GeometricBasics.cpp

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#include "GeometricBasics.h"
 
#include <Eigen/Dense>
 
#include "BaseLib/Logging.h"
#include "Point3d.h"
 
namespace MathLib
{
double orientation3d(MathLib::Point3d const& p,
                     MathLib::Point3d const& a,
                     MathLib::Point3d const& b,
                     MathLib::Point3d const& c)
{
    auto const pp = Eigen::Map<Eigen::Vector3d const>(p.getCoords());
    auto const pa = Eigen::Map<Eigen::Vector3d const>(a.getCoords());
    auto const pb = Eigen::Map<Eigen::Vector3d const>(b.getCoords());
    auto const pc = Eigen::Map<Eigen::Vector3d const>(c.getCoords());
 
    Eigen::Vector3d const u = pp - pa;
    Eigen::Vector3d const v = pp - pb;
    Eigen::Vector3d const w = pp - pc;
    return u.cross(v).dot(w);
}
 
double calcTetrahedronVolume(MathLib::Point3d const& a,
                             MathLib::Point3d const& b,
                             MathLib::Point3d const& c,
                             MathLib::Point3d const& d)
{
    auto const va = Eigen::Map<Eigen::Vector3d const>(a.getCoords());
    auto const vb = Eigen::Map<Eigen::Vector3d const>(b.getCoords());
    auto const vc = Eigen::Map<Eigen::Vector3d const>(c.getCoords());
    auto const vd = Eigen::Map<Eigen::Vector3d const>(d.getCoords());
    Eigen::Vector3d const w = vb - va;
    Eigen::Vector3d const u = vc - va;
    Eigen::Vector3d const v = vd - va;
    return std::abs(u.cross(v).dot(w)) / 6.0;
}
 
double calcTriangleArea(MathLib::Point3d const& a, MathLib::Point3d const& b,
                        MathLib::Point3d const& c)
{
    auto const va = Eigen::Map<Eigen::Vector3d const>(a.getCoords());
    auto const vb = Eigen::Map<Eigen::Vector3d const>(b.getCoords());
    auto const vc = Eigen::Map<Eigen::Vector3d const>(c.getCoords());
    Eigen::Vector3d const u = vc - va;
    Eigen::Vector3d const v = vb - va;
    Eigen::Vector3d const w = u.cross(v);
    return 0.5 * w.norm();
}
 
bool isPointInTetrahedron(MathLib::Point3d const& p, MathLib::Point3d const& a,
                          MathLib::Point3d const& b, MathLib::Point3d const& c,
                          MathLib::Point3d const& d, double eps)
{
    double const d0(MathLib::orientation3d(d, a, b, c));
    // if tetrahedron is not coplanar
    if (std::abs(d0) > std::numeric_limits<double>::epsilon())
    {
        bool const d0_sign(d0 > 0);
        // if p is on the same side of bcd as a
        double const d1(MathLib::orientation3d(d, p, b, c));
        if (!(d0_sign == (d1 >= 0) || std::abs(d1) < eps))
        {
            return false;
        }
        // if p is on the same side of acd as b
        double const d2(MathLib::orientation3d(d, a, p, c));
        if (!(d0_sign == (d2 >= 0) || std::abs(d2) < eps))
        {
            return false;
        }
        // if p is on the same side of abd as c
        double const d3(MathLib::orientation3d(d, a, b, p));
        if (!(d0_sign == (d3 >= 0) || std::abs(d3) < eps))
        {
            return false;
        }
        // if p is on the same side of abc as d
        double const d4(MathLib::orientation3d(p, a, b, c));
        return d0_sign == (d4 >= 0) || std::abs(d4) < eps;
    }
    return false;
}
 
