psimpl.h

/* ***** BEGIN LICENSE BLOCK *****
 * Version: MPL 1.1
 *
 * The contents of this file are subject to the Mozilla Public License Version
 * 1.1 (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.mozilla.org/MPL/
 *
 * Software distributed under the License is distributed on an "AS IS" basis,
 * WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
 * for the specific language governing rights and limitations under the
 * License.
 *
 * The Original Code is
 * 'psimpl - generic n-dimensional polyline simplification'.
 *
 * The Initial Developer of the Original Code is
 * Elmar de Koning.
 * Portions created by the Initial Developer are Copyright (C) 2010-2011
 * the Initial Developer. All Rights Reserved.
 *
 * Contributor(s):
 *
 * ***** END LICENSE BLOCK ***** */

/*
    psimpl - generic n-dimensional polyline simplification
    Copyright (C) 2010-2011 Elmar de Koning, edekoning@gmail.com

    This file is part of psimpl, and is hosted at SourceForge:
    http://sourceforge.net/projects/psimpl/
*/

/*!
    \mainpage psimpl - generic n-dimensional polyline simplification

<pre>
    Author  - Elmar de Koning
    Support - edekoning@gmail.com
    Website - http://psimpl.sf.net
    Article - http://www.codeproject.com/KB/recipes/PolylineSimplification.aspx
    License - MPL 1.1
</pre><br>

    \section sec_psimpl psimpl
<pre>
    'psimpl' is a c++ polyline simplification library that is generic, easy to use, and supports
    the following algorithms:

    Simplification
    + Nth point - A naive algorithm that keeps only each nth point
    + Distance between points - Removes successive points that are clustered together
    + Perpendicular distance - Removes points based on their distance to the line segment defined
      by their left and right neighbors
    + Reumann-Witkam - Shifts a strip along the polyline and removes points that fall outside
    + Opheim - A constrained version of Reumann-Witkam
    + Lang - Similar to the Perpendicular distance routine, but instead of looking only at direct
      neighbors, an entire search region is processed
    + Douglas-Peucker - A classic simplification algorithm that provides an excellent approximation
      of the original line
    + A variation on the Douglas-Peucker algorithm - Slower, but yields better results at lower resolutions

    Errors
    + positional error - Distance of each polyline point to its simplification

    All the algorithms have been implemented in a single standalone C++ header using an STL-style
    interface that operates on input and output iterators. Polylines can be of any dimension, and
    defined using floating point or signed integer data types.
</pre><br>
    
    \section sec_changelog changelog
<pre>
    28-09-2010 - Initial version
    23-10-2010 - Changed license from CPOL to MPL
    26-10-2010 - Clarified input (type) requirements, and changed the behavior of the algorithms
                 under invalid input
    01-12-2010 - Added the nth point, perpendicular distance and Reumann-Witkam routines; moved all
                 functions related to distance calculations to the math namespace
    10-12-2010 - Fixed a bug in the perpendicular distance routine
    27-02-2011 - Added Opheim simplification, and functions for computing positional errors due to
                 simplification; renamed simplify_douglas_peucker_alt to simplify_douglas_peucker_n
    18-06-2011 - Added Lang simplification; fixed divide by zero bug when using integers; fixed a
                 bug where incorrect output iterators were returned under invalid input; fixed a bug
                 in douglas_peucker_n where an incorrect number of points could be returned; fixed a
                 bug in compute_positional_errors2 that required the output and input iterator types
                 to be the same; fixed a bug in compute_positional_error_statistics where invalid
                 statistics could be returned under questionable input; documented input iterator
                 requirements for each algorithm; miscellaneous refactoring of most algorithms.
</pre>
*/

#ifndef PSIMPL_GENERIC
#define PSIMPL_GENERIC


#include <queue>
#include <stack>
#include <numeric>
#include <algorithm>
#include <cmath>


/*!
    \brief Root namespace of the polyline simplification library.
*/
namespace psimpl
{
    /*!
        \brief Contains utility functions and classes.
    */
    namespace util
    {
        /*!
            \brief A smart pointer for holding a dynamically allocated array.

            Similar to boost::scoped_array.
        */
        template <typename T>
        class scoped_array
        {
        public:
            scoped_array (unsigned n) {
                array = new T [n];
            }

            ~scoped_array () {
                delete [] array;
            }

            T& operator [] (int offset) {
                return array [offset];
            }

            const T& operator [] (int offset) const {
                return array [offset];
            }

            T* get () const {
                return array;
            }

            void swap (scoped_array& b) {
                T* tmp = b.array;
                b.array = array;
                array = tmp;
            }

        private:
            scoped_array (const scoped_array&);
            scoped_array& operator= (const scoped_array&);

        private:
            T* array;
        };

        template <typename T> inline void swap (scoped_array <T>& a, scoped_array <T>& b) {
            a.swap (b);
        }
    }

    /*!
        \brief Contains functions for calculating statistics and distances between various geometric entities.
    */
    namespace math
    {
        /*!
            \brief POD structure for storing several statistical values
        */
        struct Statistics
        {
            Statistics () :
                max (0),
                sum (0),
                mean (0),
                std (0)
            {}

            double max;
            double sum;
            double mean;
            double std;     //! standard deviation
        };

        /*!
            \brief Determines if two points have the exact same coordinates.

            \param[in] p1       the first coordinate of the first point
            \param[in] p2       the first coordinate of the second point
            \return             true when the points are equal; false otherwise
        */
        template <unsigned DIM, class InputIterator>
        inline bool equal (
            InputIterator p1,
            InputIterator p2)
        {
            for (unsigned d = 0; d < DIM; ++d) {
                if (*p1 != *p2) {
                    return false;
                }
                ++p1;
                ++p2;
            }
            return true;
        }

        /*!
            \brief Creates a vector from two points.

            \param[in] p1       the first coordinate of the first point
            \param[in] p2       the first coordinate of the second point
            \param[in] result   the resulting vector (p2-p1)
            \return             one beyond the last coordinate of the resulting vector
        */
        template <unsigned DIM, class InputIterator, class OutputIterator>
        inline OutputIterator make_vector (
            InputIterator p1,
            InputIterator p2,
            OutputIterator result)
        {
            for (unsigned d = 0; d < DIM; ++d) {
                *result = *p2 - *p1;
                ++result;
                ++p1;
                ++p2;
            }
            return result;
        }

        /*!
            \brief Computes the dot product of two vectors.

            \param[in] v1   the first coordinate of the first vector
            \param[in] v2   the first coordinate of the second vector
            \return         the dot product (v1 * v2)
        */
        template <unsigned DIM, class InputIterator>
        inline typename std::iterator_traits <InputIterator>::value_type dot (
            InputIterator v1,
            InputIterator v2)
        {
            typename std::iterator_traits <InputIterator>::value_type result = 0;
            for (unsigned d = 0; d < DIM; ++d) {
                result += (*v1) * (*v2);
                ++v1;
                ++v2;
            }
            return result;
        }

        /*!
            \brief Peforms linear interpolation between two points.

