# Tcl Library Source Code

math::geometry - Tcl Math Library
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# math::geometry(n) 1.3.0 tcllib "Tcl Math Library"

## Name

math::geometry - Geometrical computations

## 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::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

list angle

Angle in degrees

list angle

Convenience procedure to create a circle from a point and a radius.

list centre

Coordinates of the circle centre

::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

## 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. 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.

Mathematics