What are the three skew lines to cross a ruling of a one-sheeted hyperboloid

Question

As the book Geometry and the Imagination said, any one-sheeted hyperboloid is a ruled surface and a ruling (the line on the surface) crosses three skew lines (any two of them are not co-planar to each other). I am wondering what are the three skew lines to define a one-sheeted hyperboloid. For example, given the equation of a one-sheeted hyperboloic: $x^2+y^2−z^2 = 1$. What are the three skew lines that define the surface?

Answer 1

Some elements of reflection.

First of all, this answer of mine some years ago can be of some help. It deals with the determination of a ruled quadric knowing three of its 3 lines.

Preliminary material: let us recall how 2 families of skew lines generating a given hyperboloid with one sheet can be obtained.

Dealing with the example you give, with equation written under the form

$$y^2-z^2=1-x^2 \ \iff $$

$$\text{Surface} \ H \ : (y-z)(y+z)=(1-x)(1+x),\tag{1}$$

the two families of lines can be given the following equations:

$$\text{Lines} \ L_a : \ \begin{cases}y-z&=&a(1-x)\\y+z&=&\dfrac{1}{a}(1+x)\end{cases}\tag{2}$$

$$\text{Lines} \ L'_b : \ \begin{cases}y-z&=&b(1+x)\\y+z&=&\dfrac{1}{b}(1-x)\end{cases}\tag{3}$$

for any non-zero real number $a$ or $b$.

Remark: please note that, by multiplication of its 2 equations, (2) $\implies$ (1) ; implication of equations meaning inclusion of corresponding geometric entities ($\forall a, L_a \subset H$) as desired. For the same reason, $\forall b, L'_b \subset H$.

$H$ can be defined by selecting any 3 lines of a family, say $L_{a_1},L_{a_2},L_{a_3}$ or 2 in a family and the third in the other one.

But taking only two of these lines, for example $L_1$ and $L'_1$:

$$L_1 : \ \begin{cases}y-z&=&1-x\\y+z&=&1+x\end{cases} \ \text{and} \ L'_1 : \ \begin{cases}y-z&=&1+x\\y+z&=&1-x\end{cases}\tag{4}$$

whose intersection point is $(x,y,z)=(0,1,0)$ is not enough to define in a unique way a hyperboloid with one sheet (think for example, among other solutions, to the hyperboloid with one sheet symmetrical to $H$ with respect to the plane defined by $L_1 \cup L'_1$). Besides, there are other candidates: there exists Hyperbolic Paraboloids that can pass through these 2 lines.