nh-rep/code/pre_processing/ParametricSample/Mathematics/DarbouxFrame.h

// David Eberly, Geometric Tools, Redmond WA 98052
// Copyright (c) 1998-2021
// Distributed under the Boost Software License, Version 1.0.
// https://www.boost.org/LICENSE_1_0.txt
// https://www.geometrictools.com/License/Boost/LICENSE_1_0.txt
// Version: 4.0.2019.08.13

#pragma once

#include <Mathematics/Matrix2x2.h>
#include <Mathematics/ParametricSurface.h>
#include <Mathematics/Vector3.h>
#include <memory>

namespace gte
{
    template <typename Real>
    class DarbouxFrame3
    {
    public:
        // Construction.  The curve must persist as long as the DarbouxFrame3
        // object does.
        DarbouxFrame3(std::shared_ptr<ParametricSurface<3, Real>> const& surface)
            :
            mSurface(surface)
        {
        }

        // Get a coordinate frame, {T0, T1, N}.  At a nondegenerate surface
        // points, dX/du and dX/dv are linearly independent tangent vectors.
        // The frame is constructed as
        //   T0 = (dX/du)/|dX/du|
        //   N  = Cross(dX/du,dX/dv)/|Cross(dX/du,dX/dv)|
        //   T1 = Cross(N, T0)
        // so that {T0, T1, N} is a right-handed orthonormal set.
        void operator()(Real u, Real v, Vector3<Real>& position, Vector3<Real>& tangent0,
            Vector3<Real>& tangent1, Vector3<Real>& normal) const
        {
            std::array<Vector3<Real>, 3> jet;
            mSurface->Evaluate(u, v, 1, jet.data());
            position = jet[0];
            tangent0 = jet[1];
            Normalize(tangent0);
            tangent1 = jet[2];
            Normalize(tangent1);
            normal = UnitCross(tangent0, tangent1);
            tangent1 = Cross(normal, tangent0);
        }

        // Compute the principal curvatures and principal directions.
        void GetPrincipalInformation(Real u, Real v, Real& curvature0, Real& curvature1,
            Vector3<Real>& direction0, Vector3<Real>& direction1) const
        {
            // Tangents:  T0 = (x_u,y_u,z_u), T1 = (x_v,y_v,z_v)
            // Normal:    N = Cross(T0,T1)/Length(Cross(T0,T1))
            // Metric Tensor:    G = +-                      -+
            //                       | Dot(T0,T0)  Dot(T0,T1) |
            //                       | Dot(T1,T0)  Dot(T1,T1) |
            //                       +-                      -+
            //
            // Curvature Tensor:  B = +-                          -+
            //                        | -Dot(N,T0_u)  -Dot(N,T0_v) |
            //                        | -Dot(N,T1_u)  -Dot(N,T1_v) |
            //                        +-                          -+
            //
            // Principal curvatures k are the generalized eigenvalues of
            //
            //     Bw = kGw
            //
            // If k is a curvature and w=(a,b) is the corresponding solution
            // to Bw = kGw, then the principal direction as a 3D vector is
            // d = a*U+b*V.
            //
            // Let k1 and k2 be the principal curvatures.  The mean curvature
            // is (k1+k2)/2 and the Gaussian curvature is k1*k2.

            // Compute derivatives.
            std::array<Vector3<Real>, 6> jet;
            mSurface->Evaluate(u, v, 2, jet.data());
            Vector3<Real> derU = jet[1];
            Vector3<Real> derV = jet[2];
            Vector3<Real> derUU = jet[3];
            Vector3<Real> derUV = jet[4];
            Vector3<Real> derVV = jet[5];

            // Compute the metric tensor.
            Matrix2x2<Real> metricTensor;
            metricTensor(0, 0) = Dot(jet[1], jet[1]);
            metricTensor(0, 1) = Dot(jet[1], jet[2]);
            metricTensor(1, 0) = metricTensor(0, 1);
            metricTensor(1, 1) = Dot(jet[2], jet[2]);

            // Compute the curvature tensor.
            Vector3<Real> normal = UnitCross(jet[1], jet[2]);
            Matrix2x2<Real> curvatureTensor;
            curvatureTensor(0, 0) = -Dot(normal, derUU);
            curvatureTensor(0, 1) = -Dot(normal, derUV);
            curvatureTensor(1, 0) = curvatureTensor(0, 1);
            curvatureTensor(1, 1) = -Dot(normal, derVV);

            // Characteristic polynomial is 0 = det(B-kG) = c2*k^2+c1*k+c0.
            Real c0 = Determinant(curvatureTensor);
            Real c1 = (Real)2 * curvatureTensor(0, 1) * metricTensor(0, 1) -
                curvatureTensor(0, 0) * metricTensor(1, 1) -
                curvatureTensor(1, 1) * metricTensor(0, 0);
            Real c2 = Determinant(metricTensor);

            // Principal curvatures are roots of characteristic polynomial.
            Real temp = std::sqrt(std::max(c1 * c1 - (Real)4 * c0 * c2, (Real)0));
            Real mult = (Real)0.5 / c2;
            curvature0 = -mult * (c1 + temp);
            curvature1 = -mult * (c1 - temp);

            // Principal directions are solutions to (B-kG)w = 0,
            // w1 = (b12-k1*g12,-(b11-k1*g11)) OR (b22-k1*g22,-(b12-k1*g12)).
            Real a0 = curvatureTensor(0, 1) - curvature0 * metricTensor(0, 1);
            Real a1 = curvature0 * metricTensor(0, 0) - curvatureTensor(0, 0);
            Real length = std::sqrt(a0 * a0 + a1 * a1);
            if (length > (Real)0)
            {
                direction0 = a0 * derU + a1 * derV;
            }
            else
            {
                a0 = curvatureTensor(1, 1) - curvature0 * metricTensor(1, 1);
                a1 = curvature0 * metricTensor(0, 1) - curvatureTensor(0, 1);
                length = std::sqrt(a0 * a0 + a1 * a1);
                if (length > (Real)0)
                {
                    direction0 = a0 * derU + a1 * derV;
                }
                else
                {
                    // Umbilic (surface is locally sphere, any direction
                    // principal).
                    direction0 = derU;
                }
            }
            Normalize(direction0);

            // Second tangent is cross product of first tangent and normal.
            direction1 = Cross(direction0, normal);
        }

    private:
        std::shared_ptr<ParametricSurface<3, Real>> mSurface;
    };
}