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math::geometry

Geometrical computations

literal type exactly as shown argument replace with your value ?optional? may be omitted

Synopsis#

package require Tcl ?8.5 9?

package require math::geometry ?1.4.2?

::math::geometry::+ point1 point2

::math::geometry::- point1 point2

::math::geometry::p x y

::math::geometry::distance point1 point2

::math::geometry::length point

::math::geometry::s* factor point

::math::geometry::direction angle

::math::geometry::h length

::math::geometry::v length

::math::geometry::between point1 point2 s

::math::geometry::octant point

::math::geometry::rect nw se

::math::geometry::nwse rect

::math::geometry::angle line

::math::geometry::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:

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 [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.

Category#

Mathematics

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

Keywords#

angle · distance · line · math · plane geometry · point