Eigen/src/Geometry/Hyperplane.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2008 Benoit Jacob <jacob.benoit.1@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
// SPDX-License-Identifier: MPL-2.0

#ifndef EIGEN_HYPERPLANE_H
#define EIGEN_HYPERPLANE_H

// IWYU pragma: private
#include "./InternalHeaderCheck.h"

namespace Eigen {

/** \geometry_module \ingroup Geometry_Module
 *
 * \class Hyperplane
 *
 * \brief A hyperplane
 *
 * A hyperplane is an affine subspace of dimension n-1 in a space of dimension n.
 * For example, a hyperplane in a plane is a line; a hyperplane in 3-space is a plane.
 *
 * \tparam Scalar_ the scalar type, i.e., the type of the coefficients
 * \tparam AmbientDim_ the dimension of the ambient space, can be a compile time value or Dynamic.
 *             Notice that the dimension of the hyperplane is AmbientDim_-1.
 *
 * This class represents a hyperplane as the zero set of the implicit equation
 * \f$ n \cdot x + d = 0 \f$ where \f$ n \f$ is a unit normal vector of the plane (linear part)
 * and \f$ d \f$ is the distance (offset) to the origin.
 */
template <typename Scalar_, int AmbientDim_, int Options_>
class Hyperplane {
 public:
  EIGEN_MAKE_ALIGNED_OPERATOR_NEW_IF_VECTORIZABLE_FIXED_SIZE(Scalar_,
                                                             AmbientDim_ == Dynamic ? Dynamic : AmbientDim_ + 1)
  enum { AmbientDimAtCompileTime = AmbientDim_, Options = Options_ };
  using Scalar = Scalar_;
  using RealScalar = typename NumTraits<Scalar>::Real;
  using Index = Eigen::Index;  ///< \deprecated since Eigen 3.3
  using VectorType = Matrix<Scalar, AmbientDimAtCompileTime, 1>;
  using Coefficients =
      Matrix<Scalar, Index(AmbientDimAtCompileTime) == Dynamic ? Dynamic : Index(AmbientDimAtCompileTime) + 1, 1,
             Options>;
  using NormalReturnType = Block<Coefficients, AmbientDimAtCompileTime, 1>;
  using ConstNormalReturnType = const Block<const Coefficients, AmbientDimAtCompileTime, 1>;

  /** Default constructor without initialization */
  EIGEN_DEVICE_FUNC inline Hyperplane() {}

  template <int OtherOptions>
  EIGEN_DEVICE_FUNC Hyperplane(const Hyperplane<Scalar, AmbientDimAtCompileTime, OtherOptions>& other)
      : m_coeffs(other.coeffs()) {}

  /** Constructs a dynamic-size hyperplane with \a _dim the dimension
   * of the ambient space */
  EIGEN_DEVICE_FUNC inline explicit Hyperplane(Index _dim) : m_coeffs(_dim + 1) {}

  /** Construct a plane from its normal \a n and a point \a e onto the plane.
   * \warning the vector normal is assumed to be normalized.
   */
  EIGEN_DEVICE_FUNC inline Hyperplane(const VectorType& n, const VectorType& e) : m_coeffs(n.size() + 1) {
    normal() = n;
    offset() = -n.dot(e);
  }

  /** Constructs a plane from its normal \a n and distance to the origin \a d
   * such that the algebraic equation of the plane is \f$ n \cdot x + d = 0 \f$.
   * \warning the vector normal is assumed to be normalized.
   */
  EIGEN_DEVICE_FUNC inline Hyperplane(const VectorType& n, const Scalar& d) : m_coeffs(n.size() + 1) {
    normal() = n;
    offset() = d;
  }

  /** Constructs a hyperplane passing through the two points. If the dimension of the ambient space
   * is greater than 2, then there isn't uniqueness, so an arbitrary choice is made.
   */
  EIGEN_DEVICE_FUNC static inline Hyperplane Through(const VectorType& p0, const VectorType& p1) {
    Hyperplane result(p0.size());
    result.normal() = (p1 - p0).unitOrthogonal();
    result.offset() = -p0.dot(result.normal());
    return result;
  }