            \param[in] p1           the first coordinate of the first point
            \param[in] p2           the first coordinate of the second point
            \param[in] fraction     the fraction used during interpolation
            \param[in] result       the interpolation result (p1 + fraction * (p2 - p1))
            \return                 one beyond the last coordinate of the interpolated point
        */
        template <unsigned DIM, class InputIterator, class OutputIterator>
        inline OutputIterator interpolate (
            InputIterator p1,
            InputIterator p2,
            float fraction,
            OutputIterator result)
        {
            typedef typename std::iterator_traits <InputIterator>::value_type value_type;

            for (unsigned d = 0; d < DIM; ++d) {
                *result = *p1 + static_cast <value_type> (fraction * (*p2 - *p1));
                ++result;
                ++p1;
                ++p2;
            }
            return result;
        }

        /*!
            \brief Computes the squared distance of two points

            \param[in] p1   the first coordinate of the first point
            \param[in] p2   the first coordinate of the second point
            \return         the squared distance
        */
        template <unsigned DIM, class InputIterator1, class InputIterator2>
        inline typename std::iterator_traits <InputIterator1>::value_type point_distance2 (
            InputIterator1 p1,
            InputIterator2 p2)
        {
            typename std::iterator_traits <InputIterator1>::value_type result = 0;
            for (unsigned d = 0; d < DIM; ++d) {
                result += (*p1 - *p2) * (*p1 - *p2);
                ++p1;
                ++p2;
            }
            return result;
        }

        /*!
            \brief Computes the squared distance between an infinite line (l1, l2) and a point p

            \param[in] l1   the first coordinate of the first point on the line
            \param[in] l2   the first coordinate of the second point on the line
            \param[in] p    the first coordinate of the test point
            \return         the squared distance
        */
        template <unsigned DIM, class InputIterator>
        inline typename std::iterator_traits <InputIterator>::value_type line_distance2 (
            InputIterator l1,
            InputIterator l2,
            InputIterator p)
        {
            typedef typename std::iterator_traits <InputIterator>::value_type value_type;

            value_type v [DIM];                 // vector l1 --> l2
            value_type w [DIM];                 // vector l1 --> p

            make_vector <DIM> (l1, l2, v);
            make_vector <DIM> (l1, p,  w);

            value_type cv = dot <DIM> (v, v);   // squared length of v
            value_type cw = dot <DIM> (w, v);   // project w onto v

            // avoid problems with divisions when value_type is an integer type
            float fraction = cv == 0 ? 0 : static_cast <float> (cw) / static_cast <float> (cv);

            value_type proj [DIM];              // p projected onto line (l1, l2)
            interpolate <DIM> (l1, l2, fraction, proj);

            return point_distance2 <DIM> (p, proj);
        }

        /*!
            \brief Computes the squared distance between a line segment (s1, s2) and a point p

            \param[in] s1   the first coordinate of the start point of the segment
            \param[in] s2   the first coordinate of the end point of the segment
            \param[in] p    the first coordinate of the test point
            \return         the squared distance
        */
        template <unsigned DIM, class InputIterator>
        inline typename std::iterator_traits <InputIterator>::value_type segment_distance2 (
            InputIterator s1,
            InputIterator s2,
            InputIterator p)
        {
            typedef typename std::iterator_traits <InputIterator>::value_type value_type;

            value_type v [DIM];        // vector s1 --> s2
            value_type w [DIM];        // vector s1 --> p

            make_vector <DIM> (s1, s2, v);
            make_vector <DIM> (s1, p,  w);

            value_type cw = dot <DIM> (w, v);   // project w onto v
            if (cw <= 0) {
                // projection of w lies to the left of s1
                return point_distance2 <DIM> (p, s1);
            }

            value_type cv = dot <DIM> (v, v);   // squared length of v
            if (cv <= cw) {
                // projection of w lies to the right of s2
                return point_distance2 <DIM> (p, s2);
            }

            // avoid problems with divisions when value_type is an integer type
            float fraction = cv == 0 ? 0 : static_cast <float> (cw) / static_cast <float> (cv);

            value_type proj [DIM];    // p projected onto segement (s1, s2)
            interpolate <DIM> (s1, s2, fraction, proj);

            return point_distance2 <DIM> (p, proj);
        }

        /*!
            \brief Computes the squared distance between a ray (r1, r2) and a point p

            \param[in] r1   the first coordinate of the start point of the ray
            \param[in] r2   the first coordinate of a point on the ray
            \param[in] p    the first coordinate of the test point
            \return         the squared distance
        */
        template <unsigned DIM, class InputIterator>
        inline typename std::iterator_traits <InputIterator>::value_type ray_distance2 (
            InputIterator r1,
            InputIterator r2,
            InputIterator p)
        {
            typedef typename std::iterator_traits <InputIterator>::value_type value_type;

            value_type v [DIM];        // vector r1 --> r2
            value_type w [DIM];        // vector r1 --> p

            make_vector <DIM> (r1, r2, v);
            make_vector <DIM> (r1, p,  w);

            value_type cv = dot <DIM> (v, v);    // squared length of v
            value_type cw = dot <DIM> (w, v);    // project w onto v

            if (cw <= 0) {
                // projection of w lies to the left of r1 (not on the ray)
                return point_distance2 <DIM> (p, r1);
            }

            // avoid problems with divisions when value_type is an integer type
            float fraction = cv == 0 ? 0 : static_cast <float> (cw) / static_cast <float> (cv);

            value_type proj [DIM];    // p projected onto ray (r1, r2)
            interpolate <DIM> (r1, r2, fraction, proj);

            return point_distance2 <DIM> (p, proj);
        }

        /*!
            \brief Computes various statistics for the range [first, last)

            \param[in] first   the first value
            \param[in] last    one beyond the last value
            \return            the calculated statistics
        */
        template <class InputIterator>
        inline Statistics compute_statistics (
            InputIterator first,
            InputIterator last)
        {
            typedef typename std::iterator_traits <InputIterator>::value_type value_type;
            typedef typename std::iterator_traits <InputIterator>::difference_type diff_type;

            Statistics stats;

            diff_type count = std::distance (first, last);
            if (count == 0) {
                return stats;
            }

            value_type init = 0;
            stats.max = static_cast <double> (*std::max_element (first, last));
            stats.sum = static_cast <double> (std::accumulate (first, last, init));
            stats.mean = stats.sum / count;
            std::transform (first, last, first, std::bind2nd (std::minus <value_type> (), stats.mean));
            stats.std = std::sqrt (static_cast <double> (std::inner_product (first, last, first, init)) / count);
            return stats;
        }
    }

    /*!
        \brief Provides various simplification algorithms for n-dimensional simple polylines.

        A polyline is simple when it is non-closed and non-selfintersecting. All algorithms
        operate on input iterators and output iterators. Note that unisgned integer types are
        NOT supported.
    */
    template <unsigned DIM, class InputIterator, class OutputIterator>
    class PolylineSimplification
    {
        typedef typename std::iterator_traits <InputIterator>::difference_type diff_type;
        typedef typename std::iterator_traits <InputIterator>::value_type value_type;
        typedef typename std::iterator_traits <const value_type*>::difference_type ptr_diff_type;

    public:
        /*!
            \brief Performs the nth point routine (NP).

            NP is an O(n) algorithm for polyline simplification. It keeps only the first, last and
            each nth point. As an example, consider any random line of 8 points. Using n = 3 will
            always yield a simplification consisting of points: 1, 4, 7, 8

            \image html psimpl_np.png

            NP is applied to the range [first, last). The resulting simplified polyline is copied
            to the output range [result, result + m*DIM), where m is the number of vertices of the
            simplified polyline. The return value is the end of the output range: result + m*DIM.

            Input (Type) requirements:
            1- DIM is not 0, where DIM represents the dimension of the polyline
            2- The InputIterator type models the concept of a forward iterator
            3- The input iterator value type is convertible to a value type of the OutputIterator
            4- The range [first, last) contains only vertex coordinates in multiples of DIM, f.e.:
               x, y, z, x, y, z, x, y, z when DIM = 3
            5- The range [first, last) contains at least 2 vertices
            6- n is not 0

            In case these requirements are not met, the entire input range [first, last) is copied
            to the output range [result, result + (last - first)) OR compile errors may occur.

            \param[in] first    the first coordinate of the first polyline point
            \param[in] last     one beyond the last coordinate of the last polyline point
            \param[in] n        specifies 'each nth point'
            \param[in] result   destination of the simplified polyline
            \return             one beyond the last coordinate of the simplified polyline
        */
        OutputIterator NthPoint (
            InputIterator first,
            InputIterator last,
            unsigned n,
            OutputIterator result)
        {
            diff_type coordCount = std::distance (first, last);
            diff_type pointCount = DIM              // protect against zero DIM
                                   ? coordCount / DIM
                                   : 0;

            // validate input and check if simplification required
            if (coordCount % DIM || pointCount < 3 || n < 2) {
                return std::copy (first, last, result);
            }

            unsigned remaining = pointCount - 1;    // the number of points remaining after key
            InputIterator key = first;              // indicates the current key

            // the first point is always part of the simplification
            CopyKey (key, result);

            // copy each nth point
            while (Forward (key, n, remaining)) {
                CopyKey (key, result);
            }

            return result;
        }

        /*!
            \brief Performs the (radial) distance between points routine (RD).

