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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.

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1 Answer

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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$$\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=1.$$

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