% POLYGONS Manipulation of planar polygons and polylines
% Version 1.6 21-Mar-2011 .
%
%   The 'polygons' module contains functions operating on shapes composed
%   of a vertex list, like polygons or polylines.
%
%   We call 'polyline' the curve defined by a series of vertices.
%   A polyline can be either closed or open, depending on whether the last
%   vertex is connected to the first one or not. This can be given as an
%   option is some functions in the module.
%   A 'polygon' is the planar domain delimited by a closed polyline. We
%   sometimes want to consider 'complex polygons', whose boundary is
%   composed of several disjoint domains. The domain defined by a single
%   closed polyline is called 'simple polygon'.
%   We call 'curve' a polyline with many vertices, such that the polyline
%   can be considered as a discrete approximation of a "real" curve.
%
%   A simple polygon or polyline is represented by a N-by-2 array, each row
%   of the array representing the coordinates of a vertex. 
%   Simple polygons are assumed to be closed, so there is no need to repeat
%   the first vertex at the end. 
%   As both polygons and polylines can be represented by a list of vertex
%   coordinates, some functions also consider the vertex list itself. Such
%   functions are prefixed by 'pointSet'. Also, many functions prefixed by
%   'polygon' or 'polyline' works also on the other type of shape.
%
%   For multiple-connected polygons, the different connected boundaries are
%   separated by a row [NaN NaN].
%
%   For some functions, the orientation of the polygon can be relevant: CCW
%   stands for 'Conter-Clockwise' (positive orientation), CW stands for
%   'Clockwise'.
%
%   Polylines are parametrized in the following way:
%   * the i-th vertex is located at position i-1
%   * points of the i-th edge have positions ranging linearly from i-1 to i
%   The parametrization domain for an open polyline is from 0 to Nv-1, and
%   from 0 to Nv for a closed polyline (positions 0 and Nv correspond to
%   the same point).
%
%   Example:
%   % Simple polygon:
%   P1 = [1 1;2 1;2 2;1 2];
%   drawPolygon(P1);
%   axis([0 5 0 5]);
%   % Multiple polygon:
%   P2 = [10 10;40 10;40 40;10 40;NaN NaN;20 20;20 30;30 30;30 20];
%   figure;drawPolygon(P2); axis([0 50 0 50]);
%
%
% Polylines
%   polylinePoint             - Extract a point from a polyline
%   polylineLength            - Return length of a polyline given as a list of points
%   polylineCentroid          - Compute centroid of a curve defined by a series of points
%   polylineSubcurve          - Extract a portion of a polyline
%   resamplePolyline          - Distribute N points equally spaced on a polyline
%   resamplePolylineByLength  - Resample a polyline with a fixed sampling step
%   reversePolyline           - Reverse a polyline, by iterating vertices from the end
%   isPointOnPolyline         - Test if a point belongs to a polyline
%   projPointOnPolyline       - Compute position of a point projected on a polyline
%   distancePointPolyline     - Compute shortest distance between a point and a polyline
%   distancePolylines         - Compute the shortest distance between 2 polylines
%   intersectLinePolyline     - Intersection points between a line and a polyline
%   intersectPolylines        - Find the common points between 2 polylines
%   polylineSelfIntersections - Find self-intersection points of a polyline
%   simplifyPolyline          - Douglas-Peucker simplification of a polyline
%   smoothPolyline            - Smooth a polyline using local averaging
%   removeMultipleVertices    - Remove multiple vertices of a polygon or polyline
%
% Polygon basic manipulation
%   polygonPoint              - Extract a point from a polygon
%   polygonSubcurve           - Extract a portion of a polygon
%   polygonEdges              - Return the edges of a simple or multiple polygon
%   reversePolygon            - Reverse a polygon, by iterating vertices from the end
%   smoothPolygon             - Smooth a polygon using local averaging
%   simplifyPolygon           - Douglas-Peucker simplification of a polygon
%   projPointOnPolygon        - Compute position of a point projected on a polygon
%   splitPolygons             - Convert a NaN separated polygon list to a cell array of polygons
%   polygonLoops              - Divide a possibly self-intersecting polygon into a set of simple loops
%
% Polygon clipping and intersections
%   intersectLinePolygon      - Intersection points between a line and a polygon
%   intersectRayPolygon       - Intersection points between a ray and a polygon
