libcruft-util/geom/ellipse.hpp

/*
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *     http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 *
 * Copyright 2015-2018 Danny Robson <danny@nerdcruft.net>
 */

#ifndef __UTIL_GEOM_ELLIPSE_HPP
#define __UTIL_GEOM_ELLIPSE_HPP

#include "fwd.hpp"

#include "../view.hpp"
#include "../point.hpp"
#include "../vector.hpp"

#include <cstdlib>
#include <random>
#include <iosfwd>


namespace util::geom {
    ///////////////////////////////////////////////////////////////////////////
    template <size_t S, typename ValueT>
    struct ellipse {
        // the centre point of the ellipsoid
        util::point<S,ValueT> origin;

        // the distance from the centre along each axis to the shape's edge
        util::vector<S,ValueT> radius;

        // the orientation of up for the shape
        util::vector<S,ValueT> up;
    };

    using ellipse2f = ellipse<2,float>;
    using ellipse3f = ellipse<3,float>;


    template <size_t S, typename T>
    bool
    intersects (ellipse<S,T>, point<S,T>);


    /// returns the approximate surface area of the ellipse
    ///
    /// the general form involves _substantially_ more expensive and
    /// complicated maths which is prohibitive right now.
    ///
    /// the relative error should be at most 1.061%
    inline float
    area (ellipse3f self)
    {
        auto const semiprod = self.radius * self.radius.indices<1,2,0> ();
        auto const semipow  = pow (semiprod, 1.6f);

        return 4 * pi<float> * std::pow (sum (semipow) / 3, 1/1.6f);
    }


    template <typename T>
    T
    area (ellipse<2,T> self)
    {
        return pi<T> * product (self.radius);
    }


    template <size_t S, typename T>
    T
    volume (ellipse<S,T> self)
    {
        return 4 / T{3} * pi<T> * product (self.radius);
    }


    template <size_t DimensionV, typename ValueT>
    point<DimensionV,ValueT>
    project (
        ray<DimensionV,ValueT>,
        ellipse<DimensionV,ValueT>
    );


    /// returns the distance along a ray to the surface of an ellipse.
    ///
    /// returns infinity if there is no intersection
    template <size_t DimensionV, typename ValueT>
    ValueT
    distance (
        ray<DimensionV,ValueT>,
        ellipse<DimensionV,ValueT>
    );


    // generate a covering ellipsoid for an arbitrary set of points
    //
    // this isn't guaranteed to be optimal in any specific sense. but it
    // ought not be outrageously inefficient...
    ellipse3f
    cover (util::view<const point3f*>);


    /// returns a point that is uniformly distributed about the ellipse
    /// surface.
    ///
    /// NOTE: I don't have a strong proof that the below is in fact properly
    /// uniformly distributed, so if you need a strong guarantee for your work
    /// then it might not be the best option. But visual inspection appears to
    /// confirm there aren't obvious patterns.
    ///
    /// the concept was taken from: https://math.stackexchange.com/a/2514182
    template <typename RandomT>
    util::point3f
    sample_surface (ellipse3f self, RandomT &generator)
    {
        const auto [a, b, c] = self.radius;
        const auto a2 = a * a;
        const auto b2 = b * b;
        const auto c2 = c * c;

        // generate a direction vector from a normally distributed random variable
        auto const x = std::normal_distribution<float> (0, a2) (generator);
        auto const y = std::normal_distribution<float> (0, b2) (generator);
        auto const z = std::normal_distribution<float> (0, c2) (generator);

        // find the distance to the surface along the direction vector
        auto const d = std::sqrt (x * x / a2 + y * y / b2 + z * z / c2);

        return self.origin + util::vector3f {x,y,z} / d;
    }


    template <size_t S, typename T>
    std::ostream&
    operator<< (std::ostream&, ellipse<S,T>);
}


///////////////////////////////////////////////////////////////////////////////
#include "sample.hpp"

#include <cmath>
#include <random>

namespace util::geom {
    template <typename T, template <size_t,typename> class K, typename G>
    struct sampler<2,T,K,G>
    {
        static util::point<2,T> fn (K<2,T> k, G &g)
        {
            std::uniform_real_distribution<T> dist;

            float phi = dist (g) * 2 * pi<T>;
            float rho = std::sqrt (dist (g));

            return util::point<2,T> {
                std::cos (phi),
                std::sin (phi)
            } * rho * k.radius + k.origin.template as<util::vector> ();
        }
    };
}

#endif