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







math::geometry - Geometrical computations








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












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:









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).







  

  
Coordinates of the line









  
::math::geometry::calculateDistanceToLine P line

  
Calculate the distance of point P to the (infinite) line and return the
      result







  

  
List of two numbers, the coordinates of the point

  

  
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 of two numbers, the coordinates of the point

  

  
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 of two numbers, the coordinates of the point

  

  
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 of two numbers, the coordinates of the point

  

  
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 of two numbers, the coordinates of the point

  

  
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 of two numbers, the coordinates of the point

  

  
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 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).







  

  
Coordinates of the point to be moved

  

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

  

  
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).







  

  
First line segment

  

  
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).







  

  
First line segment

  

  
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).







  

  
First line

  

  
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).







  

  
First polyline

  

  
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.

  







  

  
First polyline

  

  
Second polyline

  

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









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

  
Check if two intervals overlap.







  

  
Begin and end of first interval

  

  
Begin and end of second interval

  

  
Check for strict or non-strict overlap









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

  
Check if two rectangles overlap.







  

  
upper-left corner of the first rectangle

  

  
lower-right corner of the first rectangle

  

  
upper-left corner of the second rectangle

  

  
lower-right corner of the second rectangle

  

  
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.







  

  
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.







  

  
Coordinates of the point

  

  
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.







  

  
Upper-left corner of the rectangle

  

  
Lower-right corner of the rectangle
    

  

  

  
The polygon in question









  
::math::geometry::areaPolygon polygon

  
Calculate the area of a polygon.







  

  
The polygon in question













  
[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/










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.








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








Mathematics








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








  

    1.1.2
    math