  /** Constructs a hyperplane passing through the three points. The dimension of the ambient space
   * is required to be exactly 3.
   */
  EIGEN_DEVICE_FUNC static inline Hyperplane Through(const VectorType& p0, const VectorType& p1, const VectorType& p2) {
    EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 3)
    Hyperplane result(p0.size());
    VectorType v0(p2 - p0), v1(p1 - p0);
    result.normal() = v0.cross(v1);
    RealScalar norm = result.normal().norm();
    if (norm <= v0.norm() * v1.norm() * NumTraits<RealScalar>::epsilon()) {
      Matrix<Scalar, 2, 3> m;
      m << v0.transpose(), v1.transpose();
      JacobiSVD<Matrix<Scalar, 2, 3>, ComputeFullV> svd(m);
      result.normal() = svd.matrixV().col(2);
    } else
      result.normal() /= norm;
    result.offset() = -p0.dot(result.normal());
    return result;
  }

  /** Constructs a hyperplane passing through the parametrized line \a parametrized.
   * If the dimension of the ambient space is greater than 2, then there isn't uniqueness,
   * so an arbitrary choice is made.
   */
  // FIXME: for consistency, consider implementing as a static Through function.
  EIGEN_DEVICE_FUNC explicit Hyperplane(const ParametrizedLine<Scalar, AmbientDimAtCompileTime>& parametrized)
      : m_coeffs(parametrized.dim() + 1) {
    normal() = parametrized.direction().unitOrthogonal();
    offset() = -parametrized.origin().dot(normal());
  }

  /** \returns the dimension in which the plane holds */
  EIGEN_DEVICE_FUNC inline Index dim() const {
    return AmbientDimAtCompileTime == Dynamic ? m_coeffs.size() - 1 : Index(AmbientDimAtCompileTime);
  }

  /** normalizes \c *this */
  EIGEN_DEVICE_FUNC void normalize(void) { m_coeffs /= normal().norm(); }

  /** \returns the signed distance between the plane \c *this and a point \a p.
   * \sa absDistance()
   */
  EIGEN_DEVICE_FUNC inline Scalar signedDistance(const VectorType& p) const { return normal().dot(p) + offset(); }

  /** \returns the absolute distance between the plane \c *this and a point \a p.
   * \sa signedDistance()
   */
  EIGEN_DEVICE_FUNC inline Scalar absDistance(const VectorType& p) const { return numext::abs(signedDistance(p)); }

  /** \returns the projection of a point \a p onto the plane \c *this.
   */
  EIGEN_DEVICE_FUNC inline VectorType projection(const VectorType& p) const { return p - signedDistance(p) * normal(); }

  /** \returns a constant reference to the unit normal vector of the plane, which corresponds
   * to the linear part of the implicit equation.
   */
  EIGEN_DEVICE_FUNC inline ConstNormalReturnType normal() const {
    return ConstNormalReturnType(m_coeffs, 0, 0, dim(), 1);
  }

  /** \returns a non-constant reference to the unit normal vector of the plane, which corresponds
   * to the linear part of the implicit equation.
   */
  EIGEN_DEVICE_FUNC inline NormalReturnType normal() { return NormalReturnType(m_coeffs, 0, 0, dim(), 1); }

  /** \returns the distance to the origin, which is also the "constant term" of the implicit equation
   * \warning the vector normal is assumed to be normalized.
   */
  EIGEN_DEVICE_FUNC inline const Scalar& offset() const { return m_coeffs.coeff(dim()); }

  /** \returns a non-constant reference to the distance to the origin, which is also the constant part
   * of the implicit equation */
  EIGEN_DEVICE_FUNC inline Scalar& offset() { return m_coeffs(dim()); }

  /** \returns a constant reference to the coefficients c_i of the plane equation:
   * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$
   */
  EIGEN_DEVICE_FUNC inline const Coefficients& coeffs() const { return m_coeffs; }

