# Tutorial: Porting the Inversive-Distance Functional (Phase 9a)

This is a complete, step-by-step example of how to add a new functional
to conformallab++.  It ports `InversiveDistanceFunctional.java` from the
Java reference implementation (Luo 2004).

**Prerequisite:** Read [doc/api/extending.md](../api/extending.md) first
for the general pattern.  This tutorial fills in every detail for one
specific case.

---

## Mathematical background

Inversive distance (Luo 2004) uses a different edge-length update formula:

```
Given inversive distances I_{ij} ∈ ℝ for each edge {i,j}:

  cosh(l̃_{ij}) = I_{ij} · cosh((u_i + u_j) / 2)
               + (cosh²((u_i - u_j) / 2) - 1) · ...
```

For the Euclidean version the angle formula is the same law of cosines,
but the log-lengths Λ̃ are computed differently from the u-vector.

**Reference:** Luo, F. (2004). *Combinatorial Yamabe Flow on Surfaces*.
Communications in Contemporary Mathematics, 6(5), 765–780.

**Java source:** `de.varylab.discreteconformal.functional.InversiveDistanceFunctional`

---

## Step 1 — Create the header

```bash
cp code/include/euclidean_functional.hpp \
   code/include/inversive_distance_functional.hpp
```

Edit the new file:

```cpp
#pragma once
// inversive_distance_functional.hpp
//
// Phase 9a — Inversive-distance discrete conformal functional (Luo 2004).
// Ported from de.varylab.discreteconformal.functional.InversiveDistanceFunctional.
//
// Usage: identical to euclidean_functional.hpp — replace lambda0 with
// inversive_distance0 (the initial inversive distances per edge).

#include "conformal_mesh.hpp"
#include <Eigen/Dense>
#include <cmath>

namespace conformallab {

struct InversiveDistanceMaps {
    CGAL::Surface_mesh<CGAL::Simple_cartesian<double>::Point_3>
        ::Property_map<Edge_index, double>   inv_dist0; ///< I_{ij} per edge
    CGAL::Surface_mesh<CGAL::Simple_cartesian<double>::Point_3>
        ::Property_map<Vertex_index, double> theta_v;   ///< target corner angles Θ_v
    CGAL::Surface_mesh<CGAL::Simple_cartesian<double>::Point_3>
        ::Property_map<Vertex_index, int>    v_idx;     ///< DOF index (-1 = pinned)
};

// ... (follow euclidean_functional.hpp exactly, replacing lambda0 with inv_dist0
//      and the angle formula with the Luo (2004) version)
```

---

## Step 2 — Implement the energy, gradient, and angle formula

The Euclidean angle formula uses the law of cosines on edge lengths
derived from log-lengths.  For inversive distance, the edge lengths
are derived from inversive distances I_{ij} and the u-vector:

```cpp
// Euclidean (reference):
//   lambda_ij = lambda0_ij + u_i + u_j
//   l_ij      = exp(lambda_ij / 2)

// Inversive distance (Luo 2004):
//   cosh(l̃_ij) = I_ij * cosh((u_i + u_j) / 2)   [simplified form]
//   l̃_ij       = acosh(I_ij * cosh((u_i + u_j) / 2))
```

Then the **corner angle** at vertex k opposite edge {i,j} in triangle {i,j,k}
is computed by the standard law of cosines from l̃_{ij}, l̃_{jk}, l̃_{ki}.

The **gradient** is the same as Euclidean:

```cpp
G_v = Σ_{faces containing v} alpha_v(face, u)  −  Theta_v
```

---

## Step 3 — Write the gradient-check test

Create `code/tests/cgal/test_inversive_distance.cpp`:

```cpp
#include "inversive_distance_functional.hpp"
#include "mesh_factory.hpp"
#include <gtest/gtest.h>

// Finite-difference gradient check: copy the pattern from
// test_euclidean_functional.cpp :: EuclideanFunctional.GradientCheck_Triangle
TEST(InversiveDistance, GradientCheck_Triangle)
{
    // Build a single equilateral triangle (open mesh, no boundary issues).
    ConformalMesh mesh = MeshFactory::make_open_mesh(MeshFactory::Kind::triangle);
    InversiveDistanceMaps maps = setup_inversive_distance_maps(mesh);
    compute_inversive_distance0_from_mesh(mesh, maps);  // I_ij from 3D edge lengths

    const int n = /* count free DOFs */;
    std::vector<double> x0(n, 0.0);
    constexpr double eps = 1e-5;

    auto G  = inversive_distance_gradient(mesh, x0, maps);

    for (int i = 0; i < n; ++i) {
        auto xp = x0; xp[i] += eps;
        auto xm = x0; xm[i] -= eps;
        double Ep = inversive_distance_energy(mesh, xp, maps);
        double Em = inversive_distance_energy(mesh, xm, maps);
        double fd = (Ep - Em) / (2.0 * eps);
        EXPECT_NEAR(G[i], fd, 1e-6) << "gradient mismatch at DOF " << i;
    }
}
```

Run with:

```bash
./build/conformallab_cgal_tests --gtest_filter="InversiveDistance.*"
```

---

## Step 4 — Register in CMakeLists

Add to `code/tests/cgal/CMakeLists.txt` inside the `add_executable` block:

```cmake
    # ── Phase 9a: Inversive-distance functional ────────────────────────────────
    test_inversive_distance.cpp
```

---

## Step 5 — Add a Newton wrapper

In `code/include/newton_solver.hpp` add:

```cpp
/// Solve the inversive-distance conformal problem.
/// \see newton_euclidean() — identical structure.
inline NewtonResult newton_inversive_distance(
    ConformalMesh&                  mesh,
    std::vector<double>             x0,
    const InversiveDistanceMaps&    m,
    double                          tol      = 1e-8,
    int                             max_iter = 200)
{
    // Copy newton_euclidean() exactly, replacing euclidean_* with
    // inversive_distance_*.  The Hessian is PSD (Luo 2004, Thm. 1),
    // so SimplicialLDLT with SparseQR fallback applies directly.
}
```

---

## Step 6 — Verify against Java output

The Java library outputs text results for a triangulated torus.  To compare:

1. Run Java ConformalLab on the same OFF mesh with Inversive-Distance mode.
2. Record the final gradient norm and angle sums.
3. Run the C++ equivalent and check:

```cpp
EXPECT_LT(res.grad_inf_norm, 1e-8);
EXPECT_LT(res.iterations, 50);  // inversive-distance converges faster than hyper-ideal
```

**Java reference:** `InversiveDistanceFunctionalTest.java` in `de.varylab.discreteconformal.test`

---

## Checklist

- [ ] `code/include/inversive_distance_functional.hpp` compiles
- [ ] `GradientCheck_Triangle` passes
- [ ] `GradientCheck_OpenMesh` passes (copy from Euclidean tests)
- [ ] Newton converges on `torus_4x4.off`
- [ ] Result matches Java output (gradient norm < 1e-8 on same mesh)
- [ ] Registered in `CMakeLists.txt`
- [ ] `doc/roadmap/java-parity.md` updated: Phase 9a → ✅