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math::geometry(3tcl) Tcl Math Library math::geometry(3tcl) 





NAME


math::geometry - Geometrical computations






SYNOPSIS


package require Tcl ?8.5?


package require math::geometry ?1.4.1?


::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::angleBetween vector1
    vector2


::math::geometry::inproduct vector1
  vector2


::math::geometry::areaParallellogram vector1
    vector2


::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::overlapBBox polyline1
    polyline2 ?strict?


::math::geometry::pointInsideBBox bbox
  point


::math::geometry::cathetusPoint pa pb
    cathetusLength ?location?


::math::geometry::parallel line offset
    ?orient?


::math::geometry::unitVector line


::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::rotateAbout p angle
    polyline


::math::geometry::reflect angle polyline


::math::geometry::degToRad angle


::math::geometry::radToDeg angle


::math::geometry::circle centre radius


::math::geometry::circleTwoPoints point1
    point2


::math::geometry::pointInsideCircle point
    circle


::math::geometry::lineIntersectsCircle line
    circle


::math::geometry::lineSegmentIntersectsCircle
    segment circle


::math::geometry::intersectionLineWithCircle line
    circle


::math::geometry::intersectionCircleWithCircle
    circle1 circle2


::math::geometry::tangentLinesToCircle point
    circle


::math::geometry::intersectionPolylines polyline1
    polyline2 ?mode? ?granularity?


::math::geometry::intersectionPolylineCircle
    polyline circle ?mode? ?granularity?


::math::geometry::polylineCutOrigin polyline1
    polyline2 ?granularity?


::math::geometry::polylineCutEnd polyline1
    polyline2 ?granularity?


::math::geometry::splitPolyline polyline
    numberVertex


::math::geometry::enrichPolyline polyline
    accuracy


::math::geometry::cleanupPolyline polyline










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.


Note: In version 1.4.0 an inconsistency was repaired - see
    https://core.tcl-lang.org/tcllib/tktview?name=fb4812f82b. More in
    COORDINATE SYSTEM


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.
    • 
  
  • circle - a list of three numbers, the first two are the coordinates
      of the centre and the third is the radius.
    • 







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 (east), 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::angleBetween vector1 vector2

  
Calculate the angle between two vectors (in degrees)







  

  
First vector

  

  
Second vector







  
::math::geometry::inproduct vector1 vector2

  
Calculate the inner product of two vectors







  

  
First vector

  

  
Second vector







  
::math::geometry::areaParallellogram vector1
    vector2

  
Calculate the area of the parallellogram with the two vectors as its
    sides







  

  
First vector

  

  
Second vector







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







  

  
List of two numbers, the coordinates of the point

  

  
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 of two numbers, the coordinates of the point

  

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







  

  
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::overlapBBox polyline1 polyline2
    ?strict?

  
Check if the bounding boxes of two polylines overlap or not.
    
Arguments:

  







  

  
The first polyline

  

  
The second polyline

  

  
Whether strict overlap is to checked (1) or if the bounding boxes may
      touch (0, default)







  
::math::geometry::pointInsideBBox bbox point

  

    
Check if the point is inside or on the bounding box or not.
        Arguments:

  







  

  
The bounding box given as a list of x/y coordinates

  

  
The point to be checked







  
::math::geometry::cathetusPoint pa pb
    cathetusLength ?location?

  
Return the third point of the rectangular triangle defined by the two
      given end points of the hypothenusa. The triangle's side from point A (or
      B, if the location is given as "b") to the third point is the
      cathetus length. If the cathetus' length is lower than the length of the
      hypothenusa, an empty list is returned.
    
Arguments:

  







  

  
The starting point on hypotenuse

  

  
The ending point on hypotenuse

  

  
The length of the cathetus of the triangle

  

  
The location of the given cathetus, "a" means given cathetus
      shares point pa (default) "b" means given cathetus shares point
      pb







  
::math::geometry::parallel line offset ?orient?

  
Return a line parallel to the given line, with a distance
      "offset". The orientation is determined by the two points
      defining the line.
    
Arguments:

  







  

  
The given line

  

  
The distance to the given line

  

  
Orientation of the new line with respect to the given line (defaults to
      "right")









  
::math::geometry::unitVector line

  
Return a unit vector from the given line or direction, if the line
      argument is a single point (then a line through the origin is assumed)
      Arguments:







  

  
The line in question (or a single point, implying a line through the
      origin)







  
::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::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.







  

  
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







  
::math::geometry::translate vector polyline

  
Translate a polyline over a given vector







  

  
Translation vector

  

  
The polyline to be translated







  
::math::geometry::rotate angle polyline

  
Rotate a polyline over a given angle (degrees) around the origin







  

  
Angle over which to rotate the polyline (degrees)

  

  
The polyline to be rotated







  
::math::geometry::rotateAbout p angle
    polyline

  
Rotate a polyline around a given point p and return the new polyline.
    
Arguments:

  







  

  
The point of rotation

  

  
The angle over which to rotate the polyline (degrees)

  

  
The polyline to be rotated







  
::math::geometry::reflect angle polyline

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







  

  
Angle of the line of reflection (degrees)

  

  
The polyline to be reflected







  
::math::geometry::degToRad angle

  
Convert from degrees to radians







  

  
Angle in degrees







  
::math::geometry::radToDeg angle

  
Convert from radians to degrees







  

  
Angle in radians







  
::math::geometry::circle centre radius

  
Convenience procedure to create a circle from a point and a radius.







