What type of quadric is this?

Question

I am trying to do a StreamPlot 3D in mathematica with the following RegionPlot

$-x^2+2y^2+z\leq1.$

I was also looking at quadrics and this one does not seem to be on the list. Looking at the shape of the region is what made me curious about what type of quadric might be.

Thanks in advance.

Answer 1

Equality

$$-x^2+2y^2+z=1\tag{1}$$ can be written under the form:

$$\underbrace{1-z}_Z=\underbrace{(\sqrt{2}y-x)}_X\underbrace{(\sqrt{2}y+x)}_Y,$$

which means that, taking this affine transformation

$$\begin{cases}X&=&-x&+\sqrt{2}y&\\Y&=& \ \ \ x&+\sqrt{2}y&\\Z&=&&&-z+1\end{cases}$$

we obtain the standard equation $Z=XY$ of a Hyperbolic Paraboloid ("manta ray" shape)

Remark: if instead of an equal sign in (1), one has a $\leq$, it will give one of the two regions delimited by this surface, precisely the one that contains the origin.