Surface of a polynomial

Question

How can I find the surface represented by the polynomial $$x^2-y^2-2xz=0$$ any clue please?? I have tried to plot it using Maple

Answer 1

This is a quadric surface, degenerate because it has no constant term. The corresponding matrix $$ \pmatrix{1 & 0 & -1\cr 0 & -1 & 0\cr -1 & 0 & 0\cr} $$ has eigenvalues $-1$ and $(1\pm\sqrt{5})/2$, two negative and one positive. So it is an elliptic cone. A parametrization is $$ \eqalign{x &= (1+\sqrt{5}) (\cos(\theta)-1) r\cr y &= (1+\sqrt{5}) 5^{1/4} \sin(\theta) r\cr z &= \left((3 + \sqrt{5}) \cos(\theta) + 2\right) r\cr} $$ And here's a picture:

Of course this is only part of it: the cone extends infinitely in both directions.