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math::geometry - Geometrical computations

- 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*

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.

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

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.

**::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

- list
**::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

- list
**::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

- list
**::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

- list
**::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

- list
**::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

- list
**::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

- list
**::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

- list
**::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

- list
**::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

- list
**::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

- list
**::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

- list
**::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.

- list
**::math::geometry::polylinesIntersect***polyline1**polyline2*Check if two polylines intersect or not (in two dimensions only).

- list
*polyline1* First polyline

- list
*polyline2* Second polyline

- list
**::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)

- list
**::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

- double
**::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

- list
**::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

- list
**::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

- list
**::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

- list
**::math::geometry::areaPolygon***polygon*Calculate the area of a polygon.

- list
*polygon* The polygon in question

- list

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.

Mathematics

Copyright © 2001 by Ideogramic ApS and other parties

Copyright © 2004 by Arjen Markus

Copyright © 2010 by Andreas Kupries

Copyright © 2010 by Kevin Kenny