triangle_quadrature.h

#pragma once

#include <functional>
#include <vector>

#include "drake/common/eigen_types.h"
#include "drake/multibody/triangle_quadrature/triangle_quadrature_rule.h"

namespace drake {
namespace multibody {

/// A class for integrating a function using numerical quadrature over
/// triangular domains.
///
/// @tparam NumericReturnType the output type of the function being integrated.
///         Commonly will be a IEEE floating point scalar (e.g., `double`), but
///         could be an Eigen::VectorXd, a multibody::SpatialForce, or any
///         other numeric type that supports both scalar multiplication
///         (i.e., operator*(const NumericReturnType&, double) and addition with
///         another of the same type (i.e.,
///         operator+(const NumericReturnType&, const NumericReturnType&)).
/// @tparam T the scalar type of the function being integrated over. Supported
///         types are currently only IEEE floating point scalars.
template <typename NumericReturnType, typename T>
class TriangleQuadrature {
 public:
  /// Numerically integrates the function f over a triangle using the given
  /// quadrature rule and the initial value.
  /// @param f(p) a function that returns a numerical value for point p in the
  ///        domain of the triangle, specified in barycentric coordinates.
  ///        The barycentric coordinates are given by
  ///        (p[0], p[1], 1 - p[0] - p[1]).
  /// @param area the area of the triangle.
  static NumericReturnType Integrate(
      const std::function<NumericReturnType(const Vector2<T>&)>& f,
      const TriangleQuadratureRule& rule,
      const T& area);

  /// Alternative signature for Integrate() that uses three-dimensional
  /// barycentric coordinates.
  /// @param f(p) a function that returns a numerical value for point p in the
  ///        domain of the triangle, specified in barycentric coordinates.
  ///        The barycentric coordinates are given by
  ///        (p[0], p[1], p[2]).
  /// @param area the area of the triangle.
  static NumericReturnType Integrate(
      const std::function<NumericReturnType(const Vector3<T>&)>& f,
      const TriangleQuadratureRule& rule,
      const T& area);
};

template <typename NumericReturnType, typename T>
NumericReturnType TriangleQuadrature<NumericReturnType, T>::Integrate(
    const std::function<NumericReturnType(const Vector2<T>&)>& f,
    const TriangleQuadratureRule& rule,
    const T& area) {
  // Get the quadrature points and weights.
  const std::vector<Eigen::Vector2d>& barycentric_coordinates =
      rule.quadrature_points();
  const std::vector<double>& weights = rule.weights();
  DRAKE_DEMAND(barycentric_coordinates.size() == weights.size());
  DRAKE_DEMAND(weights.size() >= 1);

  // Sum the weighted function evaluated at the transformed quadrature points.
  // The looping is done in this particular way so that the return type can
  // be either a traditional scalar (e.g., `double`), an Eigen Vector, or
  // some other numeric type, without having to worry about the numerous
  // possible ways to initialize to zero (e.g., `double integral = 0.0` vs.
  // `Eigen::VectorXd = Eigen::VectorXd::Zero())` or having to know the
  // dimension of the NumericReturnType (i.e., scalar or vector dimension).
  NumericReturnType integral = f(barycentric_coordinates[0]) * weights[0];
  for (int i = 1; i < static_cast<int>(weights.size()); ++i)
    integral += f(barycentric_coordinates[i]) * weights[i];

  return integral * area;
}

template <typename NumericReturnType, typename T>
NumericReturnType TriangleQuadrature<NumericReturnType, T>::Integrate(
    const std::function<NumericReturnType(const Vector3<T>&)>& f,
    const TriangleQuadratureRule& rule,
    const T& area) {
  // TODO(edrumwri) These composite functions might be slowing computation.
  //     Consider optimizing this.

  // Create a function fprime accepting a 2D barycentric representation but
  // using it to call the given 3D function f.
  std::function<NumericReturnType(const Vector2<T>&)> fprime = [&f](
      const Vector2<T>& barycentric_2D) {
    return f(Vector3<T>(
        barycentric_2D[0],
        barycentric_2D[1],
        T(1.0) - barycentric_2D[0] - barycentric_2D[1]));
  };

  return Integrate(fprime, rule, area);
}

}  // namespace multibody
}  // namespace drake