ConformalLabpp/code/include/hyper_ideal_geometry.hpp

#pragma once
// hyper_ideal_geometry.hpp
//
// Pure-math building blocks for the hyper-ideal discrete conformal map.
// Ported from de.varylab.discreteconformal.functional.HyperIdealUtility
// and the private helpers of HyperIdealFunctional (lij, αij, σi, σij).
//
// All functions are independent of the mesh type.
//
// Notation follows the original Java / paper:
//   b_i, b_j  – vertex variables (log scale factors, hyper-ideal vertices)
//   a_ij      – edge variable (intersection angle between horocycles)
//   l_ij      – effective hyperbolic edge length in the auxiliary triangle
//   β_i       – interior angle of the hyperbolic triangle at vertex i
//   α_ij      – dihedral angle of the tetrahedron at edge ij

#include <cmath>
#include <algorithm>

namespace conformallab {

constexpr double PI = 3.14159265358979323846264338328;

// ── Length functions ─────────────────────────────────────────────────────────

// ζ(x,y,z)  — interior angle in a hyperbolic triangle with edge lengths
//             x, y, z, opposite to the side of length z.
// Ports HyperIdealUtility.ζ(x, y, z).
inline double zeta(double x, double y, double z)
{
    double cx = std::cosh(x), cy = std::cosh(y), cz = std::cosh(z);
    double sx = std::sinh(x), sy = std::sinh(y);
    double nbd = (cx*cy - cz) / (sx*sy);
    nbd = std::clamp(nbd, -1.0, 1.0);   // guard floating-point rounding
    return std::acos(nbd);
}

// ζ₁₃(x,y,z) — third edge length in a right-angled hyperbolic hexagon.
// Ports HyperIdealUtility.ζ_13(x, y, z).
inline double zeta13(double x, double y, double z)
{
    double cx = std::cosh(x), cy = std::cosh(y), cz = std::cosh(z);
    double sx = std::sinh(x), sy = std::sinh(y);
    return std::acosh((cx*cy + cz) / (sx*sy));
}

// ζ₁₄(x,y) — edge length in a hyperbolic pentagon with one ideal vertex.
// Ports HyperIdealUtility.ζ_14(x, y).
inline double zeta14(double x, double y)
{
    double cy = std::cosh(y), sy = std::sinh(y);
    return std::acosh((std::exp(x) + cy) / sy);
}

// ζ₁₅(x) — length in a hyperbolic quadrilateral with two ideal vertices.
// Ports HyperIdealUtility.ζ_15(x).
inline double zeta15(double x)
{
    return 2.0 * std::asinh(std::exp(x / 2.0));
}

// ── Effective edge length ─────────────────────────────────────────────────────

// l_ij: effective hyperbolic length of edge ij.
//   b_i, b_j – vertex log scale factors  (used only if vertex is hyper-ideal)
//   a_ij     – edge intersection-angle variable
//   vi_var   – true if vertex i is hyper-ideal (has a DOF b_i)
//   vj_var   – true if vertex j is hyper-ideal
// Ports HyperIdealFunctional.lij().
inline double lij(double bi, double bj, double aij, bool vi_var, bool vj_var)
{
    if (vi_var && vj_var) return zeta13(bi, bj, aij);
    if (vi_var)           return zeta14(aij, bi);
    if (vj_var)           return zeta14(aij, bj);
    return zeta15(aij);
}

// ── Auxiliary angle functions ─────────────────────────────────────────────────

// σᵢ(aᵢⱼ, aₖᵢ, aⱼₖ, vj_var, vk_var) — intermediate half-length at vertex i.
// Ports HyperIdealFunctional.σi().
inline double sigma_i(double aij, double aki, double ajk, bool vj_var, bool vk_var)
{
    if (vj_var && vk_var) return zeta13(aij, aki, ajk);
    if (vj_var)           return zeta14(ajk - aki, aij);
    if (vk_var)           return zeta14(ajk - aij, aki);
    return zeta15(ajk - aij - aki);
}

// σᵢⱼ(aᵢⱼ, bᵢ, bⱼ, vj_var) — intermediate half-length for edge ij from vertex i.
// Ports HyperIdealFunctional.σij().
inline double sigma_ij(double aij, double bi, double bj, bool vj_var)
{
    if (vj_var) return zeta13(aij, bi, bj);
    return zeta14(-aij, bi);
}

// α_ij: computed dihedral angle at edge ij in the face with vertices i, j, k.
//
// Arguments (cyclic role assignment):
//   aij, ajk, aki  – edge variables
//   bi, bj, bk     – vertex variables
//   βi, βj, βk     – interior angles of the auxiliary hyperbolic triangle
//   vi_var, vj_var, vk_var – which vertices are hyper-ideal
//
// Ports HyperIdealFunctional.αij() (the private helper).
// Note: the vk_var case recurses once (never more than one level deep).
inline double alpha_ij(
    double aij, double ajk, double aki,
    double bi,  double bj,  double bk,
    double beta_i, double beta_j, double beta_k,
    bool vi_var, bool vj_var, bool vk_var)
{
    if (vi_var) {
        double si  = sigma_i (aij, aki, ajk, vj_var, vk_var);
        double sij = sigma_ij(aij, bi, bj, vj_var);
        double sik = sigma_ij(aki, bi, bk, vk_var);
        return zeta(si, sij, sik);
    }
    if (vj_var) {
        double sj  = sigma_i (ajk, aij, aki, vk_var, vi_var);
        double sjk = sigma_ij(ajk, bj, bk, vk_var);
        double sji = sigma_ij(aij, bj, bi, vi_var);
        return zeta(sj, sji, sjk);
    }
    if (vk_var) {
        // Derive α_ij from α_jk (one level of recursion).
        double a_jk = alpha_ij(ajk, aki, aij,
                               bj, bk, bi,
                               beta_j, beta_k, beta_i,
                               vj_var, vk_var, vi_var);
        return PI - a_jk - beta_j;
    }
    // All ideal: closed-form formula.
    return 0.5 * (PI + beta_k - beta_i - beta_j);
}

} // namespace conformallab