Draft code for finding Stokes lines via relaxing ropes.

1 files changed, 99 insertions, 0 deletions

diff --git a/stokes.hpp b/stokes.hpp
new file mode 100644
index 0000000..b60ae0e
--- /dev/null
+++ b/stokes.hpp
@@ -0,0 +1,99 @@
+#include "stereographic.hpp"
+
+template <class Scalar>
+class Rope {
+  private:
+    std::vector<Vector<Scalar>> z;
+
+    template <int p>
+    Vector<Scalar> δz(const Tensor<Scalar, p>& J) const {
+      Vector<Scalar> dz(z.size());
+
+      for (unsigned i = 1; i < z.size() - 1; i++) {
+        Vector<Scalar> z12 = (z[i] + z[i - 1]) / 2.0;
+        Vector<Scalar> g;
+        std::tie(std::ignore, g, std::ignore) = stereographicHamGradHessHess(J, z12);
+
+        Vector<Scalar> Δ = z[i] - z[i - 1];
+
+        g.normalize();
+        Δ.normalize();
+
+        if (abs(arg(g.dot(Δ))) < M_PI / 2) {
+          dz[i] = g - Δ;
+        } else {
+          dz[i] = -g - Δ;
+        }
+      }
+
+      return dz;
+    }
+
+    template <int p>
+    void perturb(const Tensor<Scalar, p>& J, double δ) {
+      z += δ * this->δz(J);
+    }
+
+    void spread() {
+      std::vector<double> d(z.size() - 1);
+      double l = 0;
+
+      for (unsigned i= 0; i < z.size() - 1; i++) {
+        double Δ = (z[i + 1] - z[i]).norm();
+        d[i] = Δ;
+        l += Δ;
+      }
+
+      double a = 0;
+      unsigned pos = 0;
+
+      for (unsigned i = 1; i < z.size() - 1; i++) {
+        double b = i * l / (z.size() - 1);
+        while (b >= a) {
+          a += d[pos];
+          pos++;
+        } 
+
+        z[i] = z[pos] - (a - b) * (z[pos] - z[pos - 1]).normalized();
+      }
+    }
+
+  public:
+    Rope(unsigned N, const Vector<Scalar>& z1, const Vector<Scalar>& z2) : z(N + 2) {
+      for (unsigned i = 0; i < N + 2; i++) {
+        z[i] = z1 + (z2 - z1) * ((double)i / (N + 1.0));
+      }
+    }
+
+    template <int p>
+    void step(const Tensor<Scalar, p>& J, double δ) {
+      this->perturb(J, δ);
+      this->spread();
+    }
+
+    template <int p>
+    double cost(const Tensor<Scalar, p>& J) const {
+      return this->δz(J).norm();
+    }
+
+    template <int p>
+    void relax(const Tensor<Scalar, p>& J, unsigned N, double δ) {
+      for (unsigned i = 0; i < N; i++) {
+        this->step(J, δ);
+      }
+    }
+
+    Rope<Scalar> interpolate() const {
+      Rope<Scalar> r(2 * z.size() - 1);
+
+      for (unsigned i = 0; i < z.size(); i++) {
+        r.z[2 * i] = z[i];
+      }
+
+      for (unsigned i = 0; i < z.size() - 1; i++) {
+        r.z[2 * i + 1] = (z[i] + z[i + 1]) / 2.0;
+      }
+
+      return r;
+    }
+};