            RD is a brute-force O(n) algorithm for polyline simplification. It reduces successive
            vertices that are clustered too closely to a single vertex, called a key. The resulting
            keys form the simplified polyline.

            \image html psimpl_rd.png

            RD is applied to the range [first, last) using the specified tolerance tol. The
            resulting simplified polyline is copied to the output range [result, result + m*DIM),
            where m is the number of vertices of the simplified polyline. The return value is the
            end of the output range: result + m*DIM.

            Input (Type) requirements:
            1- DIM is not 0, where DIM represents the dimension of the polyline
            2- The InputIterator type models the concept of a forward iterator
            3- The input iterator value type is convertible to a value type of the output iterator
            4- The range [first, last) contains only vertex coordinates in multiples of DIM, f.e.:
               x, y, z, x, y, z, x, y, z when DIM = 3
            5- The range [first, last) contains at least 2 vertices
            6- tol is not 0

            In case these requirements are not met, the entire input range [first, last) is copied
            to the output range [result, result + (last - first)) OR compile errors may occur.

            \param[in] first    the first coordinate of the first polyline point
            \param[in] last     one beyond the last coordinate of the last polyline point
            \param[in] tol      radial (point-to-point) distance tolerance
            \param[in] result   destination of the simplified polyline
            \return             one beyond the last coordinate of the simplified polyline
        */
        OutputIterator RadialDistance (
            InputIterator first,
            InputIterator last,
            value_type tol,
            OutputIterator result)
        {
            diff_type coordCount = std::distance (first, last);
            diff_type pointCount = DIM      // protect against zero DIM
                                   ? coordCount / DIM
                                   : 0;
            value_type tol2 = tol * tol;    // squared distance tolerance

            // validate input and check if simplification required
            if (coordCount % DIM || pointCount < 3 || tol2 == 0) {
                return std::copy (first, last, result);
            }

            InputIterator current = first;  // indicates the current key
            InputIterator next = first;     // used to find the next key

            // the first point is always part of the simplification
            CopyKeyAdvance (next, result);

            // Skip first and last point, because they are always part of the simplification
            for (diff_type index = 1; index < pointCount - 1; ++index) {
                if (math::point_distance2 <DIM> (current, next) < tol2) {
                    Advance (next);
                    continue;
                }
                current = next;
                CopyKeyAdvance (next, result);
            }
            // the last point is always part of the simplification
            CopyKeyAdvance (next, result);

            return result;
        }

        /*!
            \brief Repeatedly performs the perpendicular distance routine (PD).

            The algorithm stops after calling the PD routine 'repeat' times OR when the
            simplification does not improve. Note that this algorithm will need to store
            up to two intermediate simplification results.

            \sa PerpendicularDistance(InputIterator, InputIterator, value_type, OutputIterator)

            \param[in] first    the first coordinate of the first polyline point
            \param[in] last     one beyond the last coordinate of the last polyline point
            \param[in] tol      perpendicular (segment-to-point) distance tolerance
            \param[in] repeat   the number of times to successively apply the PD routine
            \param[in] result   destination of the simplified polyline
            \return             one beyond the last coordinate of the simplified polyline
        */
        OutputIterator PerpendicularDistance (
            InputIterator first,
            InputIterator last,
            value_type tol,
            unsigned repeat,
            OutputIterator result)
        {
            if (repeat == 1) {
                // single pass
                return PerpendicularDistance (first, last, tol, result);
            }
            // only validate repeat; other input is validated by simplify_perpendicular_distance
            if (repeat < 1) {
                return std::copy (first, last, result);
            }
            diff_type coordCount = std::distance (first, last);

            // first pass: [first, last) --> temporary array 'tempPoly'
            util::scoped_array <value_type> tempPoly (coordCount);
            PolylineSimplification <DIM, InputIterator, value_type*> psimpl_to_array;
            diff_type tempCoordCount = std::distance (tempPoly.get (),
                psimpl_to_array.PerpendicularDistance (first, last, tol, tempPoly.get ()));

            // check if simplification did not improved
            if (coordCount == tempCoordCount) {
                return std::copy (tempPoly.get (), tempPoly.get () + coordCount, result);
            }
            std::swap (coordCount, tempCoordCount);
            --repeat;

            // intermediate passes: temporary array 'tempPoly' --> temporary array 'tempResult'
            if (1 < repeat) {
                util::scoped_array <value_type> tempResult (coordCount);
                PolylineSimplification <DIM, value_type*, value_type*> psimpl_arrays;

                while (--repeat) {
                    tempCoordCount = std::distance (tempResult.get (),
                        psimpl_arrays.PerpendicularDistance (
                            tempPoly.get (), tempPoly.get () + coordCount, tol, tempResult.get ()));

                    // check if simplification did not improved
                    if (coordCount == tempCoordCount) {
                        return std::copy (tempPoly.get (), tempPoly.get () + coordCount, result);
                    }
                    util::swap (tempPoly, tempResult);
                    std::swap (coordCount, tempCoordCount);
                }
            }

            // final pass: temporary array 'tempPoly' --> result
            PolylineSimplification <DIM, value_type*, OutputIterator> psimpl_from_array;
            return psimpl_from_array.PerpendicularDistance (
                tempPoly.get (), tempPoly.get () + coordCount, tol, result);
        }

        /*!
            \brief Performs the perpendicular distance routine (PD).

            PD is an O(n) algorithm for polyline simplification. It computes the perpendicular
            distance of each point pi to the line segment S(pi-1, pi+1). Only when this distance is
            larger than the given tolerance will pi be part of the simpification. Note that the
            original polyline can only be reduced by a maximum of 50%. Multiple passes are required
            to achieve higher points reductions.

            \image html psimpl_pd.png

            PD is applied to the range [first, last) using the specified tolerance tol. The
            resulting simplified polyline is copied to the output range [result, result + m*DIM),
            where m is the number of vertices of the simplified polyline. The return value is the
            end of the output range: result + m*DIM.

            Input (Type) requirements:
            1- DIM is not 0, where DIM represents the dimension of the polyline
            2- The InputIterator type models the concept of a forward iterator
            3- The input iterator value type is convertible to a value type of the output iterator
            4- The range [first, last) contains only vertex coordinates in multiples of DIM, f.e.:
               x, y, z, x, y, z, x, y, z when DIM = 3
            5- The range [first, last) contains at least 2 vertices
            6- tol is not 0

            In case these requirements are not met, the entire input range [first, last) is copied
            to the output range [result, result + (last - first)) OR compile errors may occur.

            \param[in] first    the first coordinate of the first polyline point
            \param[in] last     one beyond the last coordinate of the last polyline point
            \param[in] tol      perpendicular (segment-to-point) distance tolerance
            \param[in] result   destination of the simplified polyline
            \return             one beyond the last coordinate of the simplified polyline
        */
        OutputIterator PerpendicularDistance (
            InputIterator first,
            InputIterator last,
            value_type tol,
            OutputIterator result)
        {
            diff_type coordCount = std::distance (first, last);
            diff_type pointCount = DIM      // protect against zero DIM
                                   ? coordCount / DIM
                                   : 0;
            value_type tol2 = tol * tol;    // squared distance tolerance

            // validate input and check if simplification required
            if (coordCount % DIM || pointCount < 3 || tol2 == 0) {
                return std::copy (first, last, result);
            }

            InputIterator p0 = first;
            InputIterator p1 = AdvanceCopy(p0);
            InputIterator p2 = AdvanceCopy(p1);

            // the first point is always part of the simplification
            CopyKey (p0, result);

            while (p2 != last) {
                // test p1 against line segment S(p0, p2)
                if (math::segment_distance2 <DIM> (p0, p2, p1) < tol2) {
                    CopyKey (p2, result);
                    // move up by two points
                    p0 = p2;
                    Advance (p1, 2);
                    if (p1 == last) {
                        // protect against advancing p2 beyond last
                        break;
                    }
                    Advance (p2, 2);
                }
                else {
                    CopyKey (p1, result);
                    // move up by one point
                    p0 = p1;
                    p1 = p2;
                    Advance (p2);
                }
            }
            // make sure the last point is part of the simplification
            if (p1 != last) {
                CopyKey (p1, result);
            }
            return result;
        }

        /*!
            \brief Performs Reumann-Witkam approximation (RW).

