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NAME
math::geometry  Geometrical computations
Table Of Contents
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 twodimensional 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 ycoordinate of a first point and the x and ycoordinates of a second point on the line.
Note: In version 1.4.0 an inconsistency was repaired  see https://core.tcllang.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 ycoordinates respectively.
line  a list of four coordinates, interpreted as the x and ycoordinates of two distinct points on the line.
line segment  a list of four coordinates, interpreted as the x and ycoordinates 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 ycoordinates 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.

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).

Returns a horizontal vector on the Xaxis of the specified length. Positive lengths point to the right (east).

Returns a vertical vector on the Yaxis 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
 east
 northeast
 north
 northwest
 west
 southwest
 south
 southeast
Each octant is the arc of the circle +/ 22.5 degrees from the cardinal direction the octant is named for.

Construct a rectangle from its northwest and southeast corners and return it as the result of the command.

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

Calculate the angle from the positive xaxis to a given line (in two dimensions only).
list line
Coordinates of the line
::math::geometry::angleBetween vector1 vector2
Calculate the angle between two vectors (in degrees)
list vector1
First vector
list vector2
Second vector
::math::geometry::inproduct vector1 vector2
Calculate the inner product of two vectors
list vector1
First vector
list vector2
Second vector
::math::geometry::areaParallellogram vector1 vector2
Calculate the area of the parallellogram with the two vectors as its sides
list vector1
First vector
list vector2
Second vector
::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 nonstrict overlap
::math::geometry::rectanglesOverlap P1 P2 Q1 Q2 strict
Check if two rectangles overlap.
list P1
upperleft corner of the first rectangle
list P2
lowerright corner of the first rectangle
list Q1
upperleft corner of the second rectangle
list Q2
lowerright corner of the second rectangle
list strict
choosing strict or nonstrict interpretation
::math::geometry::bbox polyline
Calculate the bounding box of a polyline. Returns a list of four coordinates: the upperleft and the lowerright corner of the box.
list polyline
The polyline to be examined
::math::geometry::overlapBBox polyline1 polyline2 ?strict?
Check if the bounding boxes of two polylines overlap or not.
Arguments:
list polyline1
The first polyline
list polyline1
The second polyline
int strict
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:
list bbox
The bounding box given as a list of x/y coordinates
list point
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:
list pa
The starting point on hypotenuse
list pb
The ending point on hypotenuse
float cathetusLength
The length of the cathetus of the triangle
string location
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:
list line
The given line
float offset
The distance to the given line
string orient
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:
list line
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.
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 socalled winding number to determine this. It handles selfintersecting 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
Upperleft corner of the rectangle
list P2
Lowerright 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 translated
::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 rotated
::math::geometry::rotateAbout p angle polyline
Rotate a polyline around a given point p and return the new polyline.
Arguments:
list p
The point of rotation
float angle
The angle over which to rotate the polyline (degrees)
list polyline
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 xaxis
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
::math::geometry::circle centre radius
Convenience procedure to create a circle from a point and a radius.
list centre
Coordinates of the circle centre
list radius
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.
list point1
First point
list point2
Second point
::math::geometry::pointInsideCircle point circle
Determine if the given point is inside the circle or on the circumference (1) or outside (0).
list point
Point to be checked
list circle
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).
list line
Line to be checked
list circle
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).
list segment
Line segment to be checked
list circle
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 1.0e10 times the radius is used to determine this situation.)
list line
Line to be checked
list circle
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 1.0e10 times the mean of the radii of the two circles is used to determine this situation.)
list circle1
First circle
list circle2
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 1.0e10 times the radius of the circle is used to determine this situation.)
list point
Point in question
list circle
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:
list polyline1
The first polyline
list polyline2
The second polyline
string mode
Whether to return only the first (default) or to return all intersection points ("all")
int granularity
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:
list polyline
The polyline that may intersect the circle
list circle
The circle in question
string mode
Whether to return only the first (default) or to return all intersection points ("all")
int granularity
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:
list polyline1
The first polyline (from which a part is to be returned)
list polyline2
The second polyline
int granularity
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:
list polyline1
The first polyline (from which a part is to be returned)
list polyline2
The second polyline
int granularity
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:
list polyline
The polyline to be split up
int numberVertex
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:
list polyline
The polyline to be refined
int accuracy
The number of subsegments to be created
::math::geometry::cleanupPolyline polyline
Remove duplicate neighbouring vertices and return the result.
Arguments:
list polyline
The polyline to be cleaned up
COORDINATE SYSTEM
The coordinate system used by the package is the ordinary cartesian system, where the positive xaxis is directed to the right and the positive yaxis is directed upwards. Angles and directions are defined with respect to the positive xaxis in a counterclockwise direction, so that an angle of 90 degrees is the direction of the positive yaxis. 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
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. 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 leftmost button in the secondary navigation bar.
KEYWORDS
angle, distance, line, math, plane geometry, point
CATEGORY
Mathematics
COPYRIGHT
Copyright © 2001 by Ideogramic ApS and other parties
Copyright © 2010 by Andreas Kupries
Copyright © 2010 by Kevin Kenny
Copyright © 2018 by Arjen Markus
Copyright © 2020 by Manfred Rosenberger