math::geometry(n)              Tcl Math Library              math::geometry(n)



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NAME
       math::geometry - Geometrical computations

SYNOPSIS
       package require Tcl  ?8.3?

       package require math::geometry  ?1.1.2?

       ::math::geometry::+ point1 point2

       ::math::geometry::- point1 point2

       ::math::geometry::p x y

       ::math::geometry::distance point1 point2

       ::math::geometry::length point

       ::math::geometry::s* factor point

       ::math::geometry::direction angle

       ::math::geometry::h length

       ::math::geometry::v length

       ::math::geometry::between point1 point2 s

       ::math::geometry::octant point

       ::math::geometry::rect nw se

       ::math::geometry::nwse rect

       ::math::geometry::angle line

       ::math::geometry::calculateDistanceToLine P line

       ::math::geometry::calculateDistanceToLineSegment P linesegment

       ::math::geometry::calculateDistanceToPolyline P polyline

       ::math::geometry::findClosestPointOnLine P line

       ::math::geometry::findClosestPointOnLineSegment P linesegment

       ::math::geometry::findClosestPointOnPolyline P polyline

       ::math::geometry::lengthOfPolyline polyline

       ::math::geometry::movePointInDirection P direction dist

       ::math::geometry::lineSegmentsIntersect linesegment1 linesegment2

       ::math::geometry::findLineSegmentIntersection linesegment1 linesegment2

       ::math::geometry::findLineIntersection line1 line2

       ::math::geometry::polylinesIntersect polyline1 polyline2

       ::math::geometry::polylinesBoundingIntersect polyline1 polyline2
       granularity

       ::math::geometry::intervalsOverlap y1 y2 y3 y4 strict

       ::math::geometry::rectanglesOverlap P1 P2 Q1 Q2 strict

       ::math::geometry::bbox polyline

       ::math::geometry::pointInsidePolygon P polyline

       ::math::geometry::rectangleInsidePolygon P1 P2 polyline

       ::math::geometry::areaPolygon polygon

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DESCRIPTION
       The math::geometry package is a collection of functions for
       computations and manipulations on two-dimensional geometrical objects,
       such as points, lines and polygons.

       The geometrical objects are implemented as plain lists of coordinates.
       For instance a line is defined by a list of four numbers, the x- and y-
       coordinate of a first point and the x- and y-coordinates of a second
       point on the line.

       The various types of object are recognised by the number of coordinate
       pairs and the context in which they are used: a list of four elements
       can be regarded as an infinite line, a finite line segment but also as
       a polyline of one segment and a point set of two points.

       Currently the following types of objects are distinguished:

       o      point - a list of two coordinates representing the x- and y-
              coordinates respectively.

       o      line - a list of four coordinates, interpreted as the x- and y-
              coordinates of two distinct points on the line.

       o      line segment - a list of four coordinates, interpreted as the x-
              and y-coordinates of the first and the last points on the line
              segment.

       o      polyline - a list of an even number of coordinates, interpreted
              as the x- and y-coordinates of an ordered set of points.

       o      polygon - like a polyline, but the implicit assumption is that
              the polyline is closed (if the first and last points do not
              coincide, the missing segment is automatically added).

       o      point set - again a list of an even number of coordinates, but
              the points are regarded without any ordering.

PROCEDURES
       The package defines the following public procedures:

       ::math::geometry::+ point1 point2
              Compute the sum of the two vectors given as points and return
              it.  The result is a vector as well.

       ::math::geometry::- point1 point2
              Compute the difference (point1 - point2) of the two vectors
              given as points and return it. The result is a vector as well.

       ::math::geometry::p x y
              Construct a point from its coordinates and return it as the
              result of the command.

       ::math::geometry::distance point1 point2
              Compute the distance between the two points and return it as the
              result of the command. This is in essence the same as

                  math::geometry::length [math::geomtry::- point1 point2]


       ::math::geometry::length point
              Compute the length of the vector and return it as the result of
              the command.

