interpolator.h

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// © Copyright 2010 - 2014 BlackTopp Studios Inc.
/* This file is part of The Mezzanine Engine.

    The Mezzanine Engine is free software: you can redistribute it and/or modify
    it under the terms of the GNU General Public License as published by
    the Free Software Foundation, either version 3 of the License, or
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    GNU General Public License for more details.

    You should have received a copy of the GNU General Public License
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*/
/* The original authors have included a copy of the license specified above in the
   'Docs' folder. See 'gpl.txt'
*/
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   Joseph Toppi - toppij@gmail.com
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*/
#ifndef _interpolator_h
#define _interpolator_h

/// @file
/// @brief Helper classes to assist in generating data points between two other data points.

#include "datatypes.h"
#include "exception.h"
#include "cubicspline.h"

#ifndef SWIG // STD headers are bad for Swig
    #include <cmath>
    #include <iterator>

    #include <iostream>
    #include <typeinfo>
#endif

namespace Mezzanine
{

    // Forward declaration
    template <typename InterpolatableType, typename InterpolatorType> class TrackStorage;

    /// @brief A simple functor for interpolating data points in a simple way.
    /// @details This interpolator provides certain guarantees.
    ///     - All data points will be valid interpolated values.
    ///     - There are "corners".
    ///     - Interpolation will be constant/O(1) time (and as fast as reasonable).
    ///     - This shape defined by interpolating a set of these will not leave a Convex Hull(or Axis Aligned Bounding Box) that could contain the data
    /// Corners can be thought of as any non-smooth change, and may not be intuitively in some interpolatable
    /// types.
    template <typename T>
    class MEZZ_LIB LinearInterpolator
    {
        public:
            /// @brief The type this will interpolate
            /// @note All Interpolators need to declare an InterpolatableType
            typedef T InterpolatableType;
            /// @brief The storage to use with thison tracks
            typedef std::vector<InterpolatableType> Storage;

            /// @brief Get a value at a given location between exactly two others.
            /// @param Begin The data point at one end of line segment
            /// @param End  The data point at the other end of line segment
            /// @param Location A value between 0.0 and 1.0 indicate what point on the line segment defined by Begin and End you want.
            /// @return A Value equal to Begin if 0.0 is passed, equal to End if 1.0 is passed equal to a point exactly in the middle if 0.5 is passed.
            static T InterpolateMath(T Begin, T End, Real Location)
                { return ((End-Begin)*Location)+Begin; }

            /// @brief Handles Interpolation of multiple points.
            /// @details This will treat each data point as if it were equidistant from its
            /// neighbors and find the datasegment the desired point resides in. Then it will
            /// return a data point partway through that data segment. For example if you have
            /// three Vector2's defining two data segments as follows:
            /// @n 0,0 - 1,1 - 2,0
            /// @n @n
            /// Requesting the following locations would return the following data points:
            /// @n 0.0 = 0,0
            /// @n 0.5 = 1,1
            /// @n 1.0 = 2,0
            /// @n
            /// @n 0.25 = 0.5,0.5
            /// @n 0.75 = 1.5,0.5
            /// @n @n Any floating point location between 0 and 1 could be requested following
            /// this pattern.
            /// @n @n Should the data segments not actually be equal in size that they will each
            /// still be treated as an equal length when calcuting size. For example if you
            /// have 2 data segments like the previous example, then .25 will be halfway through
            /// the first segment and .75 will be halfwy through the second. If you are
            /// interpolating a series of points through segment like these the larger data
            /// segment will be traversed faster. For example consider the following Real value
            /// values as data points:
            /// @n -5, 0, 100
            /// Requesting the following locations would return the following data points:
            /// @n 0.0 = -5
            /// @n 0.5 = 0
            /// @n 1.0 = 100
            /// @n
            /// @n 0.25 = -2.5
            /// @n 0.75 = 50
            /// @n
            /// @n 0.0 = -5
            /// @n 0.1 = -4
            /// @n 0.2 = -3
            /// @n 0.3 = -2
            /// @n 0.4 = -1
            /// @n 0.5 = 0
            /// @n 0.6 = 20
            /// @n 0.7 = 40
            /// @n 0.8 = 60
            /// @n 0.9 = 80
            /// @n 1.0 = 100
            /// @n @n Note how even though the location increase by only 0.1 each step
            /// the resulting interpolated data point move relative to the length of the
            /// segment.
            /// @param Begin An iterator at the beginning of a rande of data point
            /// @param End An iterator one past the end of the data range to interpolate.
            /// @param Location A value between 0.0 and 1.0 that represents
            /// @return A T at a location along the data segments defined by Begin and End.
            template<typename TIterator>
            static T GetInterpolatedFromMultiple(TIterator Begin, TIterator End, Real Location)
            {
                Whole DataPointCount = std::distance(Begin,End);
                Whole LineSegmentCount = DataPointCount-1;

