hyperbola curve formula in 3 dimensions 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?
 A: 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.
A: 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 $$\left\{(x',y',z)\left|\frac {z^2}{d^2}-\frac {x^2+y^2}{e^2}=1,y=-\frac ba x,x'=x+,y'=y+\frac ab\frac c{\sqrt {1+(\frac ab)^2}}\right.\right\}$$
A: Write your hyperbola on z=0 plane. Then apply shifting the origin and rotating around all three axis. You should multiply your (x,y,z) vector by a 3x3 matrix for rotations in one step.
