 math::geometry(3) Geometrical computations

SYNOPSIS

package require Tcl ?8.3?

package require math::geometry ?1.0.3?

::math::geometry::angle line

::math::geometry::calculateDistanceToLine P line

::math::geometry::calculateDistanceToLineSegment P linesegment

::math::geometry::calculateDistanceToPolyline P polyline

::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::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.
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::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).
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

::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

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 SF Trackers [http://sourceforge.net/tracker/?group_id=12883]. Please also report any ideas for enhancements you may have for either package and/or documentation.

KEYWORDS

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

Mathematics