                if(LineSegmentCount<1)
                    { MEZZ_EXCEPTION(Exception::PARAMETERS_RANGE_EXCEPTION,"Cannot GetPercentageThroughSegment in GenericLinearInterpolator without a data segment. There must be two or more data points."); }
                if(1==LineSegmentCount)
                    { return InterpolateMath(*(Begin), *(Begin+1), Location); }

                Whole UsingLineSegment = Location * Real(LineSegmentCount); // Pick a Line Segment
                if(UsingLineSegment>=LineSegmentCount)                      // We are past or at the end of the line segments
                    { return *(End-1); }

                Real LocalPercentage = std::fmod(PreciseReal(Location), PreciseReal(1.0/PreciseReal(LineSegmentCount)))*LineSegmentCount;

                return InterpolateMath(*(Begin+UsingLineSegment),           // The first point of the line segment
                                       *(Begin+UsingLineSegment+1),
                                       LocalPercentage);                    // The percentage we are through this line segment
            }

            /// @brief This will interpolates data points with GetInterpolatedFromMultiple or InterpolateMath a required
            /// @details read about GetInterpolatedFromMultiple or InterpolateMath to see what kinds of results this
            /// can produce.
            /// @param Begin An iterator at the beginning of a range of data point
            /// @param End An iterator one past the end of the data range to interpolate.
            /// @param Location A value between 0.0 and 1.0 that represents
            /// @return A T at a location along the data segments defined by Begin and End.
            template<typename TIterator>
            static T Interpolate(TIterator Begin, TIterator End, Real Location)
            {
                if(Begin==End)
                    { MEZZ_EXCEPTION(Exception::PARAMETERS_RANGE_EXCEPTION,"Requires at least 2 data points for linear interpolation, was provided 0."); }
                if(Begin+1==End)
                    { MEZZ_EXCEPTION(Exception::PARAMETERS_RANGE_EXCEPTION,"Requires at least 2 data points for linear interpolation, was provided 1."); }

                if(Begin+2==End)
                    { return InterpolateMath(*Begin, *(Begin+1), Location); }

                return GetInterpolatedFromMultiple(Begin, End, Location);
            }
    };

    /// @brief A simple functor for interpolating data points in a simple way.
    /// @details This interpolator provides different guarantees different from the linear one:
    ///     - Data points, might not be valid interpolated values
    ///     - There are no "Corners".
    ///     - Execution could be as bad as O^N.
    ///     - This shape defined by interpolating a set of these will not leave a Convex Hull(or Axis Aligned Bounding Box) that could contain the data.
    ///     - Will be able to provide interpolated values for a small set of data points.
    /// There might be corners when connecting 2 different bezier curves if not careful, any
    /// bezier implementation taking a finite amount of points cannot help this.
    template <typename T>
    class MEZZ_LIB BezierInterpolator
    {
        public:
            /// @brief The type this will interpolate
            /// @note All Interpolators need to declare an InterpolatableType
            typedef T InterpolatableType;
            /// @brief The storage to use with thison tracks
            typedef std::vector<InterpolatableType> Storage;

