General Background
Suppose I have one side of a two-sheet hyperboloid as a general three dimensional shape, where the symmetry axis is along, say, my x-axis ($\hat{\mathbf{x}}$) in my chosen coordinate basis. If I look at the intersection line in either the XY- or YZ-planes, the line can be given (in polar coordinates) by: $$ r + x_{o} = \frac{ L }{ 1 + e \cos{\theta} } \tag{1} $$ where $r$ is the distance from the focus to the path, $x_{o}$ is the displacement of the focus along $\hat{\mathbf{x}}$ relative to the origin, $L = a \ \left( 1 - e^{2} \right) = b^{2}/a$ is the semi-latus rectum, $e$ is the eccentricity, $\theta$ is the polar angle, $a$ is the semi-major axis and $b$ the semi-minor axis. Since $e > 1$, the line demarking the intersection of the hyperboloid with the plane of projection is clearly a hyperbola.
Assume that so long as the plane of projection contains the symmetry axis, Equation 1 will always describe the intersection of the hyperboloid with the plane of projection (i.e., just assuming azimuthal symmetry). In other words, Equation 1 will generate a two dimensional surface if rotated about the $\hat{\mathbf{x}}$-axis.
Problem
The Equation 1 works if the plane of projection contains the symmetry axis. However, suppose I want to find a general equation, of similar form to Equation 1, that defines the intersection line for arbitrary offsets of the plane of projection. To keep things simple, assume that the offset is only along either $\hat{\mathbf{y}}$ or $\hat{\mathbf{z}}$, not both simultaneously (i.e., not looking for a completely general form allowing for arbitrary offsets in any direction, just one direction at a time).
Questions
- Is there a closed form describing the intersection line for arbitrary offsets along only the $\hat{\mathbf{y}}$- or $\hat{\mathbf{z}}$-directions?
- If no to the first question, then is there an abbreviated form that would simplify any numerical analysis?
- If yes to the first question, what is the result and how is it derived?
Side Note
My purpose for looking into this is to create a simple routine that will plot the location of a bow shock given an arbitrary plane of projection offset along either the $\hat{\mathbf{y}}$- or $\hat{\mathbf{z}}$-directions.
Thus, as the magnitude of the offset increases from zero I expect the magnitude of $r + x_{o}$ to change (and perhaps even the shape of the curve?). I would guess that one could construct a general form where $r = r(y_{o},z_{o})$ and $x_{o} = x_{o}(y_{o},z_{o})$ in that plane, but I am not sure how.