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# math::geometry(n) 1.4.1 tcllib "Tcl Math Library"

## 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).

  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 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::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 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 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 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

::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 1.0e-10 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 1.0e-10 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 1.0e-10 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 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

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. 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 © 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