Hyperboloid Equation canonical hyperbola equation:
$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 $$
or
$$ d_{kl}=\sqrt{(x-x_{k})^2+(y-y_{k})^2}-\sqrt{(x-x_{l})^2+(y-y_{l})^2} $$
where the focus of hyperbola are $(x_{k},y_{k})$ and $(x_{l},y_{l})$
canonical hyperboloid equation is:
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1 $$
can I use
$$ d_{kl}=\sqrt{(x-x_{k})^2+(y-y_{k})^2+(z-z_{k})^2}-\sqrt{(x-x_{l})^2+(y-y_{l})^2+(z-z_{l})^2} $$
as hyperboloid equation? If can, which one is it: one sheet or two sheet?
Or, is it another shape equation?
Thank you.
 A: Note that the equation
$$ d_{kl} = \left\lvert\sqrt{(x-x_k)^2+(y-y_k)^2}
             - \sqrt{(x-x_l)^2+(y-y_l)^2}\right\rvert $$
is simply
$d_{kl} = d(\mathbf x, F_1) - d(\mathbf x, F_2)$
where $F_1$ and $F_2$ are the foci of the hyperbola,
$\mathbf x = (x,y)$ is any point on the hyperbola,
and $d(P,Q)$ is the function giving the distance between points $P$ and $Q.$
(Note the absolute value on the right-hand side, which enables the equation to product both branches of the hyperbola.)
The equation
$$ d_{kl} = \left\lvert\sqrt{(x-x_k)^2+(y-y_k)^2+(z-z_k)^2}
                         - \sqrt{(x-x_l)^2+(y-y_l)^2+(z-z_l)^2}\right\rvert \tag1 $$
also is simply $d_{kl} = d(\mathbf x, F_1) - d(\mathbf x, F_2)$,
except that now $\mathbf x = (x,y,z)$ is a point in three dimensions instead of two.
This gives the equation of a hyperboloid produced by taking a hyperbola with foci $F_1$ and $F_2$ in some plane through the two foci and rotating that hyperbola about its transverse axis (the axis through the foci).
The result is a hyperboloid of two sheets contained within a double cone.
The equation
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1, \tag2$$
on the other hand, is a hyperboloid of one sheet.
If $\lvert a\rvert = \lvert b\rvert$ the hyperboloid can be produced by rotating a hyperbola around its conjugate axis, which we can see by writing its equation in cylindrical coordinates (with $z$ as the cylindrical axis) as
$$\frac{r^2}{a^2}-\frac{z^2}{c^2}=1.$$
If $\lvert a\rvert \neq \lvert b\rvert,$ however,
the hyperboloid is not a surface of revolution at all,
although it still has only one sheet.
So I don't think you'll have much luck finding the equivalence between Equations $(1)$ and $(2).$ But you might try
$$\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{b^2}=1,$$
or in cylindrical coordinates (with the $x$ axis as the cylindrical axis),
$$\frac{x^2}{a^2}-\frac{r^2}{b^2}=1.$$
For a more general hyperboloid of two sheets (not equivalent to a surface of revolution), try
$$\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=1.$$