            The O(n) RW routine uses a point-to-line (perpendicular) distance tolerance. It defines
            a line through the first two vertices of the original polyline. For each successive
            vertex vi its perpendicular distance to this line is calculated. A new key is found at
            vi-1, when this distance exceeds the specified tolerance. The vertices vi and vi+1 are
            then used to define a new line, and the process repeats itself.

            \image html psimpl_rw.png

            RW routine is applied to the range [first, last) using the specified perpendicular
            distance tolerance tol. The resulting simplified polyline is copied to the output range
            [result, result + m*DIM), where m is the number of vertices of the simplified polyline.
            The return value is the end of the output range: result + m*DIM.

            Input (Type) Requirements:
            1- DIM is not 0, where DIM represents the dimension of the polyline
            2- The InputIterator type models the concept of a forward iterator
            3- The input iterator value type is convertible to a value type of the output iterator
            4- The range [first, last) contains vertex coordinates in multiples of DIM,
               f.e.: x, y, z, x, y, z, x, y, z when DIM = 3
            5- The range [first, last) contains at least 2 vertices
            6- tol is not 0

            In case these requirements are not met, the entire input range [first, last) is copied
            to the output range [result, result + (last - first)) OR compile errors may occur.

            \param[in] first    the first coordinate of the first polyline point
            \param[in] last     one beyond the last coordinate of the last polyline point
            \param[in] tol      perpendicular (point-to-line) distance tolerance
            \param[in] result   destination of the simplified polyline
            \return             one beyond the last coordinate of the simplified polyline
        */
        OutputIterator ReumannWitkam (
            InputIterator first,
            InputIterator last,
            value_type tol,
            OutputIterator result)
        {
            diff_type coordCount = std::distance (first, last);
            diff_type pointCount = DIM      // protect against zero DIM
                                   ? coordCount / DIM
                                   : 0;
            value_type tol2 = tol * tol;    // squared distance tolerance

            // validate input and check if simplification required
            if (coordCount % DIM || pointCount < 3 || tol2 == 0) {
                return std::copy (first, last, result);
            }

            // define the line L(p0, p1)
            InputIterator p0 = first;               // indicates the current key
            InputIterator p1 = AdvanceCopy (first); // indicates the next point after p0

            // keep track of two test points
            InputIterator pi = p1;     // the previous test point
            InputIterator pj = p1;     // the current test point (pi+1)

            // the first point is always part of the simplification
            CopyKey (p0, result);

            // check each point pj against L(p0, p1)
            for (diff_type j = 2; j < pointCount; ++j) {
                pi = pj;
                Advance (pj);

                if (math::line_distance2 <DIM> (p0, p1, pj) < tol2) {
                    continue;
                }
                // found the next key at pi
                CopyKey (pi, result);
                // define new line L(pi, pj)
                p0 = pi;
                p1 = pj;
            }
            // the last point is always part of the simplification
            CopyKey (pj, result);

            return result;
        }

        /*!
            \brief Performs Opheim approximation (OP).

            The O(n) OP routine is very similar to the Reumann-Witkam (RW) routine, and can be seen
            as a constrained version of that RW routine. OP uses both a minimum and a maximum
            distance tolerance to constrain the search area. For each successive vertex vi, its
            radial distance to the current key vkey (initially v0) is calculated. The last point
            within the minimum distance tolerance is used to define a ray R (vkey, vi). If no
            such vi exists, the ray is defined as R(vkey, vkey+1). For each successive vertex vj
            beyond vi its perpendicular distance to the ray R is calculated. A new key is found at
            vj-1, when this distance exceeds the minimum tolerance Or when the radial distance
            between vj and the vkey exceeds the maximum tolerance. After a new key is found, the
            process repeats itself.

            \image html psimpl_op.png

            OP routine is applied to the range [first, last) using the specified distance tolerances
            min_tol and max_tol. The resulting simplified polyline is copied to the output range
            [result, result + m*DIM), where m is the number of vertices of the simplified polyline.
            The return value is the end of the output range: result + m*DIM.

            Input (Type) Requirements:
            1- DIM is not 0, where DIM represents the dimension of the polyline
            2- The InputIterator type models the concept of a forward iterator
            3- The input iterator value type is convertible to a value type of the output iterator
            4- The range [first, last) contains vertex coordinates in multiples of DIM,
               f.e.: x, y, z, x, y, z, x, y, z when DIM = 3
            5- The range [first, last) contains at least 2 vertices
            6- min_tol is not 0
            7- max_tol is not 0

            In case these requirements are not met, the entire input range [first, last) is copied
            to the output range [result, result + (last - first)) OR compile errors may occur.

            \param[in] first    the first coordinate of the first polyline point
            \param[in] last     one beyond the last coordinate of the last polyline point
            \param[in] min_tol  radial and perpendicular (point-to-ray) distance tolerance
            \param[in] max_tol  radial distance tolerance
            \param[in] result   destination of the simplified polyline
            \return             one beyond the last coordinate of the simplified polyline
        */
        OutputIterator Opheim (
            InputIterator first,
            InputIterator last,
            value_type min_tol,
            value_type max_tol,
            OutputIterator result)
        {
            diff_type coordCount = std::distance (first, last);
            diff_type pointCount = DIM                    // protect against zero DIM
                                   ? coordCount / DIM
                                   : 0;
            value_type min_tol2 = min_tol * min_tol;    // squared minimum distance tolerance
            value_type max_tol2 = max_tol * max_tol;    // squared maximum distance tolerance

            // validate input and check if simplification required
            if (coordCount % DIM || pointCount < 3 || min_tol2 == 0 || max_tol2 == 0) {
                return std::copy (first, last, result);
            }

            // define the ray R(r0, r1)
            InputIterator r0 = first;  // indicates the current key and start of the ray
            InputIterator r1 = first;  // indicates a point on the ray
            bool rayDefined = false;

            // keep track of two test points
            InputIterator pi = r0;     // the previous test point
            InputIterator pj =         // the current test point (pi+1)
                AdvanceCopy (pi);

            // the first point is always part of the simplification
            CopyKey (r0, result);

            for (diff_type j = 2; j < pointCount; ++j) {
                pi = pj;
                Advance (pj);

                if (!rayDefined) {
                    // discard each point within minimum tolerance
                    if (math::point_distance2 <DIM> (r0, pj) < min_tol2) {
                        continue;
                    }
                    // the last point within minimum tolerance pi defines the ray R(r0, r1)
                    r1 = pi;
                    rayDefined = true;
                }

                // check each point pj against R(r0, r1)
                if (math::point_distance2 <DIM> (r0, pj) < max_tol2 &&
                    math::ray_distance2 <DIM> (r0, r1, pj) < min_tol2)
                {
                    continue;
                }
                // found the next key at pi
                CopyKey (pi, result);
                // define new ray R(pi, pj)
                r0 = pi;
                rayDefined = false;
            }
            // the last point is always part of the simplification
            CopyKey (pj, result);

            return result;
        }

        /*!
            \brief Performs Lang approximation (LA).

            The LA routine defines a fixed size search-region. The first and last points of that
            search region form a segment. This segment is used to calculate the perpendicular
            distance to each intermediate point. If any calculated distance is larger than the
            specified tolerance, the search region will be shrunk by excluding its last point. This
            process will continue untill all calculated distances fall below the specified tolerance
            , or there are no more intermediate points. At this point all intermediate points are
            removed and a new search region is defined starting at the last point from old search
            region.
            Note that the size of the search region (look_ahead parameter) controls the maximum
            amount of simplification, e.g.: a size of 20 will always result in a simplification that
            contains at least 5% of the original points.

            \image html psimpl_la.png

            LA routine is applied to the range [first, last) using the specified tolerance and
            look ahead values. The resulting simplified polyline is copied to the output range
            [result, result + m*DIM), where m is the number of vertices of the simplified polyline.
            The return value is the end of the output range: result + m*DIM.

