How do I handle this single-variable function being called with two variables?

Question

As part of a personal project, I found the paper "On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg, which discusses deriving equations for a 3D surface I want to parameterize. However, despite my efforts to decipher the paper, I can't follow everything it is describing. The paper plays fast and loose with notation and relies on conventions I haven't encountered before. I have been unable to find other resources about the same surface that are easier to parse.

I don't need a full understanding of all the involved concepts for the project I'm working on, I just need to know how to work the algebra to get the surface I want.

What I'm Trying to Understand

The key section of the paper for my purposes is 4.3 Discrete Symmetries:

4.3 Discrete Symmetries

Consider a discrete subgroup of isometries $G ⊂ O(3)$, and suppose that $r_0$ is the support function of a surface of constant width $w$.

Proposition 4.6: The surface determined by the support function $$ r(ξ, \bar{ξ}) = \frac{1}{\#G} \sum_{g ∈ G} r_0\bigl(g(ξ), g(\bar{ξ})\bigr), $$ is a surface of constant width $w$ which is invariant under $G$.

Proof: This follows from the fact that the antipodal map commutes with elements of $O(3)$. Q.E.D.

Applying this approach to the case of $r_0$ being equal to the support function in Example 4.2 and $G$ being the tetrahedral group, we can construct closed convex surfaces of constant width with tetrahedral symmetry. The results are shown in Figure 5, where both a surface (left) and its focal set (right) are presented.

$$ \text{Fɪɢᴜʀᴇ 5.} $$

From a previous question I know that with "$G$ being the tetrahedral group" we have $\frac{1}{\#G} = \frac{1}{12}$, and $g ∈ G$ refers to the following set of twelve functions in a complex variable $z$, where $U_n = e^{\frac{n}{3} ⋅ 2 π i}$ is the $n$th cube root of unity:

• $g_{01}(z) = U_0 ⋅ z = z$
• $g_{02}(z) = U_1 ⋅ z = -\frac{1 - \sqrt{3} i}{2} ⋅ z$
• $g_{03}(z) = U_2 ⋅ z = -\frac{1 + \sqrt{3} i}{2} ⋅ z$
• $g_{04}(z) = \frac{\sqrt{2} ⋅ U_0 - 1 ⋅ z}{\sqrt{2} ⋅ U_0 ⋅ z + 1} = \frac{\sqrt{2} - z}{\sqrt{2} z + 1}$
• $g_{05}(z) = \frac{\sqrt{2} ⋅ U_1 - 1 ⋅ z}{\sqrt{2} ⋅ U_2 ⋅ z + 1} = \frac{(\sqrt{2} - \sqrt{6} i) + 2 ⋅ z}{(\sqrt{2} + \sqrt{6} i) ⋅ z - 2}$
• $g_{06}(z) = \frac{\sqrt{2} ⋅ U_2 - 1 ⋅ z}{\sqrt{2} ⋅ U_1 ⋅ z + 1} = \frac{(\sqrt{2} + \sqrt{6} i) + 2 ⋅ z}{(\sqrt{2} - \sqrt{6} i) ⋅ z - 2}$
• $g_{07}(z) = U_1 ⋅ \frac{\sqrt{2} - 1 ⋅ z}{\sqrt{2} ⋅ z + 1} = -\frac{1 - \sqrt{3} i}{2} ⋅ \frac{\sqrt{2} - z}{\sqrt{2} ⋅ z + 1}$
• $g_{08}(z) = U_2 ⋅ \frac{\sqrt{2} - 1 ⋅ z}{\sqrt{2} ⋅ z + 1} = -\frac{1 + \sqrt{3} i}{2} ⋅ \frac{\sqrt{2} - z}{\sqrt{2} ⋅ z + 1}$
• $g_{09}(z) = \frac{\sqrt{2} - 1 ⋅ U_1 ⋅ z}{\sqrt{2} ⋅ z + 1 ⋅ U_2} = \frac{2 \sqrt{2} + (1 - \sqrt{3} i) ⋅ z}{2 \sqrt{2} ⋅ z - (1 + \sqrt{3} i)}$
• $g_{10}(z) = \frac{\sqrt{2} - 1 ⋅ U_2 ⋅ z}{\sqrt{2} ⋅ z + 1 ⋅ U_1} = \frac{2 \sqrt{2} + (1 + \sqrt{3} i) ⋅ z}{2 \sqrt{2} ⋅ z - (1 - \sqrt{3} i)}$
• $g_{11}(z) = \frac{\sqrt{2} + U_1 ⋅ (1 ⋅ z + \sqrt{2})}{\sqrt{2} ⋅ z - U_2 ⋅ (1 - \sqrt{2} ⋅ z)} = \frac{-(z - \sqrt{2}) + \sqrt{3} i (z + \sqrt{2})}{ (1 + \sqrt{2} z) + \sqrt{3} i (1 - \sqrt{2} z)}$
• $g_{12}(z) = \frac{\sqrt{2} + U_2 ⋅ (1 ⋅ z + \sqrt{2})}{\sqrt{2} ⋅ z - U_1 ⋅ (1 - \sqrt{2} ⋅ z)} = \frac{ (z - \sqrt{2}) + \sqrt{3} i (z + \sqrt{2})}{-(1 + \sqrt{2} z) + \sqrt{3} i (1 - \sqrt{2} z)}$

However, the sum in section 4.3 calls $r_0$ as a two-variable function $r_0\bigl(g(ξ), g(\bar{ξ})\bigr)$, yet it also states that $r_0$ is "equal to the support function in Example 4.2". The support function in question is a single-variable function in $R ∈ ℝ$ with three real parameters $a$, $b$, and $C$, giving a rotationally-symmetric surface of constant width $1 + 2 C$: $$ r(R) = \frac{a + b R^2 + (3 - b) R^4 + (1 - a) R^6}{(1 + R^2)^3} + C $$

Central Question: How do I go from $r(R)$ to $r(ξ, \bar{ξ})$?

