# Finding parameterization of surface that maps parameter distance to arc length

I have a surface in $\mathbb{R}^3$ defined by four corner points $p_i$ and with known normals at each corner $n_i$. I've also constrained the contour of each edge to be a circular arc, which can be derived in a straightforward manner from the points and normals.

What I'm trying to do now is define a mapping $M: s \to p$ with $s=(u,v)$ and $p=(x,y,z)$ s.t. for any pair of points $s_a$, $s_b$ the standard Euclidean distance $\lvert s_b - s_a \rvert$ is always equal to the arc length along the surface between the mapped 3d points $p_a$ to $p_b$.

I've arbitrarily defined the origin so that $M(0, 0) = p_1$ and so that the $u$ and $v$ axes line up with the arcs from ($p_1$ to $p_2$) and ($p_1$ to $p_4$) respectively. This makes my surface region of interest a quadrilateral in $(u,v)$ space and I've worked out how to compute its four corner points.

I've found a lot of information on equidistant projections for spheres (for mapping applications related to the Earth), but I've been having a harder time tracking down how to use equidistance as a constraint to determine a surface. I know the boundary conditions for my surface and I know I want an equidistant projection, so I want to find what shape of surface can allow that.

My guess is that the problem will eventually boil down to a system of partial differential equations (hopefully linear), which can then be solved to fully determine the injective surface mapping function. This is where I've reached the limit of my current knowledge. What should I be researching next to resolve this problem?

Firstly, I don't see how it's possible to construct a circular arc edge from two points and two normals, in general. I think it's possible only if the two normals make the same angle with the chord joining the two points. So, if you can always construct circular edges, there must be something special about your point/normal data that you haven't mentioned to us.

There are not many types of surfaces whose isoparametric curves in both directions are circular. Spheres and tori certainly, and maybe cyclides. Or, if you regard a straight line as a degenerate circle, then circular cylinders and cones and planes could be included.

Among these, the only ones that satisfy your arclength condition are cylinders and planes.

• Thanks, I see you addressed my other question as well and you're correct, I was mistakenly adding symmetries that didn't necessarily exist. I'll have to do some research on Bezier curves to see if that makes things more tractable. – Dan Bryant Apr 6 '16 at 14:50
• In the CAD literature, there's a lot of techniques for interpolating points and normals, But in most cases they use triangular patches, rather than rectangular ones. There are a couple of good surveys by Broschiroli and Funfzig that will give you a place to start: researchgate.net/publication/… – bubba Apr 6 '16 at 15:03
• This has put me on the right track to resolving the true underlying problem, so marking this as the answer. I think I can use a quadratic Bézier surface using the technique you mentioned in the other answer to define the control points. I still need to figure out how to reparameterize it in terms of arc length, which I may raise as a different question. – Dan Bryant Apr 6 '16 at 17:38

HINT:

These lines should be both geodesic and have zero normal curvature which should be from hyperbolic geometry (K<0).

My hunch is that the special case of axi-symmetric surfaces is a hyperboloid of one sheet with the skew lines as ruled generators. The patch is a skew quadrilateral chosen from these generators. The hyperbolic paraboloid can be also a candidate.