            Input (Type) Requirements:
            1- DIM is not 0, where DIM represents the dimension of the polyline
            2- The InputIterator type models the concept of a bidirectional iterator
            3- The InputIterator value type is convertible to a value type of the output iterator
            4- The range [first, last) contains vertex coordinates in multiples of DIM,
               f.e.: x, y, z, x, y, z, x, y, z when DIM = 3
            5- The range [first, last) contains at least 2 vertices
            6- tol is not 0
            7- look_ahead is not zero

            In case these requirements are not met, the entire input range [first, last) is copied
            to the output range [result, result + (last - first)) OR compile errors may occur.

            \param[in] first      the first coordinate of the first polyline point
            \param[in] last       one beyond the last coordinate of the last polyline point
            \param[in] tol        perpendicular (point-to-segment) distance tolerance
            \param[in] look_ahead defines the size of the search region
            \param[in] result     destination of the simplified polyline
            \return               one beyond the last coordinate of the simplified polyline
        */
        OutputIterator Lang (
            InputIterator first,
            InputIterator last,
            value_type tol,
            unsigned look_ahead,
            OutputIterator result)
        {
            diff_type coordCount = std::distance (first, last);
            diff_type pointCount = DIM      // protect against zero DIM
                                   ? coordCount / DIM
                                   : 0;
            value_type tol2 = tol * tol;    // squared minimum distance tolerance
            
            // validate input and check if simplification required
            if (coordCount % DIM || pointCount < 3 || look_ahead < 2 || tol2 == 0) {
                return std::copy (first, last, result);
            }

            InputIterator current = first;          // indicates the current key
            InputIterator next = first;             // used to find the next key

            unsigned remaining = pointCount - 1;    // the number of points remaining after current
            unsigned moved = Forward (next, look_ahead, remaining);

            // the first point is always part of the simplification
            CopyKey (current, result);

            while (moved) {
                value_type d2 = 0;
                InputIterator p = AdvanceCopy (current);

                while (p != next) {
                    d2 = std::max (d2, math::segment_distance2 <DIM> (current, next, p));
                    if (tol2 < d2) {
                        break;
                    }
                    Advance (p);
                }
                if (d2 < tol2) {
                    current = next;
                    CopyKey (current, result);
                    moved = Forward (next, look_ahead, remaining);
                }
                else {
                    Backward (next, remaining);
                }
            }
            return result;
        }

        /*!
            \brief Performs Douglas-Peucker approximation (DP).

            The DP algorithm uses the RadialDistance (RD) routine O(n) as a preprocessing step.
            After RD the algorithm is O (n m) in worst case and O(n log m) on average, where m < n
            (m is the number of points after RD).

            The DP algorithm starts with a simplification that is the single edge joining the first
            and last vertices of the polyline. The distance of the remaining vertices to that edge
            are computed. The vertex that is furthest away from theedge (called a key), and has a
            computed distance that is larger than a specified tolerance, will be added to the
            simplification. This process will recurse for each edge in the current simplification,
            untill all vertices of the original polyline are within tolerance.

            \image html psimpl_dp.png

            Note that this algorithm will create a copy of the input polyline during the vertex
            reduction step.

            RD followed by DP is applied to the range [first, last) using the specified tolerance
            tol. The resulting simplified polyline is copied to the output range
            [result, result + m*DIM), where m is the number of vertices of the simplified polyline.
            The return value is the end of the output range: result + m*DIM.

            Input (Type) requirements:
            1- DIM is not 0, where DIM represents the dimension of the polyline
            2- The InputIterator type models the concept of a forward iterator
            3- The InputIterator value type is convertible to a value type of the output iterator
            4- The range [first, last) contains vertex coordinates in multiples of DIM, f.e.:
               x, y, z, x, y, z, x, y, z when DIM = 3
            5- The range [first, last) contains at least 2 vertices
            6- tol is not 0

            In case these requirements are not met, the entire input range [first, last) is copied
            to the output range [result, result + (last - first)) OR compile errors may occur.

            \param[in] first    the first coordinate of the first polyline point
            \param[in] last     one beyond the last coordinate of the last polyline point
            \param[in] tol      perpendicular (point-to-segment) distance tolerance
            \param[in] result   destination of the simplified polyline
            \return             one beyond the last coordinate of the simplified polyline
        */
        OutputIterator DouglasPeucker (
            InputIterator first,
            InputIterator last,
            value_type tol,
            OutputIterator result)
        {
            diff_type coordCount = std::distance (first, last);
            diff_type pointCount = DIM      // protect against zero DIM
                                   ? coordCount / DIM
                                   : 0;
            // validate input and check if simplification required
            if (coordCount % DIM || pointCount < 3 || tol == 0) {
                return std::copy (first, last, result);
            }
            // radial distance routine as preprocessing
            util::scoped_array <value_type> reduced (coordCount);   // radial distance results
            PolylineSimplification <DIM, InputIterator, value_type*> psimpl_to_array;
            ptr_diff_type reducedCoordCount = std::distance (reduced.get (),
                psimpl_to_array.RadialDistance (first, last, tol, reduced.get ()));
            ptr_diff_type reducedPointCount = reducedCoordCount / DIM;

            // douglas-peucker approximation
            util::scoped_array <unsigned char> keys (pointCount);         // douglas-peucker results
            DPHelper::Approximate (reduced.get (), reducedCoordCount, tol, keys.get ());

            // copy all keys
            const value_type* current = reduced.get ();
            for (ptr_diff_type p=0; p<reducedPointCount; ++p, current += DIM) {
                if (keys [p]) {
                    for (unsigned d = 0; d < DIM; ++d) {
                        *result = current [d];
                        ++result;
                    }
                }
            }
            return result;
        }

        /*!
            \brief Performs a Douglas-Peucker approximation variant (DPn).

            This algorithm is a variation of the original implementation. Instead of considering
            one polyline segment at a time, all segments of the current simplified polyline are
            evaluated at each step. Only the vertex with the maximum distance from its edge is
            added to the simplification. This process will recurse untill the the simplification
            contains the desired amount of vertices.

            The algorithm, which does not use the (radial) distance between points routine as a
            preprocessing step, is O(n2) in worst case and O(n log n) on average.

            Note that this algorithm will create a copy of the input polyline for performance
            reasons.

            DPn is applied to the range [first, last). The resulting simplified polyline consists
            of count vertices and is copied to the output range [result, result + count). The
            return value is the end of the output range: result + count.

            Input (Type) requirements:
            1- DIM is not 0, where DIM represents the dimension of the polyline
            2- The InputIterator type models the concept of a forward iterator
            3- The InputIterator value type is convertible to a value type of the output iterator
            4- The range [first, last) contains vertex coordinates in multiples of DIM, f.e.:
               x, y, z, x, y, z, x, y, z when DIM = 3
            5- The range [first, last) contains a minimum of count vertices
            6- count is at least 2

            In case these requirements are not met, the entire input range [first, last) is copied
            to the output range [result, result + (last - first)) OR compile errors may occur.

            \sa DouglasPeucker

            \param[in] first    the first coordinate of the first polyline point
            \param[in] last     one beyond the last coordinate of the last polyline point
            \param[in] count    the maximum number of points of the simplified polyline
            \param[in] result   destination of the simplified polyline
            \return             one beyond the last coordinate of the simplified polyline
        */
        OutputIterator DouglasPeuckerN (
            InputIterator first,
            InputIterator last,
            unsigned count,
            OutputIterator result)
        {
            diff_type coordCount = std::distance (first, last);
            diff_type pointCount = DIM      // protect against zero DIM
                                   ? coordCount / DIM
                                   : 0;
            // validate input and check if simplification required
            if (coordCount % DIM || pointCount <= static_cast <diff_type> (count) || count < 2) {
                return std::copy (first, last, result);
            }

            // copy coords
            util::scoped_array <value_type> coords (coordCount);
            for (ptr_diff_type c=0; c<coordCount; ++c) {
                coords [c] = *first;
                ++first;
            }

            // douglas-peucker approximation
            util::scoped_array <unsigned char> keys (pointCount);
            DPHelper::ApproximateN (coords.get (), coordCount, count, keys.get ());

            // copy keys
            const value_type* current = coords.get ();
            for (ptr_diff_type p=0; p<pointCount; ++p, current += DIM) {
                if (keys [p]) {
                    for (unsigned d = 0; d < DIM; ++d) {
                        *result = current [d];
                        ++result;
                    }
                }
            }
            return result;
        }

        /*!
            \brief Computes the squared positional error between a polyline and its simplification.