  

  
Coordinates of the circle centre

  

  
Radius of the circle







  
::math::geometry::circleTwoPoints point1 point2

  
Convenience procedure to create a circle from two points on its
      circumference The centre is the point between the two given points, the
      radius is half the distance between them.







  

  
First point

  

  
Second point







  
::math::geometry::pointInsideCircle point circle

  
Determine if the given point is inside the circle or on the circumference
      (1) or outside (0).







  

  
Point to be checked

  

  
Circle that may or may not contain the point







  
::math::geometry::lineIntersectsCircle line
    circle

  
Determine if the given line intersects the circle or touches it (1) or
      does not (0).







  

  
Line to be checked

  

  
Circle that may or may not be intersected







  
::math::geometry::lineSegmentIntersectsCircle segment
    circle

  
Determine if the given line segment intersects the circle or touches it
      (1) or does not (0).







  

  
Line segment to be checked

  

  
Circle that may or may not be intersected







  
::math::geometry::intersectionLineWithCircle line
    circle

  
Determine the points at which the given line intersects the circle. There
      can be zero, one or two points. (If the line touches the circle or is
      close to it, then one point is returned. An arbitrary margin of 1.0e-10
      times the radius is used to determine this situation.)







  

  
Line to be checked

  

  
Circle that may or may not be intersected







  
::math::geometry::intersectionCircleWithCircle circle1
    circle2

  
Determine the points at which the given two circles intersect. There can
      be zero, one or two points. (If the two circles touch the circle or are
      very close, then one point is returned. An arbitrary margin of 1.0e-10
      times the mean of the radii of the two circles is used to determine this
      situation.)







  

  
First circle

  

  
Second circle







  
::math::geometry::tangentLinesToCircle point
    circle

  
Determine the tangent lines from the given point to the circle. There can
      be zero, one or two lines. (If the point is on the cirucmference or very
      close to the circle, then one line is returned. An arbitrary margin of
      1.0e-10 times the radius of the circle is used to determine this
      situation.)







  

  
Point in question

  

  
Circle to which the tangent lines are to be determined







  
::math::geometry::intersectionPolylines polyline1
    polyline2 ?mode? ?granularity?

  
Return the first point or all points where the two polylines intersect. If
      the number of points in the polylines is large, you can use the
      granularity to get an approximate answer faster.
    
Arguments:

  







  

  
The first polyline

  

  
The second polyline

  

  
Whether to return only the first (default) or to return all intersection
      points ("all")

  

  
The number of points that will be skipped plus 1 in the search for
      intersection points (1 or smaller means an exact answer is returned)







  
::math::geometry::intersectionPolylineCircle polyline
    circle ?mode? ?granularity?

  
Return the first point or all points where the polyline intersects the
      circle. If the number of points in the polyline is large, you can use the
      granularity to get an approximate answer faster.
    
Arguments:

  







  

  
The polyline that may intersect the circle

  

  
The circle in question

  

  
Whether to return only the first (default) or to return all intersection
      points ("all")

  

  
The number of points that will be skipped plus 1 in the search for
      intersection points (1 or smaller means an exact answer is returned)







  
::math::geometry::polylineCutOrigin polyline1
    polyline2 ?granularity?

  
Return the part of the first polyline from the origin up to the first
      intersection with the second. If the number of points in the polyline is
      large, you can use the granularity to get an approximate answer faster.
    
Arguments:

  







  

  
The first polyline (from which a part is to be returned)

  

  
The second polyline

  

  
The number of points that will be skipped plus 1 in the search for
      intersection points (1 or smaller means an exact answer is returned)







  
::math::geometry::polylineCutEnd polyline1 polyline2
    ?granularity?

  
Return the part of the first polyline from the last intersection point
      with the second to the end. If the number of points in the polyline is
      large, you can use the granularity to get an approximate answer faster.
    
Arguments:

  







  

  
The first polyline (from which a part is to be returned)

  

  
The second polyline

  

  
The number of points that will be skipped plus 1 in the search for
      intersection points (1 or smaller means an exact answer is returned)







  
::math::geometry::splitPolyline polyline
    numberVertex

  
Split the poyline into a set of polylines where each separate polyline
      holds "numberVertex" vertices between the two end points.
    
Arguments:

  







  

  
The polyline to be split up

  

  
The number of "internal" vertices







  
::math::geometry::enrichPolyline polyline
    accuracy

  
Split up each segment of a polyline into a number of smaller segments and
      return the result.
    
Arguments:

  







  

  
The polyline to be refined

  

  
The number of subsegments to be created







  
::math::geometry::cleanupPolyline polyline

  
Remove duplicate neighbouring vertices and return the result.
    
Arguments:

  







  

  
The polyline to be cleaned up










COORDINATE SYSTEM


The coordinate system used by the package is the ordinary
    cartesian system, where the positive x-axis is directed to the right and the
    positive y-axis is directed upwards. Angles and directions are defined with
    respect to the positive x-axis in a counter-clockwise direction, so that an
    angle of 90 degrees is the direction of the positive y-axis. Note that the
    Tk canvas coordinates differ from this, as there the origin is located in
    the upper left corner of the window. Up to and including version 1.3, the
    direction and octant procedures of this package used this convention
    inconsistently.






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) 2010 by Andreas Kupries
Copyright (c) 2010 by Kevin Kenny
Copyright (c) 2018 by Arjen Markus
Copyright (c) 2020 by Manfred Rosenberger








  

    1.4.1
    tcllib