%   intersectEdgePolygon      - Intersection point of an edge with a polygon
%   polygonSelfIntersections  - Find self-intersection points of a polygon
%   clipPolygon               - Clip a polygon with a rectangular box
%   clipPolygonHP             - Clip a polygon with a Half-plane defined by a directed line
%
% Point Sets
%   pointSetsAverage          - Compute the average of several point sets
%   minimumCaliperDiameter    - Minimum caliper diameter of a set of points
%   findPoint                 - Find index of a point in an set from its coordinates
%   convexHull                - Convex hull of a set of points
%   randomPointInPolygon      - Generate random point(s) in a polygon
%
% Measures on Polygons
%   isPointInPolygon          - Test if a point is located inside a polygon
%   polygonContains           - Test if a point is contained in a multiply connected polygon
%   polygonCentroid           - Compute the centroid (center of mass) of a polygon
%   polygonArea               - Compute the signed area of a polygon
%   polygonInertiaEllipse     - Compute ellipse with same inertia moments as polygon
%   polygonSecondAreaMoments  - Compute second-order area moments of a polygon
%   polygonLength             - Perimeter of a polygon
%   polygonNormalAngle        - Compute the normal angle at a vertex of the polygon
%   polygonBounds             - Compute the bounding box of a polygon
%   polygonOuterNormal        - Outer normal vector for a given vertex(ices)
%   distancePointPolygon      - Shortest distance between a point and a polygon
%   distancePolygons          - Compute the shortest distance between 2 polygons
%   distancePolygonsNoCross   - Compute the shortest distance between 2 polygons
%   polygonSignature          - Polar signature of a polygon (polar distance to origin)
%   signatureToPolygon        - Reconstruct a polygon from its polar signature
%
% More complex operations on polygons
%   resamplePolygon           - Distribute N points equally spaced on a polygon
%   resamplePolygonByLength   - Resample a polygon with a fixed sampling step
%   densifyPolygon            - Add several points on each edge of the polygon
%   expandPolygon             - Expand a polygon by a given (signed) distance
%   triangulatePolygon        - Compute a triangulation of the polygon
%   polygonSymmetryAxis       - Try to identify symmetry axis of polygon
%   medialAxisConvex          - Compute medial axis of a convex polygon
%
% Curves (polylines with lot of vertices)
%   parametrize               - Parametrization of a polyline, based on edges lengths
%   curvature                 - Estimate curvature of a polyline defined by points
%   cart2geod                 - Convert cartesian coordinates to geodesic coord.
%   geod2cart                 - Convert geodesic coordinates to cartesian coord.
%   curveMoment               - Compute inertia moment of a 2D curve
%   curveCMoment              - Compute centered inertia moment of a 2D curve
%   curveCSMoment             - Compute centered scaled moment of a 2D curve
%
% Functions from stochastic geometry
%   steinerPoint              - Compute steiner point (weighted centroid) of a polygon
%   steinerPolygon            - Create a Steiner polygon from a set of vectors
%   supportFunction           - Compute support function of a polygon
%   convexification           - Compute the convexification of a polygon
%
% Input, Output and conversions
%   polygonToRow              - Convert polygon coordinates to a row vector
%   rowToPolygon              - Create a polygon from a row vector
%   contourMatrixToPolylines  - Converts a contour matrix array into a polyline set
%   readPolygonSet            - Read a set of simple polygons stored in a file
%   writePolygonSet           - Write a set of simple polygons into a file
%
% Drawing functions
%   drawPolyline              - Draw a polyline specified by a list of points
%   drawPolygon               - Draw a polygon specified by a list of points
%   fillPolygon               - Fill a polygon specified by a list of points
%   drawVertices              - Draw the vertices of a polygon or polyline
%
%
%   Credits:
%   * function intersectPolylines uses the 'interX' contribution from "NS"
%       (file exchange 22441, called 'curve-intersections')
%
% -----
% Author: David Legland
% e-mail: david.legland@inra.fr
% created the  07/11/2005.
% Project homepage: http://github.com/mattools/matGeom
% http://www.pfl-cepia.inra.fr/index.php?page=geom2d
% Copyright INRA - Cepia Software Platform.

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