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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

____________________________________________________________________________________________________________

DESCRIPTION
       The  math::geometry  package  is a collection of functions for computations and manipulations on two-dimensional twodimensional
       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-coordi-nates y-coordinates
       nates 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:

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

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

       •      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.

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

       •      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).

       •      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://source -
       forge.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)