A Surface of Constant Width is a 3D surface with the special property that any two parallel planes which are tangent to it are always a constant distance apart, no matter the relative rotations of the surface and paired-planes. I want to construct a trio of parametric equations $\bigl(X(u, v), Y(u, v), Z(u, v)\bigr)$ that give a surface with the following properties:
- has a constant width of $S + 2 A$, where $S$ and $A$ are non-negative real numbers, $A$ is the radius of the smallest osculating circle on the surface, and $S + A$ is the radius of the largest osculating circle on the surface;
- has full tetrahedral symmetry;
- is $G^∞$-smooth for positive values of $S$ and $A$ and finite values of $u$ and $v$, meaning the surface, curvature, derivative of curvature, second derivative of curvature, and so-on are all continuous in all directions at all points on the surface; and
- linear-step interpolation between the minimum and maximum values for $u$ and $v$ produces 3D points with a tetrahedrally-symmetrical arrangement and constant (or nearly-constant) angles between the normal-vectors of neighboring points (meaning points are closer together in highly-curved areas of the surface, and farther apart in less-curved areas of the surface).
I was able to get exactly what I wanted in two dimensions with the paper "A Polynomial Curve of Constant Width" by Stanley Rabinowitz. Based on that paper, we can define the functions
$$ p(t) = \left(1 + \frac{\cos(n × t)}{n^2 - 1}\right) × \frac{S}{2} + A $$ $$ X(t) = \cos(t) × p(t) - \sin(t) × p'(t) $$ $$ Y(t) = \sin(t) × p(t) + \cos(t) × p'(t) $$
where $S$ is a positive real number, $A$ is a non-negative real number and the radius of the smallest osculating circle on the curve, $S + A$ is the radius of the largest osculating circle on the curve, and $n$ is an odd integer greater than 2.
When plotted as the parametric curve $C(t) = \bigl(X(t), Y(t)\bigr)$ with $0 ≤ t < 2 π$, these generate a smoothed-out version of a regular $n$-sided Reuleaux polygon with side-length $S$ and corner-radius $A$, which is also (for positive $A$) $G^∞$-smooth at all points. Additionally, when sampling the curve at parameter values with a step-size of $Δt = \frac{2 π}{n k}$ for integer $k$, the normal-vectors of adjacent sample-points have angles that differ by a constant $\frac{2 π}{n k}$ radians, and the arrangement of the sample-points has full $n$-gon dihedral symmetry.
I've tried to figure out how to adapt the methods in Rabinowitz's paper to three dimensions, but without success.
How do I construct such an adaptation, for the constraints I have specified?
I did find the paper "On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg, which includes a figure on page 16 that depicts the surface I want to parameterize, and discusses doing the exact 2D-to-3D conversion I want. However, despite my efforts to decipher this paper, I can't follow what it is describing, as the key part of the paper hinges on using a function defined with a single variable as a function of two variables, and I can't figure out how to handle that.