Provided by: tcllib_1.19-dfsg-2_all bug

NAME

       math::geometry - Geometrical computations

SYNOPSIS

       package require Tcl  ?8.5?

       package require math::geometry  ?1.2.3?

       ::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::calculateDistanceToPolygon P polygon

       ::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::pointInsidePolygonAlt P polyline

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

       ::math::geometry::areaPolygon polygon

       ::math::geometry::translate vector polyline

       ::math::geometry::rotate angle polyline

       ::math::geometry::reflect angle polyline

       ::math::geometry::degToRad angle

       ::math::geometry::radToDeg angle

_________________________________________________________________________________________________

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:

       •      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. Note  that
              a polyline needs not to be closed.

              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::calculateDistanceToPolygon P polygon
              Calculate the distance of point P to the polygon and return the result. If the list
              of coordinates is not closed (first and last points differ),  it  is  automatically
              closed.

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

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

       ::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). If an endpoint of one segment lies on the other
              segment (or is very close to the segment), they are considered to intersect

              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::pointInsidePolygonAlt 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.  Note:  this
              alternative  procedure  uses  the  so-called  winding  number to determine this. It
              handles self-intersecting polygons in a "natural" way.

              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

       ::math::geometry::translate vector polyline
              Translate a polyline over a given vector

              list vector
                     Translation vector

              list polyline
                     The polyline to be rotated

       ::math::geometry::rotate angle polyline
              Rotate a polyline over a given angle (degrees) around the origin

              list angle
                     Angle over which to rotate the polyline (degrees)

              list polyline
                     The polyline to be translated

       ::math::geometry::reflect angle polyline
              Reflect  a  polyline in a line through the origin at a given angle (degrees) to the
              x-axis

              list angle
                     Angle of the line of reflection (degrees)

              list polyline
                     The polyline to be reflected

       ::math::geometry::degToRad angle
              Convert from degrees to radians

              list angle
                     Angle in degrees

       ::math::geometry::radToDeg angle
              Convert from radians to degrees

              list angle
                     Angle in radians

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 Trackers
       [http://core.tcl.tk/tcllib/reportlist].  Please also report any ideas for enhancements you
       may have for either package and/or documentation.

       When proposing code changes, please provide unified diffs, i.e the output of diff -u.

       Note further that attachments are strongly preferred over inlined patches. Attachments can
       be made by going to the Edit form of the ticket immediately after its creation,  and  then
       using the left-most button in the secondary navigation bar.

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