math::geometry(3tcl) Tcl Math Library math::geometry(3tcl)
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NAME
math::geometry - Geometrical computations
SYNOPSIS
package require Tcl ?8.5?
package require math::geometry ?1.3.0?
::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::angleBetween vector1 vector2
::math::geometry::inproduct vector1 vector2
::math::geometry::areaParallellogram vector1 vector2
::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 granu-
larity
::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
::math::geometry::circle centre radius
::math::geometry::circleTwoPoints point1 point2
::math::geometry::pointInsideCircle point circle
::math::geometry::lineIntersectsCircle line circle
::math::geometry::lineSegmentIntersectsCircle segment circle
::math::geometry::intersectionLineWithCircle line circle
::math::geometry::intersectionCircleWithCircle circle1 circle2
::math::geometry::tangentLinesToCircle point circle
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DESCRIPTION
The math::geometry package is a collection of functions for computa-
tions 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:
o point - a list of two coordinates representing the x- and y-co-
ordinates respectively.
o line - a list of four coordinates, interpreted as the x- and y-
coordinates of two distinct points on the line.
o 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.
o polyline - a list of an even number of coordinates, interpreted
as the x- and y-coordinates of an ordered set of points.
o polygon - like a polyline, but the implicit assumption is that
the polyline is closed (if the first and last points do not co-
incide, the missing segment is automatically added).
o point set - again a list of an even number of coordinates, but
the points are regarded without any ordering.
o circle - a list of three numbers, the first two are the coordi-
nates of the centre and the third is the radius.
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 re-
sult 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 rela-
tive 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 con-
taining 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::angleBetween vector1 vector2
Calculate the angle between two vectors (in degrees)
list vector1
First vector
list vector2
Second vector
::math::geometry::inproduct vector1 vector2
Calculate the inner product of two vectors
list vector1
First vector
list vector2
Second vector
::math::geometry::areaParallellogram vector1 vector2
Calculate the area of the parallellogram with the two vectors as
its sides
list vector1
First vector
list vector2
Second vector
::math::geometry::calculateDistanceToLine P line
Calculate the distance of point P to the (infinite) line and re-
turn 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 re-
turn 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 co-
ordinates 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 be-
hind 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 granu-
larity
Check whether two polylines intersect, but reduce the correct-
ness 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, con-
sisting 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 in-
side 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 in-
side the polygon. Note: this alternative procedure uses the so-
called winding number to determine this. It handles self-inter-
secting 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 com-
plete 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 translated
::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 rotated
::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
::math::geometry::circle centre radius
Convenience procedure to create a circle from a point and a ra-
dius.
list centre
Coordinates of the circle centre
list radius
Radius of the circle
::math::geometry::circleTwoPoints point1 point2
Convenience procedure to create a circle from two points on its
circumference The centre is the point between the two given
points, the radius is half the distance between them.
list point1
First point
list point2
Second point
::math::geometry::pointInsideCircle point circle
Determine if the given point is inside the circle or on the cir-
cumference (1) or outside (0).
list point
Point to be checked
list circle
Circle that may or may not contain the point
::math::geometry::lineIntersectsCircle line circle
Determine if the given line intersects the circle or touches it
(1) or does not (0).
list line
Line to be checked
list circle
Circle that may or may not be intersected
::math::geometry::lineSegmentIntersectsCircle segment circle
Determine if the given line segment intersects the circle or
touches it (1) or does not (0).
list segment
Line segment to be checked
list circle
Circle that may or may not be intersected
::math::geometry::intersectionLineWithCircle line circle
Determine the points at which the given line intersects the cir-
cle. There can be zero, one or two points. (If the line touches
the circle or is close to it, then one point is returned. An ar-
bitrary margin of 1.0e-10 times the radius is used to determine
this situation.)
list line
Line to be checked
list circle
Circle that may or may not be intersected
::math::geometry::intersectionCircleWithCircle circle1 circle2
Determine the points at which the given two circles intersect.
There can be zero, one or two points. (If the two circles touch
the circle or are very close, then one point is returned. An ar-
bitrary margin of 1.0e-10 times the mean of the radii of the two
circles is used to determine this situation.)
list circle1
First circle
list circle2
Second circle
::math::geometry::tangentLinesToCircle point circle
Determine the tangent lines from the given point to the circle.
There can be zero, one or two lines. (If the point is on the
cirucmference or very close to the circle, then one line is re-
turned. An arbitrary margin of 1.0e-10 times the radius of the
circle is used to determine this situation.)
list point
Point in question
list circle
Circle to which the tangent lines are to be determined
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 out-
put 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) 2010 by Andreas Kupries
Copyright (c) 2010 by Kevin Kenny
Copyright (c) 2018 by Arjen Markus
tcllib 1.3.0 math::geometry(3tcl)