bionic (3) math_geometry.3tcl.gz
NAME
math::geometry - Geometrical computations
SYNOPSIS
package require Tcl ?8.5?
package require math::geometry ?1.2.3?
::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::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::reflect angle polyline
::math::geometry::degToRad angle
::math::geometry::radToDeg angle
________________________________________________________________________________________________________________
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::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 rotated
::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 translated
::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
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) 2004 by Arjen Markus
Copyright (c) 2010 by Andreas Kupries
Copyright (c) 2010 by Kevin Kenny