is called a homogeneous function of degree $n$ ($n$ is an integer) if $F(tx,ty,tz)=t^nF(x,y,z)$ holds for any nonzero real number t,and any possible x , y, z. In this case, $F(x,y,z)=0$ is said to be a homogeneous equation. For example, $x^2+y^2+z^2=F(x,y,z)$ is a homogeneous function of degree $2$.
- The question.
"In a rectangular coordinate system with the vertex of the conical surface as the origin,the conical surface can be represented by homogeneous equations of $x$, $y$, and $z$". This is a conclusion we learnt from our textbook on Analytic Geometry, which appears as a theorem. However, we have no idea why it's correct, whether this theory only holds in rectangular coordinate system...we can't find any proof of it in any literature,and the writer of the textbook didn't provide proof as well.
- Attempts already made
According to the rotational surface equation "$kz=±\sqrt{(x'^2+y'^2)}$" given in the rectangular coordinate system, it can be proved easily that this conclusion holds for the circular conical surface (revolution conical surface) in the rectangular coordinate system, but for other (general) conical surfaces I can’t think of a good proof method.
So if anyone understands it and knows why, please answer my question (better off if you can provide me with a rigorous thinking or proof:)) Thank you soooooo much!!