  /** \returns a non-constant reference to the coefficients c_i of the plane equation:
   * \f$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 \f$
   */
  EIGEN_DEVICE_FUNC inline Coefficients& coeffs() { return m_coeffs; }

  /** \returns the intersection of *this with \a other.
   *
   * \warning The ambient space must be a plane, i.e. have dimension 2, so that \c *this and \a other are lines.
   *
   * \note If \a other is approximately parallel to *this, this method will return any point on *this.
   */
  EIGEN_DEVICE_FUNC VectorType intersection(const Hyperplane& other) const {
    EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(VectorType, 2)
    Scalar det = coeffs().coeff(0) * other.coeffs().coeff(1) - coeffs().coeff(1) * other.coeffs().coeff(0);
    // since the line equations ax+by=c are normalized with a^2+b^2=1, the following tests
    // whether the two lines are approximately parallel.
    if (internal::isMuchSmallerThan(det, Scalar(1))) {  // special case where the two lines are approximately parallel.
                                                        // Pick any point on the first line.
      if (numext::abs(coeffs().coeff(1)) > numext::abs(coeffs().coeff(0)))
        return VectorType(coeffs().coeff(1), -coeffs().coeff(2) / coeffs().coeff(1) - coeffs().coeff(0));
      else
        return VectorType(-coeffs().coeff(2) / coeffs().coeff(0) - coeffs().coeff(1), coeffs().coeff(0));
    } else {  // general case
      Scalar invdet = Scalar(1) / det;
      return VectorType(
          invdet * (coeffs().coeff(1) * other.coeffs().coeff(2) - other.coeffs().coeff(1) * coeffs().coeff(2)),
          invdet * (other.coeffs().coeff(0) * coeffs().coeff(2) - coeffs().coeff(0) * other.coeffs().coeff(2)));
    }
  }

  /** Applies the transformation matrix \a mat to \c *this and returns a reference to \c *this.
   *
   * \param mat the Dim x Dim transformation matrix
   * \param traits specifies whether the matrix \a mat represents an #Isometry
   *               or a more generic #Affine transformation. The default is #Affine.
   */
  template <typename XprType>
  EIGEN_DEVICE_FUNC inline Hyperplane& transform(const MatrixBase<XprType>& mat, TransformTraits traits = Affine) {
    if (traits == Affine) {
      normal() = mat.inverse().adjoint() * normal();
      m_coeffs /= normal().norm();
    } else if (traits == Isometry)
      normal() = mat * normal();
    else {
      eigen_assert(0 && "invalid traits value in Hyperplane::transform()");
    }
    return *this;
  }

  /** Applies the transformation \a t to \c *this and returns a reference to \c *this.
   *
   * \param t the transformation of dimension Dim
   * \param traits specifies whether the transformation \a t represents an #Isometry
   *               or a more generic #Affine transformation. The default is #Affine.
   *               Other kind of transformations are not supported.
   */
  template <int TrOptions>
  EIGEN_DEVICE_FUNC inline Hyperplane& transform(const Transform<Scalar, AmbientDimAtCompileTime, Affine, TrOptions>& t,
                                                 TransformTraits traits = Affine) {
    transform(t.linear(), traits);
    offset() -= normal().dot(t.translation());
    return *this;
  }

  /** \returns \c *this with scalar type casted to \a NewScalarType
   *
   * Note that if \a NewScalarType is equal to the current scalar type of \c *this
   * then this function smartly returns a const reference to \c *this.
   */
  template <typename NewScalarType>
  EIGEN_DEVICE_FUNC inline
      typename internal::cast_return_type<Hyperplane,
                                          Hyperplane<NewScalarType, AmbientDimAtCompileTime, Options> >::type
      cast() const {
    return
        typename internal::cast_return_type<Hyperplane,
                                            Hyperplane<NewScalarType, AmbientDimAtCompileTime, Options> >::type(*this);
  }

