closest_line_point.h

// Description: Closest points between a line and a point

#ifndef PASTELGEOMETRY_CLOSEST_LINE_POINT_H
#define PASTELGEOMETRY_CLOSEST_LINE_POINT_H

#include "pastel/geometry/shape/line.h"

#include "pastel/sys/vector.h"

namespace Pastel
{

    //! Computes closest points between a line and a point.
    /*!
   Returns:
   A parameter t such that 'line.at(t)' and 'point' are closest points.
   */
    template <typename Real, int N>
    Real closest(
        const Line<Real, N>& line,
        const Vector<Real, N>& point)
    {
        // Parametrize the line by:
        // L(t) = P + tD
        //
        // The closest point C on the line L
        // to the given point X is given
        // by the condition that vector (X - C) must
        // be orthogonal to D.
        //
        // dot(X - C, D) = 0
        // => dot(X - (P + tD), D) = 0
        // => dot(X - P, D) - t * dot(D, D) = 0
        // => t = dot(X - P, D) / dot(D, D)
        //
        // Because D is a unit vector,
        // t = dot(X - P, D)

        return dot(point - line.position(),
            line.direction());
    }

}

#endif