bool isPointInTriangle(MathLib::Point3d const& p,
                       MathLib::Point3d const& a,
                       MathLib::Point3d const& b,
                       MathLib::Point3d const& c,
                       double eps_pnt_out_of_plane,
                       double eps_pnt_out_of_tri,
                       MathLib::TriangleTest algorithm)
{
    switch (algorithm)
    {
        case MathLib::GAUSS:
            return gaussPointInTriangle(p, a, b, c, eps_pnt_out_of_plane,
                                        eps_pnt_out_of_tri);
        case MathLib::BARYCENTRIC:
            return barycentricPointInTriangle(p, a, b, c, eps_pnt_out_of_plane,
                                              eps_pnt_out_of_tri);
        default:
            ERR("Selected algorithm for point in triangle testing not found, "
                "falling back on default.");
    }
    return gaussPointInTriangle(p, a, b, c, eps_pnt_out_of_plane,
                                eps_pnt_out_of_tri);
}
 
bool gaussPointInTriangle(MathLib::Point3d const& q,
                          MathLib::Point3d const& a,
                          MathLib::Point3d const& b,
                          MathLib::Point3d const& c,
                          double eps_pnt_out_of_plane,
                          double eps_pnt_out_of_tri)
{
    auto const pa = Eigen::Map<Eigen::Vector3d const>(a.getCoords());
    auto const pb = Eigen::Map<Eigen::Vector3d const>(b.getCoords());
    auto const pc = Eigen::Map<Eigen::Vector3d const>(c.getCoords());
    Eigen::Vector3d const v = pb - pa;
    Eigen::Vector3d const w = pc - pa;
 
    Eigen::Matrix2d mat;
    mat(0, 0) = v.squaredNorm();
    mat(0, 1) = v[0] * w[0] + v[1] * w[1] + v[2] * w[2];
    mat(1, 0) = mat(0, 1);
    mat(1, 1) = w.squaredNorm();
    Eigen::Vector2d y(
        v[0] * (q[0] - a[0]) + v[1] * (q[1] - a[1]) + v[2] * (q[2] - a[2]),
        w[0] * (q[0] - a[0]) + w[1] * (q[1] - a[1]) + w[2] * (q[2] - a[2]));
 
    y = mat.partialPivLu().solve(y);
 
    const double lower(eps_pnt_out_of_tri);
    const double upper(1 + lower);
 
    if (-lower <= y[0] && y[0] <= upper && -lower <= y[1] && y[1] <= upper &&
        y[0] + y[1] <= upper)
    {
        MathLib::Point3d const q_projected(std::array<double, 3>{
            {a[0] + y[0] * v[0] + y[1] * w[0], a[1] + y[0] * v[1] + y[1] * w[1],
             a[2] + y[0] * v[2] + y[1] * w[2]}});
        if (MathLib::sqrDist(q, q_projected) <= eps_pnt_out_of_plane)
        {
            return true;
        }
    }
 
    return false;
}
 
bool barycentricPointInTriangle(MathLib::Point3d const& p,
                                MathLib::Point3d const& a,
                                MathLib::Point3d const& b,
                                MathLib::Point3d const& c,
                                double eps_pnt_out_of_plane,
                                double eps_pnt_out_of_tri)
{
    if (std::abs(MathLib::orientation3d(p, a, b, c)) > eps_pnt_out_of_plane)
    {
        return false;
    }
 
    auto const vp = Eigen::Map<Eigen::Vector3d const>(p.getCoords());
    auto const va = Eigen::Map<Eigen::Vector3d const>(a.getCoords());
    auto const vb = Eigen::Map<Eigen::Vector3d const>(b.getCoords());
    auto const vc = Eigen::Map<Eigen::Vector3d const>(c.getCoords());
    Eigen::Vector3d const pa = va - vp;
    Eigen::Vector3d const pb = vb - vp;
    Eigen::Vector3d const pc = vc - vp;
    double const area_x_2(calcTriangleArea(a, b, c) * 2);
 