            For each point in the range [original_first, original_last) the squared distance to the
            simplification [simplified_first, simplified_last) is calculated. Each positional error
            is copied to the output range [result, result + count), where count is the number of
            points in the original polyline. The return value is the end of the output range:
            result + count.

            Note that both the original and simplified polyline must be defined using the same
            value_type.

            \image html psimpl_pos_error.png

            Input (Type) requirements:
            1- DIM is not 0, where DIM represents the dimension of the polyline
            2- The InputIterator type models the concept of a forward iterator
            3- The InputIterator value type is convertible to a value type of the output iterator
            4- The ranges [original_first, original_last) and [simplified_first, simplified_last)
               contain vertex coordinates in multiples of DIM, f.e.: x, y, z, x, y, z, x, y, z
               when DIM = 3
            5- The ranges [original_first, original_last) and [simplified_first, simplified_last)
               contain a minimum of 2 vertices
            6- The range [simplified_first, simplified_last) represents a simplification of the
               range [original_first, original_last), meaning each point in the simplification
               has the exact same coordinates as some point from the original polyline

            In case these requirements are not met, the valid flag is set to false OR
            compile errors may occur.

            \param[in] original_first   the first coordinate of the first polyline point
            \param[in] original_last    one beyond the last coordinate of the last polyline point
            \param[in] simplified_first the first coordinate of the first simplified polyline point
            \param[in] simplified_last  one beyond the last coordinate of the last simplified polyline point
            \param[in] result           destination of the squared positional errors
            \param[out] valid           [optional] indicates if the computed positional errors are valid
            \return                     one beyond the last computed positional error
        */
        OutputIterator ComputePositionalErrors2 (
            InputIterator original_first,
            InputIterator original_last,
            InputIterator simplified_first,
            InputIterator simplified_last,
            OutputIterator result,
            bool* valid=0)
        {
            diff_type original_coordCount = std::distance (original_first, original_last);
            diff_type original_pointCount = DIM     // protect against zero DIM
                                            ? original_coordCount / DIM
                                            : 0;

            diff_type simplified_coordCount = std::distance (simplified_first, simplified_last);
            diff_type simplified_pointCount = DIM   // protect against zero DIM
                                              ? simplified_coordCount / DIM
                                              : 0;

            // validate input
            if (original_coordCount % DIM || original_pointCount < 2 ||
                simplified_coordCount % DIM || simplified_pointCount < 2 ||
                original_pointCount < simplified_pointCount ||
                !math::equal <DIM> (original_first, simplified_first))
            {
                if (valid) {
                    *valid = false;
                }
                return result;
            }

            // define (simplified) line segment S(simplified_prev, simplified_first)
            InputIterator simplified_prev = simplified_first;
            std::advance (simplified_first, DIM);

            // process each simplified line segment
            while (simplified_first != simplified_last) {
                // process each original point until it equals the end of the line segment
                while (original_first != original_last &&
                       !math::equal <DIM> (original_first, simplified_first))
                {
                    *result = math::segment_distance2 <DIM> (simplified_prev, simplified_first,
                                                             original_first);
                    ++result;
                    std::advance (original_first, DIM);
                }
                // update line segment S
                simplified_prev = simplified_first;
                std::advance (simplified_first, DIM);
            }
            // check if last original point matched
            if (original_first != original_last) {
                *result = 0;
                ++result;
            }

            if (valid) {
                *valid = original_first != original_last;
            }
            return result;
        }

        /*!
            \brief Computes statistics for the positional errors between a polyline and its simplification.

            Various statistics (mean, max, sum, std) are calculated for the positional errors
            between the range [original_first, original_last) and its simplification the range
            [simplified_first, simplified_last).

            Input (Type) requirements:
            1- DIM is not 0, where DIM represents the dimension of the polyline
            2- The InputIterator type models the concept of a forward iterator
            3- The InputIterator value type is convertible to double
            4- The ranges [original_first, original_last) and [simplified_first, simplified_last)
               contain vertex coordinates in multiples of DIM, f.e.: x, y, z, x, y, z, x, y, z
               when DIM = 3
            5- The ranges [original_first, original_last) and [simplified_first, simplified_last)
               contain a minimum of 2 vertices
            6- The range [simplified_first, simplified_last) represents a simplification of the
               range [original_first, original_last), meaning each point in the simplification
               has the exact same coordinates as some point from the original polyline

            In case these requirements are not met, the valid flag is set to false OR
            compile errors may occur.

            \sa ComputePositionalErrors2

            \param[in] original_first   the first coordinate of the first polyline point
            \param[in] original_last    one beyond the last coordinate of the last polyline point
            \param[in] simplified_first the first coordinate of the first simplified polyline point
            \param[in] simplified_last  one beyond the last coordinate of the last simplified polyline point
            \param[out] valid           [optional] indicates if the computed statistics are valid
            \return                     the computed statistics
        */
        math::Statistics ComputePositionalErrorStatistics (
            InputIterator original_first,
            InputIterator original_last,
            InputIterator simplified_first,
            InputIterator simplified_last,
            bool* valid=0)
        {
            diff_type pointCount = std::distance (original_first, original_last) / DIM;
            util::scoped_array <double> errors (pointCount);
            PolylineSimplification <DIM, InputIterator, double*> ps;

            diff_type errorCount = 
                std::distance (
                    errors.get (), 
                    ps.ComputePositionalErrors2 (original_first, original_last,
                                                 simplified_first, simplified_last,
                                                 errors.get (), valid));

            std::transform (errors.get (), errors.get () + errorCount,
                            errors.get (),
                            std::ptr_fun <double, double> (std::sqrt));

            return math::compute_statistics (errors.get (), errors.get () + errorCount);
        }

    private:
        /*!
            \brief Copies the key to the output destination, and increments the iterator.

            \sa CopyKey

            \param[in,out] key      the first coordinate of the key
            \param[in,out] result   destination of the copied key
        */
        inline void CopyKeyAdvance (
            InputIterator& key,
            OutputIterator& result)
        {
            for (unsigned d = 0; d < DIM; ++d) {
                *result = *key;
                ++result;
                ++key;
            }
        }

        /*!
            \brief Copies the key to the output destination.

            \sa CopyKeyAdvance

            \param[in]     key      the first coordinate of the key
            \param[in,out] result   destination of the copied key
        */
        inline void CopyKey (
            InputIterator key,
            OutputIterator& result)
        {
            CopyKeyAdvance (key, result);
        }

        /*!
            \brief Increments the iterator by n points.

            \sa AdvanceCopy

            \param[in,out] it  iterator to be advanced
            \param[in]     n   number of points to advance
        */
        inline void Advance (
            InputIterator& it,
            diff_type n = 1)
        {
            std::advance (it, n * static_cast <diff_type> (DIM));
        }
        
        /*!
            \brief Increments a copy of the iterator by n points.

            \sa Advance

            \param[in] it   iterator to be advanced
            \param[in] n    number of points to advance
            \return         an incremented copy of the input iterator
        */
        inline InputIterator AdvanceCopy (
            InputIterator it,
            diff_type n = 1)
        {
            Advance (it, n);
            return it;
        }

        /*!
            \brief Increments the iterator by n points if possible.

            If there are fewer than n point remaining the iterator will be incremented to the last
            point.

            \sa Backward

            \param[in,out] it           iterator to be advanced
            \param[in]     n            number of points to advance
            \param[in,out] remaining    number of points remaining after it
            \return                     the actual amount of points that the iterator advanced
        */
        inline unsigned Forward (
            InputIterator& it,
            unsigned n,
            unsigned& remaining)
        {
            n = std::min (n, remaining);
            Advance (it, n);
            remaining -= n;
            return n;
        }

        /*!
            \brief Decrements the iterator by 1 point.

            \sa Forward

            \param[in,out] it            iterator to be advanced
            \param[in,out] remaining     number of points remaining after it
        */
        inline void Backward (
            InputIterator& it,
            unsigned& remaining)
        {
            Advance (it, -1);
            ++remaining;
        }

    private:
        /*!
            \brief Douglas-Peucker approximation helper class.

