distance_plane_point.h

// Description: Distance between a plane and a point

#ifndef PASTELGEOMETRY_DISTANCE_PLANE_POINT_H
#define PASTELGEOMETRY_DISTANCE_PLANE_POINT_H

#include "pastel/geometry/shape/plane.h"
#include "pastel/sys/vector.h"
#include "pastel/sys/vector/vector_tools.h"
#include "pastel/sys/math_functions.h"
#include "pastel/math/norm/euclidean_norm.h"

namespace Pastel
{

    //! Euclidean distance between a plane and a point.
    /*!
   Preconditions:
   plane.n() == point.n()

   Time complexity:
   O(plane.n())
   */
    template <typename Real, int N>
    auto distance2(
        const Plane<Real, N>& plane,
        const Vector<Real, N>& point)
    {
        PENSURE_OP(plane.n(), ==, point.n());

        // Let
        // D = plane.position() - point
        //
        // The length of the projection of D onto the
        // plane normal is
        // 
        // |D_proj| 
        // = |D| cos(alpha))
        // = (|plane.normal()| |D| cos(alpha)) / |plane.normal()|
        // = dot(plane.normal(), D) / |plane.normal()|
        //
        // where alpha is the angle between the plane normal
        // and D.
        //
        // It then follows that:
        // |D_proj|^2 = dot^2(plane.normal(), D) / dot(plane.normal())

        auto norm = Euclidean_Norm<Real>();
        return norm[
            square(
                dot(plane.normal(), 
                plane.position() - point)
            ) / dot(plane.normal())
        ];
    }

}

#endif