    double const alpha((pb.cross(pc).norm()) / area_x_2);
    if (alpha < -eps_pnt_out_of_tri || alpha > 1 + eps_pnt_out_of_tri)
    {
        return false;
    }
    double const beta((pc.cross(pa).norm()) / area_x_2);
    if (beta < -eps_pnt_out_of_tri || beta > 1 + eps_pnt_out_of_tri)
    {
        return false;
    }
    double const gamma(1 - alpha - beta);
    return !(gamma < -eps_pnt_out_of_tri || gamma > 1 + eps_pnt_out_of_tri);
}
 
bool isPointInTriangleXY(MathLib::Point3d const& p,
                         MathLib::Point3d const& a,
                         MathLib::Point3d const& b,
                         MathLib::Point3d const& c)
{
    // criterion: p-a = u0 * (b-a) + u1 * (c-a); 0 <= u0, u1 <= 1, u0+u1 <= 1
    Eigen::Matrix2d mat;
    mat(0, 0) = b[0] - a[0];
    mat(0, 1) = c[0] - a[0];
    mat(1, 0) = b[1] - a[1];
    mat(1, 1) = c[1] - a[1];
    Eigen::Vector2d y;
    y << p[0] - a[0], p[1] - a[1];
 
    y = mat.partialPivLu().solve(y);
 
    // check if u0 and u1 fulfills the condition
    return 0 <= y[0] && y[0] <= 1 && 0 <= y[1] && y[1] <= 1 && y[0] + y[1] <= 1;
}
 
bool dividedByPlane(const MathLib::Point3d& a, const MathLib::Point3d& b,
                    const MathLib::Point3d& c, const MathLib::Point3d& d)
{
    for (unsigned x = 0; x < 3; ++x)
    {
        const unsigned y = (x + 1) % 3;
        const double abc =
            (b[x] - a[x]) * (c[y] - a[y]) - (b[y] - a[y]) * (c[x] - a[x]);
        const double abd =
            (b[x] - a[x]) * (d[y] - a[y]) - (b[y] - a[y]) * (d[x] - a[x]);
 
        if ((abc > 0 && abd < 0) || (abc < 0 && abd > 0))
        {
            return true;
        }
    }
    return false;
}
 
bool isCoplanar(const MathLib::Point3d& a, const MathLib::Point3d& b,
                const MathLib::Point3d& c, const MathLib::Point3d& d)
{
    auto const pa = Eigen::Map<Eigen::Vector3d const>(a.getCoords());
    auto const pb = Eigen::Map<Eigen::Vector3d const>(b.getCoords());
    auto const pc = Eigen::Map<Eigen::Vector3d const>(c.getCoords());
    auto const pd = Eigen::Map<Eigen::Vector3d const>(d.getCoords());
 
    Eigen::Vector3d const ab = pb - pa;
    Eigen::Vector3d const ac = pc - pa;
    Eigen::Vector3d const ad = pd - pa;
 
    auto const eps_squared =
        std::pow(std::numeric_limits<double>::epsilon(), 2);
    if (ab.squaredNorm() < eps_squared || ac.squaredNorm() < eps_squared ||
        ad.squaredNorm() < eps_squared)
    {
        return true;
    }
 
    // In exact arithmetic <ac*ad^T, ab> should be zero
    // if all four points are coplanar.
    const double sqr_scalar_triple(std::pow(ac.cross(ad).dot(ab), 2));
    // Due to evaluating the above numerically some cancellation or rounding
    // can occur. For this reason a normalisation factor is introduced.
    const double normalisation_factor =
        (ab.squaredNorm() * ac.squaredNorm() * ad.squaredNorm());
 
    // tolerance 1e-11 is chosen such that
    // a = (0,0,0), b=(1,0,0), c=(0,1,0) and d=(1,1,1e-6) are considered as
    // coplanar
    // a = (0,0,0), b=(1,0,0), c=(0,1,0) and d=(1,1,1e-5) are considered as not
    // coplanar
    return (sqr_scalar_triple / normalisation_factor < 1e-11);
}
 
}  // end namespace MathLib