            Contains helper implentations for Douglas-Peucker approximation that operate solely on
            value_type arrays and value_type pointers. Note that the PolylineSimplification
            class only operates on iterators.
        */
        class DPHelper
        {
            //! \brief Defines a sub polyline.
            struct SubPoly {
                SubPoly (ptr_diff_type first=0, ptr_diff_type last=0) :
                    first (first), last (last) {}

                ptr_diff_type first;    //! coord index of the first point
                ptr_diff_type last;     //! coord index of the last point
            };

            //! \brief Defines the key of a polyline.
            struct KeyInfo {
                KeyInfo (ptr_diff_type index=0, value_type dist2=0) :
                    index (index), dist2 (dist2) {}

                ptr_diff_type index;    //! coord index of the key
                value_type dist2;       //! squared distance of the key to a segment
            };

            //! \brief Defines a sub polyline including its key.
            struct SubPolyAlt {
                SubPolyAlt (ptr_diff_type first=0, ptr_diff_type last=0) :
                    first (first), last (last) {}

                ptr_diff_type first;    //! coord index of the first point
                ptr_diff_type last;     //! coord index of the last point
                KeyInfo keyInfo;        //! key of this sub poly

                bool operator< (const SubPolyAlt& other) const {
                    return keyInfo.dist2 < other.keyInfo.dist2;
                }
            };

        public:
            /*!
                \brief Performs Douglas-Peucker approximation.

                \param[in] coords       array of polyline coordinates
                \param[in] coordCount   number of coordinates in coords []
                \param[in] tol          approximation tolerance
                \param[out] keys        indicates for each polyline point if it is a key
            */
            static void Approximate (
                const value_type* coords,
                ptr_diff_type coordCount,
                value_type tol,
                unsigned char* keys)
            {
                value_type tol2 = tol * tol;    // squared distance tolerance
                ptr_diff_type pointCount = coordCount / DIM;
                // zero out keys
                std::fill_n (keys, pointCount, 0);
                keys [0] = 1;                   // the first point is always a key
                keys [pointCount - 1] = 1;      // the last point is always a key

                typedef std::stack <SubPoly> Stack;
                Stack stack;                    // LIFO job-queue containing sub-polylines

                SubPoly subPoly (0, coordCount-DIM);
                stack.push (subPoly);           // add complete poly

                while (!stack.empty ()) {
                    subPoly = stack.top ();     // take a sub poly
                    stack.pop ();               // and find its key
                    KeyInfo keyInfo = FindKey (coords, subPoly.first, subPoly.last);
                    if (keyInfo.index && tol2 < keyInfo.dist2) {
                        // store the key if valid
                        keys [keyInfo.index / DIM] = 1;
                        // split the polyline at the key and recurse
                        stack.push (SubPoly (keyInfo.index, subPoly.last));
                        stack.push (SubPoly (subPoly.first, keyInfo.index));
                    }
                }
            }

            /*!
                \brief Performs Douglas-Peucker approximation.

                \param[in] coords       array of polyline coordinates
                \param[in] coordCount   number of coordinates in coords []
                \param[in] countTol     point count tolerance
                \param[out] keys        indicates for each polyline point if it is a key
            */
            static void ApproximateN (
                const value_type* coords,
                ptr_diff_type coordCount,
                unsigned countTol,
                unsigned char* keys)
            {
                ptr_diff_type pointCount = coordCount / DIM;
                // zero out keys
                std::fill_n (keys, pointCount, 0);
                keys [0] = 1;                   // the first point is always a key
                keys [pointCount - 1] = 1;      // the last point is always a key
                unsigned keyCount = 2;

                if (countTol == 2) {
                    return;
                }

                typedef std::priority_queue <SubPolyAlt> PriorityQueue;
                PriorityQueue queue;    // sorted (max dist2) job queue containing sub-polylines

                SubPolyAlt subPoly (0, coordCount-DIM);
                subPoly.keyInfo = FindKey (coords, subPoly.first, subPoly.last);
                queue.push (subPoly);           // add complete poly

                while (!queue.empty ()) {
                    subPoly = queue.top ();     // take a sub poly
                    queue.pop ();
                    // store the key
                    keys [subPoly.keyInfo.index / DIM] = 1;
                    // check point count tolerance
                    keyCount++;
                    if (keyCount == countTol) {
                        break;
                    }
                    // split the polyline at the key and recurse
                    SubPolyAlt left (subPoly.first, subPoly.keyInfo.index);
                    left.keyInfo = FindKey (coords, left.first, left.last);
                    if (left.keyInfo.index) {
                        queue.push (left);
                    }
                    SubPolyAlt right (subPoly.keyInfo.index, subPoly.last);
                    right.keyInfo = FindKey (coords, right.first, right.last);
                    if (right.keyInfo.index) {
                        queue.push (right);
                    }
                }
            }

        private:
            /*!
                \brief Finds the key for the given sub polyline.

                Finds the point in the range [first, last] that is furthest away from the
                segment (first, last). This point is called the key.

                \param[in] coords   array of polyline coordinates
                \param[in] first    the first coordinate of the first polyline point
                \param[in] last     the first coordinate of the last polyline point
                \return             the index of the key and its distance, or last when a key
                                    could not be found
            */
            static KeyInfo FindKey (
                const value_type* coords,
                ptr_diff_type first,
                ptr_diff_type last)
            {
                KeyInfo keyInfo;

                for (ptr_diff_type current = first + DIM; current < last; current += DIM) {
                    value_type d2 = math::segment_distance2 <DIM> (coords + first, coords + last,
                                                                   coords + current);
                    if (d2 < keyInfo.dist2) {
                        continue;
                    }
                    // update maximum squared distance and the point it belongs to
                    keyInfo.index = current;
                    keyInfo.dist2 = d2;
                }
                return keyInfo;
            }
        };
    };

    /*!
        \brief Performs the nth point routine (NP).

        This is a convenience function that provides template type deduction for
        PolylineSimplification::NthPoint.

        \param[in] first    the first coordinate of the first polyline point
        \param[in] last     one beyond the last coordinate of the last polyline point
        \param[in] n        specifies 'each nth point'
        \param[in] result   destination of the simplified polyline
        \return             one beyond the last coordinate of the simplified polyline
    */
    template <unsigned DIM, class ForwardIterator, class OutputIterator>
    OutputIterator simplify_nth_point (
        ForwardIterator first,
        ForwardIterator last,
        unsigned n,
        OutputIterator result)
    {
        PolylineSimplification <DIM, ForwardIterator, OutputIterator> ps;
        return ps.NthPoint (first, last, n, result);
    }

    /*!
        \brief Performs the (radial) distance between points routine (RD).

        This is a convenience function that provides template type deduction for
        PolylineSimplification::RadialDistance.

        \param[in] first    the first coordinate of the first polyline point
        \param[in] last     one beyond the last coordinate of the last polyline point
        \param[in] tol      radial (point-to-point) distance tolerance
        \param[in] result   destination of the simplified polyline
        \return             one beyond the last coordinate of the simplified polyline
    */
    template <unsigned DIM, class ForwardIterator, class OutputIterator>
    OutputIterator simplify_radial_distance (
        ForwardIterator first,
        ForwardIterator last,
        typename std::iterator_traits <ForwardIterator>::value_type tol,
        OutputIterator result)
    {
        PolylineSimplification <DIM, ForwardIterator, OutputIterator> ps;
        return ps.RadialDistance (first, last, tol, result);
    }

    /*!
        \brief Repeatedly performs the perpendicular distance routine (PD).

        This is a convenience function that provides template type deduction for
        PolylineSimplification::PerpendicularDistance.