       ::math::geometry::s* factor point
              Scale the vector by the factor and return it as the result of
              the command. This is a vector as well.

       ::math::geometry::direction angle
              Given the angle in degrees this command computes and returns the
              unit vector pointing into this direction. The vector for angle
              == 0 points to the right (up), and for angle == 90 up (north).

       ::math::geometry::h length
              Returns a horizontal vector on the X-axis of the specified
              length.  Positive lengths point to the right (east).

       ::math::geometry::v length
              Returns a vertical vector on the Y-axis of the specified length.
              Positive lengths point down (south).

       ::math::geometry::between point1 point2 s
              Compute the point which is at relative distance s between the
              two points and return it as the result of the command. A
              relative distance of 0 returns point1, the distance 1 returns
              point2.  Distances < 0 or > 1 extrapolate along the line between
              the two point.

       ::math::geometry::octant point
              Compute the octant of the circle the point is in and return it
              as the result of the command. The possible results are

              [1]    east

              [2]    northeast

              [3]    north

              [4]    northwest

              [5]    west

              [6]    southwest

              [7]    south

              [8]    southeast
       Each octant is the arc of the circle +/- 22.5 degrees from the cardinal
       direction the octant is named for.

       ::math::geometry::rect nw se
              Construct a rectangle from its northwest and southeast corners
              and return it as the result of the command.

       ::math::geometry::nwse rect
              Extract the northwest and southeast corners of the rectangle and
              return them as the result of the command (a 2-element list
              containing the points, in the named order).

       ::math::geometry::angle line
              Calculate the angle from the positive x-axis to a given line (in
              two dimensions only).

              list line
                     Coordinates of the line


       ::math::geometry::calculateDistanceToLine P line
              Calculate the distance of point P to the (infinite) line and
              return the result

              list P List of two numbers, the coordinates of the point

              list line
                     List of four numbers, the coordinates of two points on
                     the line


       ::math::geometry::calculateDistanceToLineSegment P linesegment
              Calculate the distance of point P to the (finite) line segment
              and return the result.

              list P List of two numbers, the coordinates of the point

              list linesegment
                     List of four numbers, the coordinates of the first and
                     last points of the line segment



       ::math::geometry::calculateDistanceToPolyline P polyline
              Calculate the distance of point P to the polyline and return the
              result.

              list P List of two numbers, the coordinates of the point

              list polyline
                     List of numbers, the coordinates of the vertices of the
                     polyline


       ::math::geometry::findClosestPointOnLine P line
              Return the point on a line which is closest to a given point.

              list P List of two numbers, the coordinates of the point

              list line
                     List of four numbers, the coordinates of two points on
                     the line


       ::math::geometry::findClosestPointOnLineSegment P linesegment
              Return the point on a line segment which is closest to a given
              point.

              list P List of two numbers, the coordinates of the point

              list linesegment
                     List of four numbers, the first and last points on the
                     line segment


       ::math::geometry::findClosestPointOnPolyline P polyline
              Return the point on a polyline which is closest to a given
              point.

              list P List of two numbers, the coordinates of the point

              list polyline
                     List of numbers, the vertices of the polyline


       ::math::geometry::lengthOfPolyline polyline
              Return the length of the polyline (note: it not regarded as a
              polygon)

              list polyline
                     List of numbers, the vertices of the polyline


       ::math::geometry::movePointInDirection P direction dist
              Move a point over a given distance in a given direction and
              return the new coordinates (in two dimensions only).

              list P Coordinates of the point to be moved

              double direction
                     Direction (in degrees; 0 is to the right, 90 upwards)

              list dist
                     Distance over which to move the point


       ::math::geometry::lineSegmentsIntersect linesegment1 linesegment2
              Check if two line segments intersect or coincide. Returns 1 if
              that is the case, 0 otherwise (in two dimensions only).

              list linesegment1
                     First line segment

              list linesegment2
                     Second line segment


       ::math::geometry::findLineSegmentIntersection linesegment1 linesegment2
              Find the intersection point of two line segments. Return the
              coordinates or the keywords "coincident" or "none" if the line
              segments coincide or have no points in common (in two dimensions
              only).

              list linesegment1
                     First line segment

              list linesegment2
                     Second line segment


       ::math::geometry::findLineIntersection line1 line2
              Find the intersection point of two (infinite) lines. Return the
              coordinates or the keywords "coincident" or "none" if the lines
              coincide or have no points in common (in two dimensions only).

              list line1
                     First line

              list line2
                     Second line
       See section References for details on the algorithm and math behind it.


