How can you prove that any two skew lines define a hyperbolic paraboloid?

Question

Obviously, the set of points equidistant to two given skew lines defines a quadric. Is it always a hyperbolic paraboloid?

Answer 1

Two skew lines are characterized by their distance and angle. WLOG (to a similarity transform), let us choose

$$Y=0,Z=1$$ and $$aX+bY=0,Z=0$$ with $a^2+b^2=1$.

The equation of the surface is

$$y^2+(z-1)^2=(ax+by)^2+z^2$$

or

$$2z=a^2(y^2-x^2-2bxy)+1\\=a^2\left(y-\left(b+\sqrt{b^2+1}\right)x\right)\left(y-\left(b-\sqrt{b^2+1}\right)x\right)+1.$$

This is clearly a paraboloid, as $z$ is a quadratic function of $x,y$, and the $z$ cross-sections are hyperbolas.