        \param[in] first    the first coordinate of the first polyline point
        \param[in] last     one beyond the last coordinate of the last polyline point
        \param[in] tol      perpendicular (segment-to-point) distance tolerance
        \param[in] repeat   the number of times to successively apply the PD routine.
        \param[in] result   destination of the simplified polyline
        \return             one beyond the last coordinate of the simplified polyline
    */
    template <unsigned DIM, class ForwardIterator, class OutputIterator>
    OutputIterator simplify_perpendicular_distance (
        ForwardIterator first,
        ForwardIterator last,
        typename std::iterator_traits <ForwardIterator>::value_type tol,
        unsigned repeat,
        OutputIterator result)
    {
        PolylineSimplification <DIM, ForwardIterator, OutputIterator> ps;
        return ps.PerpendicularDistance (first, last, tol, repeat, result);
    }

    /*!
        \brief Performs the perpendicular distance routine (PD).

        This is a convenience function that provides template type deduction for
        PolylineSimplification::PerpendicularDistance.

        \param[in] first    the first coordinate of the first polyline point
        \param[in] last     one beyond the last coordinate of the last polyline point
        \param[in] tol      perpendicular (segment-to-point) distance tolerance
        \param[in] result   destination of the simplified polyline
        \return             one beyond the last coordinate of the simplified polyline
    */
    template <unsigned DIM, class ForwardIterator, class OutputIterator>
    OutputIterator simplify_perpendicular_distance (
        ForwardIterator first,
        ForwardIterator last,
        typename std::iterator_traits <ForwardIterator>::value_type tol,
        OutputIterator result)
    {
        PolylineSimplification <DIM, ForwardIterator, OutputIterator> ps;
        return ps.PerpendicularDistance (first, last, tol, result);
    }

    /*!
        \brief Performs Reumann-Witkam polyline simplification (RW).

        This is a convenience function that provides template type deduction for
        PolylineSimplification::ReumannWitkam.

        \param[in] first    the first coordinate of the first polyline point
        \param[in] last     one beyond the last coordinate of the last polyline point
        \param[in] tol      perpendicular (point-to-line) distance tolerance
        \param[in] result   destination of the simplified polyline
        \return             one beyond the last coordinate of the simplified polyline
    */
    template <unsigned DIM, class ForwardIterator, class OutputIterator>
    OutputIterator simplify_reumann_witkam (
        ForwardIterator first,
        ForwardIterator last,
        typename std::iterator_traits <ForwardIterator>::value_type tol,
        OutputIterator result)
    {
        PolylineSimplification <DIM, ForwardIterator, OutputIterator> ps;
        return ps.ReumannWitkam (first, last, tol, result);
    }

    /*!
        \brief Performs Opheim polyline simplification (OP).

        This is a convenience function that provides template type deduction for
        PolylineSimplification::Opheim.

        \param[in] first    the first coordinate of the first polyline point
        \param[in] last     one beyond the last coordinate of the last polyline point
        \param[in] min_tol  minimum distance tolerance
        \param[in] max_tol  maximum distance tolerance
        \param[in] result   destination of the simplified polyline
        \return             one beyond the last coordinate of the simplified polyline
    */
    template <unsigned DIM, class ForwardIterator, class OutputIterator>
    OutputIterator simplify_opheim (
        ForwardIterator first,
        ForwardIterator last,
        typename std::iterator_traits <ForwardIterator>::value_type min_tol,
        typename std::iterator_traits <ForwardIterator>::value_type max_tol,
        OutputIterator result)
    {
        PolylineSimplification <DIM, ForwardIterator, OutputIterator> ps;
        return ps.Opheim (first, last, min_tol, max_tol, result);
    }

    /*!
        \brief Performs Lang polyline simplification (LA).

        This is a convenience function that provides template type deduction for
        PolylineSimplification::Lang.

        \param[in] first      the first coordinate of the first polyline point
        \param[in] last       one beyond the last coordinate of the last polyline point
        \param[in] tol        perpendicular (point-to-segment) distance tolerance
        \param[in] look_ahead defines the size of the search region
        \param[in] result     destination of the simplified polyline
        \return               one beyond the last coordinate of the simplified polyline
    */
    template <unsigned DIM, class BidirectionalIterator, class OutputIterator>
    OutputIterator simplify_lang (
        BidirectionalIterator first,
        BidirectionalIterator last,
        typename std::iterator_traits <BidirectionalIterator>::value_type tol,
        unsigned look_ahead,
        OutputIterator result)
    {
        PolylineSimplification <DIM, BidirectionalIterator, OutputIterator> ps;
        return ps.Lang (first, last, tol, look_ahead, result);
    }

    /*!
        \brief Performs Douglas-Peucker polyline simplification (DP).

        This is a convenience function that provides template type deduction for
        PolylineSimplification::DouglasPeucker.

        \param[in] first    the first coordinate of the first polyline point
        \param[in] last     one beyond the last coordinate of the last polyline point
        \param[in] tol      perpendicular (point-to-segment) distance tolerance
        \param[in] result   destination of the simplified polyline
        \return             one beyond the last coordinate of the simplified polyline
    */
    template <unsigned DIM, class ForwardIterator, class OutputIterator>
    OutputIterator simplify_douglas_peucker (
        ForwardIterator first,
        ForwardIterator last,
        typename std::iterator_traits <ForwardIterator>::value_type tol,
        OutputIterator result)
    {
        PolylineSimplification <DIM, ForwardIterator, OutputIterator> ps;
        return ps.DouglasPeucker (first, last, tol, result);
    }

    /*!
        \brief Performs a variant of Douglas-Peucker polyline simplification (DPn).

        This is a convenience function that provides template type deduction for
        PolylineSimplification::DouglasPeuckerAlt.

        \param[in] first    the first coordinate of the first polyline point
        \param[in] last     one beyond the last coordinate of the last polyline point
        \param[in] count    the maximum number of points of the simplified polyline
        \param[in] result   destination of the simplified polyline
        \return             one beyond the last coordinate of the simplified polyline
    */
    template <unsigned DIM, class ForwardIterator, class OutputIterator>
    OutputIterator simplify_douglas_peucker_n (
        ForwardIterator first,
        ForwardIterator last,
        unsigned count,
        OutputIterator result)
    {
        PolylineSimplification <DIM, ForwardIterator, OutputIterator> ps;
        return ps.DouglasPeuckerN (first, last, count, result);
    }

    /*!
        \brief Computes the squared positional error between a polyline and its simplification.

        This is a convenience function that provides template type deduction for
        PolylineSimplification::ComputePositionalErrors2.

        \param[in] original_first   the first coordinate of the first polyline point
        \param[in] original_last    one beyond the last coordinate of the last polyline point
        \param[in] simplified_first the first coordinate of the first simplified polyline point
        \param[in] simplified_last  one beyond the last coordinate of the last simplified polyline point
        \param[in] result           destination of the squared positional errors
        \param[out] valid           [optional] indicates if the computed positional errors are valid
        \return                     one beyond the last computed positional error
    */
    template <unsigned DIM, class ForwardIterator, class OutputIterator>
    OutputIterator compute_positional_errors2 (
        ForwardIterator original_first,
        ForwardIterator original_last,
        ForwardIterator simplified_first,
        ForwardIterator simplified_last,
        OutputIterator result,
        bool* valid=0)
    {
        PolylineSimplification <DIM, ForwardIterator, OutputIterator> ps;
        return ps.ComputePositionalErrors2 (original_first, original_last, simplified_first, simplified_last, result, valid);
    }

    /*!
        \brief Computes statistics for the positional errors between a polyline and its simplification.

        This is a convenience function that provides template type deduction for
        PolylineSimplification::ComputePositionalErrorStatistics.

        \param[in] original_first   the first coordinate of the first polyline point
        \param[in] original_last    one beyond the last coordinate of the last polyline point
        \param[in] simplified_first the first coordinate of the first simplified polyline point
        \param[in] simplified_last  one beyond the last coordinate of the last simplified polyline point
        \param[out] valid           [optional] indicates if the computed statistics are valid
        \return                     the computed statistics
    */
    template <unsigned DIM, class ForwardIterator>
    math::Statistics compute_positional_error_statistics (
        ForwardIterator original_first,
        ForwardIterator original_last,
        ForwardIterator simplified_first,
        ForwardIterator simplified_last,
        bool* valid=0)
    {
        PolylineSimplification <DIM, ForwardIterator, ForwardIterator> ps;
        return ps.ComputePositionalErrorStatistics (original_first, original_last, simplified_first, simplified_last, valid);
    }
}

#endif // PSIMPL_GENERIC