sphere_area.hpp

#ifndef PASTELGEOMETRY_SPHERE_AREA_HPP
#define PASTELGEOMETRY_SPHERE_AREA_HPP

#include "pastel/geometry/area/sphere_area.h"
#include "pastel/geometry/volume/sphere_volume.h"

#include <cmath>

namespace Pastel
{

    template <typename Real>
    Real lnAreaUnitSphere(integer n)
    {
        PENSURE_OP(n, >, 0);

        // See areaUnitSphere() for the derivations.

        // log(area(S))
        // = log(n * volume(S))
        // = log(n) + log(volume(S))

        if (n == 1)
        {
            // log(0) = -infinity
            return -Infinity();
        }

        return std::log((Real)n) + lnVolumeUnitSphere<Real>(n);
    }

    template <typename Real>
    Real areaUnitSphere(integer n)
    {
        PENSURE_OP(n, >, 0);

        if (n == 1)
        {
            // The surface area of the 1-dimensional
            // disk is zero. This must be taken as
            // a special case, since the general
            // formula below does not return 0 for
            // dimension 1.
            return 0;
        }

        // The volume of an R-radius n-ball is of the form
        // 
        //     V_n(R) = C_n R^n
        // 
        // The area of the n-ball is related to its volume by
        // 
        //     A_n(R) = (d/dR) V_n(R)
        //            = n C_n R^{n - 1},
        //
        // except when n = 1.

        return n * volumeUnitSphere<Real>(n);
    }

    template <typename Real, integer N>
    Real area(const Sphere<Real, N>& sphere)
    {
        return areaUnitSphere<Real>(sphere.n()) * 
            std::pow(sphere.radius(), (Real)(sphere.n() - 1));
    }

}

#endif