Section 2.2 Coordinates on defines $ξ$ thusly:

Let $ξ$ be the local complex coordinate on the unit 2-sphere in $^3$ obtained by stereographic projection from the south pole. In terms of the standard spherical polar angles $(θ, φ)$ we have $ξ = \tan\left(\frac{θ}{2}\right) e^{i φ}$.

Section 4.1 Rotational Symmetry defines $R$ thusly:

It is not hard to see that the map $ξ ↦ \bigl(ξ, η = F(ξ, \bar{ξ})\bigr)$ determining this line congruence satisfies $F = G(R) e^{i θ}$, where $G$ is a real function and $ξ = R e^{i θ}$.

(This means that $R_\text{§4} = \tan\left(\frac{θ_\text{§2}}{2}\right)$ and $θ_\text{§4} = φ_\text{§2}$.)

I've gone over this paper at least a dozen times now, and read online resources on differential geometry, and still can't figure out how I'm supposed to handle this. It makes sense that the rotationally-symmetric surface's support function is a one-variable function and the tetrahedrally-symmetric surface's is a two-variable function, but I don't see a natural way to convert the former into the latter.

Supplemental

All the other definitions of "$r$" in the paper:

• 2.3 The Correspondence Space
  • Definition 2.1: The map $Φ : × ℝ → ^3$ is defined to take $((ξ, η), r) ∈ × ℝ$ to the point in $^3$ on the oriented line $(ξ, η)$ that lies a distance $r$ from the point on the line closest to the origin (see the right of Figure 2).
• 2.4 Line Congruences
  • Given a line congruence $Σ ⊂ $, a map $r : Σ → ℝ$ determines a map $Σ → ^3$ by $(ξ, η) ↦ Φ\bigl((ξ, η), r(ξ, η)\bigr)$ for $(ξ, η) ∈ Σ$. In other words, we pick out one point on each line in the congruence (see Figure 3).
  • Theorem 2.4: A line congruence $\bigl(ξ, η = F(ξ, \bar{ξ})\bigr)$ is orthogonal to a surface in $^3$ if and only if there is a real function $r(ξ, \bar{ξ})$ satisfying $\frac{∂r}{∂\bar{ξ}} = \frac{2 F}{(1 + ξ \bar{ξ})^2}$. If there is a solution, there is a 1-parameter family generated by a real constant of integration. The function $r$ is the distance from the surface to the point on the normal line closest to the origin.
  • The surface can be reconstructed in $^3$ from this data be inserting $r = r(ξ, \bar{ξ})$ and $η = F(ξ, \bar{ξ})$ in equations (2.1). A change $r ↦ r + C$ moves the surface a distance $C$ along its normal to the “parallel” surface.
• 2.5 Focal Points of a Line Congruence
  • Theorem 2.6: The focal set of the parametric line congruence $Σ$ which is normal to a closed convex surface is given by $r = r_±(ξ, \bar{ξ}) = -ψ ± |σ|$, where the slopes ψ and σ are given by equation (2.2).
• 3.1 Oriented Normal lines
  • The set of oriented normal lines to $S$ forms a line congruence that can be parameterized by the direction of the normal. Thus the normals are given by a map $ξ ↦ \bigl(ξ, η = F(ξ, \bar{ξ})\bigr)$, and there exists a real function $r(ξ, \bar{ξ})$ satisfying equation (2.3).
• 3.3 Constant Width Surfaces with Rational Support
  • Definition 3.7: A closed convex surface has rational support if the support function is of the form $r = \frac{P(ξ, \bar{ξ})}{Q(ξ, \bar{ξ})}$ where $P$ and $Q$ are real-valued polynomials.
• 4.1 Rotational Symmetry
  • Proposition 4.2: The focal set of the oriented normals to a convex, rotationally symmetric surface with support function $r = r(R)$ is given by the surface $z = \frac{1}{2} \left(-R (1 + R^2) \frac{d^2 r}{d R^2} + (1 - 3 R^2) \frac{d r}{d R}\right) e^{i θ}$, $t = \frac{1}{4} \left(-(1 - R^4) \frac{d^2 r}{d R^2} - 2 R (3 - R^2) \frac{d r}{d R}\right)$, and the line $z = 0$, $t = -\frac{(1 + R^2)^2}{4 R} \frac{d r}{d R}$, where $z = x^1 + i x^2$ and $t = x^3$, for standard coordinates $(x^1, x^2, x^3)$ on Euclidean 3-space.
  • Proposition 4.3: The cusps on the focal set of the oriented normals to a convex, rotationally symmetric surface with support function $r = r(R)$ are solutions of the equation $(1 + R^2) \frac{d^3 r}{d R^3} + 6 R \frac{d^2 r}{d R^2} + 6 \frac{d r}{d R} = 0$.

Background

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 derive a trio of parametric equations $\bigl(X(u, v), Y(u, v), Z(u, v)\bigr)$ which together give a surface with the following properties:

• has a constant width of $w = S + 2 A$, where $S$ is a positive real number, $A$ is a non-negative real number and the radius of the smallest osculating circle on the surface, and $S + A$ is the radius of the largest osculating circle on the surface;
• both $u$ and $v$ have finite lower and upper bounds; and
• has full tetrahedral symmetry, down to the points generated on the surface with interpolation of $u$ and $v$;
• is $G^∞$-smooth for positive values of $S$ and $A$ (meaning the surface, curvature, derivative of curvature, second derivative of curvature, and so on are all continuous at all points on the surface).