            /// @brief Get a value at a given location between two others.
            /// @details This uses Linear interpolation recursively to produce a single curve
            /// following Bézier's curve algorithm. For example if interpolating the location 0.5
            /// on a set of 3 data points A,B,C and therefor 2 data segments AB and BC, you can imagine this as
            /// getting the point halfway down AB and the point halfway down down BC. Then this will
            /// get return the halfway between each of those points. This produces smooth curves but could
            /// perform slowly. For more details see the wikiedia pages on Bézier curves:
            /// http://en.wikipedia.org/wiki/Bézier_curve or http://en.wikipedia.org/wiki/B%C3%A9zier_curve
            /// @param Begin An Iterator to the begining of the range of the instances of the type to be Interpolated
            /// @param End The end (not one past the end) of the range that Begin started.
            /// @param A value between 0.0 and 1.0 indicate what point on the beziear spline defined by Begin and End you want.
            /// @return A Value equal or near to Begin if 0.0 is passed, equal to or near to End if 1.0 is passed equal to a point exactly in the middle if 0.5 is passed.
            /// @warning This is implemented as a recursive function with the only termination condition being the Edn iterator.
            template<typename TIterator>
            static T Interpolate(TIterator Begin, TIterator End, Real Location)
            {
                if(Begin==End || Begin+1==End)
                    { MEZZ_EXCEPTION(Exception::PARAMETERS_RANGE_EXCEPTION,"Requires at least 1 data points for bezier interpolation."); }

                if(Begin+2==End)
                    { return LinearInterpolator<T>::Interpolate(Begin,End,Location); }

                std::vector<T> SubSection;
                for(TIterator Iter=Begin; Iter!=End-1; Iter++)
                    { SubSection.push_back(LinearInterpolator<T>::Interpolate(Iter,(Iter+2),Location)); }
                return Interpolate(SubSection.begin(),SubSection.end(),Location);
            }
    };

    /// @brief If something specifically needs the linear interpolator for T they should use this.
    /// @details This is with be a cubic spline where applicable, and will be
    /// more smooth that the others, and be at least as intuitive as the linear version:
    ///     - Data points, will be valid interpolated values.
    ///     - There are no "Corners".
    ///     - Execution time is O^N.
    ///     - This shape defined by interpolating a set of these *will* leave a Convex Hull(or Axis Aligned Bounding Box) that could contain the data.
    ///     - Will be able to interpolated arbitrary sets of data points.
    template <typename T>
    class MEZZ_LIB SlowSplineInterpolator
    {
        public:
            /// @brief The type this will interpolate
            /// @note All Interpolators need to declare an InterpolatableType
            typedef T InterpolatableType;
            /// @brief The storage for type for a cubic spline
            typedef CubicSpline<Real,InterpolatableType> Storage;

            /// @brief Calculates the desired location on a cubic spline
            /// @param Begin An iterator to the beginning of points to interpolate
            /// @param End An iterator to the end of a data series, not one passed it.
            /// @param Location A value between 0 and 1, that indicate how far along the curve the data point to retrieve is
            /// @return This will return an value along a curve that passes smoothly through all the points passed.
            template<typename TIterator>
            static T Interpolate(TIterator Begin, TIterator End, Real Location)
            {
                /*std::vector<T> Points;
                std::cout << "Unknown Type            : " << typeid(TIterator).name() << std::endl
                          << "Vector<T>               : " << typeid(std::vector<T>).name() << std::endl
                          << "Vector<T>::iterator     : " << typeid(typename std::vector<T>::iterator).name() << std::endl
                          << "Storage                 : " << typeid(Storage).name() << std::endl
                          << "Storage::iterator       : " << typeid(typename Storage::iterator).name() << std::endl
                          << "Storage::const_iterator : " << typeid(typename Storage::const_iterator).name() << std::endl
                          << "Is the unknown type a Storage::const_iterator? " << (typeid(typename Storage::const_iterator)==typeid(TIterator)) << std::endl
                             ;
                Points.insert(Points.end(),Begin,End);// */
                std::vector<T> Points(Begin, End);
                std::vector<Real> Spacing(Points.size());
                Real JumpSize = 1/PreciseReal(Points.size()-1);
                Real CurrentJump=0;
                for(std::vector<Real>::iterator Iter=Spacing.begin();
                    Iter!=Spacing.end();
                    Iter++)
                {
                    *Iter=CurrentJump;
                    CurrentJump+=JumpSize;
                }

                CubicSpline<Real,T> Spliney(Spacing,Points);
                return Spliney.interpolate(Location);
            }
    };


} // /namespace Mezzanine

#endif // Include guard