math::geometry(3tcl)

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 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

::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

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.

•   circle - a list of three numbers, the first two are the coordinates 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 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::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 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 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 radius.

    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 circumference (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 circle. 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 arbitrary 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 arbitrary 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 returned. 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 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) 2010 by Andreas Kupries
Copyright (c) 2010 by Kevin Kenny
Copyright (c) 2018 by Arjen Markus