hyperbola curve formula in 3 dimensions

Question

Cartesian formula for 2d hyperbola curve is $x^2/a^2-y^2/b^2 = 1$.

What is the formula for a 3d hyperbola curve?

Answer 1

What is a 3D hyperbola curve? How about $\frac {x^2}{a^2}-\frac {y^2}{b^2}=1, z=0$. If you want a curve in 3D, you need two equations, just like a line. I suspect this is not what you want, so please explain.

Alternately, you might want $\frac {x^2}{a^2}-\frac {y^2+z^2}{b^2}=1$, which is a surface that rotates the 2D curve around the $x$ axis.

Added in response to comment: Given your three points, you can solve for the $a,b,c$ that are the equation of the plane. Be careful-we have been using $a,b$ as parameters of the hyperbola and now you are reusing them. The $m$ in my previous comment corresponds to $\frac ba$ here, but my plane went through the $z$ axis while if $c \ne 0$ it will miss the $z$ axis by that much.
A hyperbola with its axis parallel to $z$ that is in the plane $ax+by=0$ is $\frac {z^2}{d^2}-\frac {x^2+y^2}{e^2}=1,y=-\frac ba x$ To get it into the plane $ax+by=c$ we add $\frac c{\sqrt {1+(\frac ab)^2}}$ to $x$ and $\frac ab\frac c{\sqrt {1+(\frac ab)^2}}$ to $y$ giving $$\left\{(x',y',z)\left|\frac {z^2}{d^2}-\frac {x^2+y^2}{e^2}=1,y=-\frac ba x,x'=x+,y'=y+\frac ab\frac c{\sqrt {1+(\frac ab)^2}}\right.\right\}$$

Answer 2

What do you mean by a "3d hyperbola curve"?

If you really mean a curve, something like $$\begin{cases} \frac{x^2}{a^2}-\frac{y^2}{b^2} = 1 \\ z= 0 \end{cases}$$ is one example. (There are others.)

If you mean the "3d version" you get hyperboloids of one and two sheets respectively, but these are surfaces, not curves:

$x^2+y^2-z^2 = 1$

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

Notice that these surfaces are obtained by rotating a planar hyperbola around one of its two lines of symmetry.