Quadratic polynomial/R/3 variables/Pure form/All variables/List/Example

We make a list of real quadratic polynomials in the three variables ${}X,Y$ and ${}Z$, together with their corresponding vanishing sets, where we restrict the coefficients to ${}0,1,-1$. Moreover, we consider only such polynomials where all variables occur and the vanishing set is not empty.

• ${}Y^{2}+X^{2}-Z\,$ ${}\,$ ${}\,$ the vanishing set is a paraboloid.
• ${}Y^{2}-X^{2}-Z\,$ ${}\,$ ${}\,$ the vanishing set is a saddle surface.
• ${}X^{2}+Y^{2}+Z^{2}$ ${}\,$ ${}\,$ the only solution is the point ${}(0,0,0)$, the vanishing set is just one point.
• ${}X^{2}+Y^{2}+Z^{2}-1$ ${}\,$ ${}\,$ the vanishing set is a sphere, that is, the surface of a ball.
• ${}X^{2}+Y^{2}-Z^{2}$ ${}\,$ ${}\,$ the vanishing set is the solution set of the equation
${}Z^{2}=X^{2}+Y^{2}$. This is a (double)-cone.
• ${}X^{2}+Y^{2}-Z^{2}-1$ ${}\,$ ${}\,$ the vanishing set is a one-sheeted hyperboloid.
• ${}X^{2}+Y^{2}-Z^{2}+1$ ${}\,$ ${}\,$ the vanishing set is a two-sheeted hyperboloid.