NAME¶
math::geometry - Geometrical computations
SYNOPSIS¶
package require
Tcl ?8.3?
package require
math::geometry ?1.1.2?
::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::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::+ 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.
- 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
- 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 [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 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
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