The Problem
I have a generic surface $S_1$ in 3D space with cylindrical symmetry such that in cylindrical coordinates $(\rho,\theta,z)$, $S_1$ is defined by the smooth function
$$z = z(\rho), \hspace{10pt} z(0) = -H$$
Where $H$ is a real positive constant. I also have a unit vector $\mathbf{\hat{v}}$ which has Cartesian components
$$\mathbf{\hat{v}} = \sin(\psi)\mathbf{\hat{y}} + \cos(\psi)\mathbf{\hat{z}}$$
Where $\psi$ is a constant between $0$ and $\pi/2$.
Let $\mathbf{a}(x,y,z(\rho))$ be the vector which points from the origin to some point on $S_1$ with coordinates $(x,y,z(\rho))$. I want to find regions on $S_1$ for some given $L$ satisfying
$$L = ||\mathbf{a}|| + \mathbf{a}\cdot\mathbf{\hat{v}}$$
My ultimate goal is to be able to integrate for the surface area between different $L$ values.
My Attempts
Given $S_1$ only has one $z$ value for any given $(x,y)$, I know that I can fully describe these regions just using $x$ and $y$.
I have already found the solution to this problem for the case where $S_1$ is a plane ($z=-H$). These regions are ellipses described by
$$x = \left(\frac{\sqrt{L(L-2H\sin(\psi))}}{\sin(\psi)}\right)\cos(t)$$ $$y = \left(\frac{\sqrt{L(L-2H\sin(\psi))}}{\sin^2(\psi)}\right)\sin(t) + \left(\frac{\left(L-H\sin(\psi)\right)\cos{\psi}}{\sin^2(\psi)}\right)$$
I can then obtain the Jacobian determinant, integrate over $t \in (0,2\pi)$ and then integrate between whichever L values I'm interested in.
Assessment + Questions
I don't think a complete analytical solution for this problem exists due to how arbitrary $z(\rho)$ is. I am interested then in semi-analytical or numerical solutions. (I have had some success with almost-flat surfaces using newton's method and the above parameters $L,t$). I am also okay with restricting the form of $z(\rho)$ for simpler solutions.
To solve this problem, I feel I need to find some parameter $\alpha$ such that $(L,\alpha)$ maps 1-1 to $(x,y)$. I then need to integrate the Jacobian over $\alpha$ and $L$. I can do this all numerically.
- How do I find a good parameter $\alpha$?
- Is it a problem that sometimes there exist disconnected regions for a given $L$?
- Are there other ways of solving this that I'm missing?