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

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).

::math::geometry::angleBetween vector1 vector2

Calculate the angle between two vectors (in degrees)

::math::geometry::inproduct vector1 vector2

Calculate the inner product of two vectors

::math::geometry::areaParallellogram vector1 vector2

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

::math::geometry::calculateDistanceToLine P line

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

::math::geometry::calculateDistanceToLineSegment P linesegment

Calculate the distance of point P to the (finite) line segment and return the result.

::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.

::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.

::math::geometry::findClosestPointOnLine P line

Return the point on a line which is closest to a given point.

::math::geometry::findClosestPointOnLineSegment P linesegment

Return the point on a line segment which is closest to a given point.

::math::geometry::findClosestPointOnPolyline P polyline

Return the point on a polyline which is closest to a given point.

::math::geometry::lengthOfPolyline polyline

Return the length of the polyline (note: it not regarded as a polygon)

::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).

::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

::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).

::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).

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).

::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.

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

Check if two intervals overlap.

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

Check if two rectangles overlap.

::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.

::math::geometry::overlapBBox polyline1 polyline2 ?strict?

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

Arguments:

::math::geometry::pointInsideBBox bbox point

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

::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:

::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:

::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:

::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.

::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.

::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.

::math::geometry::areaPolygon polygon

Calculate the area of a polygon.

::math::geometry::translate vector polyline

Translate a polyline over a given vector

::math::geometry::rotate angle polyline

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

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

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

Arguments:

::math::geometry::reflect angle polyline

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

::math::geometry::degToRad angle

Convert from degrees to radians

::math::geometry::radToDeg angle

Convert from radians to degrees

::math::geometry::circle centre radius

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

::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.

::math::geometry::pointInsideCircle point circle

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

::math::geometry::lineIntersectsCircle line circle

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

::math::geometry::lineSegmentIntersectsCircle segment circle

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

::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.)

::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.)

::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.)

::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:

::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:

::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:

::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:

::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:

::math::geometry::enrichPolyline polyline accuracy

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

Arguments:

::math::geometry::cleanupPolyline polyline

Remove duplicate neighbouring vertices and return the result.

Arguments:

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