  /** Copy constructor with scalar type conversion */
  template <typename OtherScalarType, int OtherOptions>
  EIGEN_DEVICE_FUNC inline explicit Hyperplane(
      const Hyperplane<OtherScalarType, AmbientDimAtCompileTime, OtherOptions>& other) {
    m_coeffs = other.coeffs().template cast<Scalar>();
  }

  /** \returns \c true if \c *this is approximately equal to \a other, within the precision
   * determined by \a prec.
   *
   * Hyperplanes are oriented: the sign of their coefficients decides which side signedDistance()
   * reports as positive, so a hyperplane and its negation are not approximately equal here even
   * though they describe the same point set. Use isCoincident() to compare them as point sets.
   *
   * \sa isCoincident(), MatrixBase::isApprox() */
  template <int OtherOptions>
  EIGEN_DEVICE_FUNC bool isApprox(
      const Hyperplane<Scalar, AmbientDimAtCompileTime, OtherOptions>& other,
      const typename NumTraits<Scalar>::Real& prec = NumTraits<Scalar>::dummy_precision()) const {
    return m_coeffs.isApprox(other.coeffs(), prec);
  }

  /** \returns \c true if \c *this and \a other describe approximately the same set of points,
   * within the precision determined by \a prec, regardless of orientation and scale.
   *
   * Scaling the equation signedDistance() evaluates by any nonzero \f$ \gamma \f$ leaves its zero
   * set unchanged. Because dot() is conjugate-linear in the normal, that carries
   * \f$ (n, d) \f$ to \f$ (\bar{\gamma} n, \gamma d) \f$, so coincident hyperplanes need not have
   * coefficients of equal magnitude: for a real \c Scalar any nonzero real factor relates them, a
   * sign flip - which isApprox() rejects - being the norm-preserving case, and for a complex
   * \c Scalar any nonzero complex factor does.
   *
   * The comparison is therefore made on the normalized equations, which \f$ \gamma \f$ no longer
   * distinguishes beyond a unit-modulus factor: the unit normals must agree, up to that factor,
   * within \a prec, and the two distances to the origin must agree the way internal::isApprox()
   * compares scalars, that is relative to their own magnitude. Comparing the two halves separately
   * is what keeps a distant hyperplane from relaxing the comparison of the normals; it also means
   * that a hyperplane through the origin is not coincident with one that merely passes close to it.
   * The result is symmetric in the two hyperplanes, and unchanged both when either equation is
   * rescaled and when the ambient coordinates are.
   *
   * Both normals must be nonzero.
   *
   * \sa isApprox(), signedDistance(), normalize() */
  template <int OtherOptions>
  EIGEN_DEVICE_FUNC bool isCoincident(
      const Hyperplane<Scalar, AmbientDimAtCompileTime, OtherOptions>& other,
      const typename NumTraits<Scalar>::Real& prec = NumTraits<Scalar>::dummy_precision()) const {
    const RealScalar this_norm = normal().norm();
    const RealScalar other_norm = other.normal().norm();
    eigen_assert(this_norm > RealScalar(0) && other_norm > RealScalar(0));
    // Normalizing accounts for |gamma|; what is left of it is a unit-modulus factor, and the
    // normals themselves determine it, since the inner product of two coincident unit normals is
    // exactly that factor. Orthogonal normals leave it undetermined, but no factor makes those
    // agree either, so the choice made for them does not matter.
    const Scalar inner = normal().dot(other.normal());
    const RealScalar inner_norm = numext::abs(inner);
    const Scalar phase = inner_norm > RealScalar(0) ? inner / Scalar(inner_norm) : Scalar(1);
    if (!((other.normal() / Scalar(other_norm) - phase * (normal() / Scalar(this_norm))).norm() <= prec)) return false;
    return internal::isApprox(other.offset() / Scalar(other_norm), numext::conj(phase) * (offset() / Scalar(this_norm)),
                              prec);
  }

 protected:
  Coefficients m_coeffs;
};

}  // end namespace Eigen

#endif  // EIGEN_HYPERPLANE_H