       ::math::geometry::polylinesIntersect polyline1 polyline2
              Check if two polylines intersect or not (in two dimensions
              only).

              list polyline1
                     First polyline

              list polyline2
                     Second polyline


       ::math::geometry::polylinesBoundingIntersect polyline1 polyline2
       granularity
              Check whether two polylines intersect, but reduce the
              correctness of the result to the given granularity.  Use this
              for faster, but weaker, intersection checking.

              How it works:

              Each polyline is split into a number of smaller polylines,
              consisting of granularity points each. If a pair of those
              smaller lines' bounding boxes intersect, then this procedure
              returns 1, otherwise it returns 0.

              list polyline1
                     First polyline

              list polyline2
                     Second polyline

              int granularity
                     Number of points in each part (<=1 means check every
                     edge)


       ::math::geometry::intervalsOverlap y1 y2 y3 y4 strict
              Check if two intervals overlap.

              double y1,y2
                     Begin and end of first interval

              double y3,y4
                     Begin and end of second interval

              logical strict
                     Check for strict or non-strict overlap


       ::math::geometry::rectanglesOverlap P1 P2 Q1 Q2 strict
              Check if two rectangles overlap.

              list P1
                     upper-left corner of the first rectangle

              list P2
                     lower-right corner of the first rectangle

              list Q1
                     upper-left corner of the second rectangle

              list Q2
                     lower-right corner of the second rectangle

              list strict
                     choosing strict or non-strict interpretation


       ::math::geometry::bbox polyline
              Calculate the bounding box of a polyline. Returns a list of four
              coordinates: the upper-left and the lower-right corner of the
              box.

              list polyline
                     The polyline to be examined


       ::math::geometry::pointInsidePolygon P polyline
              Determine if a point is completely inside a polygon. If the
              point touches the polygon, then the point is not completely
              inside the polygon.

              list P Coordinates of the point

              list polyline
                     The polyline to be examined


       ::math::geometry::rectangleInsidePolygon P1 P2 polyline
              Determine if a rectangle is completely inside a polygon. If
              polygon touches the rectangle, then the rectangle is not
              complete inside the polygon.

              list P1
                     Upper-left corner of the rectangle

              list P2
                     Lower-right corner of the rectangle


              list polygon
                     The polygon in question


       ::math::geometry::areaPolygon polygon
              Calculate the area of a polygon.

              list polygon
                     The polygon in question

REFERENCES
       [1]    Polygon Intersection [http:/wiki.tcl.tk/12070]

       [2]    http://en.wikipedia.org/wiki/Line-line_intersection

       [3]    http://local.wasp.uwa.edu.au/~pbourke/geometry/lineline2d/

BUGS, IDEAS, FEEDBACK
       This document, and the package it describes, will undoubtedly contain
       bugs and other problems.  Please report such in the category math ::
       geometry of the Tcllib SF Trackers
       [http://sourceforge.net/tracker/?group_id=12883].  Please also report
       any ideas for enhancements you may have for either package and/or
       documentation.

KEYWORDS
       angle, distance, line, math, plane geometry, point

CATEGORY
       Mathematics

COPYRIGHT
       Copyright (c) 2001 by Ideogramic ApS and other parties
       Copyright (c) 2004 by Arjen Markus
       Copyright (c) 2010 by Andreas Kupries
       Copyright (c) 2010 by Kevin Kenny




math                                 1.1.2                   math::geometry(n)
