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# math::geometry(n) 1.1.3 tcllib "Tcl Math Library"

## Name

math::geometry - Geometrical computations

## Synopsis

- package require Tcl ?8.3?
- package require math::geometry ?1.1.3?

- ::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::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::pointInsidePolygon P polyline
- ::math::geometry::rectangleInsidePolygon P1 P2 polyline
- ::math::geometry::areaPolygon polygon

## 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.

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.

## 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 (up), 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::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::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::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

## 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.

## Keywords

angle, distance, line, math, plane geometry, point

## Category

Mathematics

## Copyright

Copyright © 2001 by Ideogramic ApS and other parties
Copyright © 2004 by Arjen Markus
Copyright © 2010 by Andreas Kupries
Copyright © 2010 by Kevin Kenny