math::geometry - Geometrical computations
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.
Note: In version 1.4.0 an inconsistency was repaired - see https://core.tcl-lang.org/tcllib/tktview?name=fb4812f82b. More in COORDINATE SYSTEM
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.
The package defines the following public procedures:
Compute the sum of the two vectors given as points and return it. The result is a vector as well.
Compute the difference (point1 - point2) of the two vectors given as points and return it. The result is a vector as well.
Construct a point from its coordinates and return it as the result of the command.
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]
Compute the length of the vector and return it as the result of the command.
Scale the vector by the factor and return it as the result of the command. This is a vector as well.
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 (east), and for angle == 90 up (north).
Returns a horizontal vector on the X-axis of the specified length. Positive lengths point to the right (east).
Returns a vertical vector on the Y-axis of the specified length. Positive lengths point down (south).
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.
Compute the octant of the circle the point is in and return it as the result of the command. The possible results are
east
northeast
north
northwest
west
southwest
south
southeast
Each octant is the arc of the circle +/- 22.5 degrees from the cardinal direction the octant is named for.
Construct a rectangle from its northwest and southeast corners and return it as the result of the command.
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).
Calculate the angle from the positive x-axis to a given line (in two dimensions only).
Coordinates of the line
Calculate the angle between two vectors (in degrees)
First vector
Second vector
Calculate the inner product of two vectors
First vector
Second vector
Calculate the area of the parallellogram with the two vectors as its sides
First vector
Second vector
Calculate the distance of point P to the (infinite) line and return the result
List of two numbers, the coordinates of the point
List of four numbers, the coordinates of two points on the line
Calculate the distance of point P to the (finite) line segment and return the result.
List of two numbers, the coordinates of the point
List of four numbers, the coordinates of the first and last points of the line segment
Calculate the distance of point P to the polyline and return the result. Note that a polyline needs not to be closed.
List of two numbers, the coordinates of the point
List of numbers, the coordinates of the vertices of the polyline
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 of two numbers, the coordinates of the point
List of numbers, the coordinates of the vertices of the polygon
Return the point on a line which is closest to a given point.
List of two numbers, the coordinates of the point
List of four numbers, the coordinates of two points on the line
Return the point on a line segment which is closest to a given point.
List of two numbers, the coordinates of the point
List of four numbers, the first and last points on the line segment
Return the point on a polyline which is closest to a given point.
List of two numbers, the coordinates of the point
List of numbers, the vertices of the polyline
Return the length of the polyline (note: it not regarded as a polygon)
List of numbers, the vertices of the polyline
Move a point over a given distance in a given direction and return the new coordinates (in two dimensions only).
Coordinates of the point to be moved
Direction (in degrees; 0 is to the right, 90 upwards)
Distance over which to move the point
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
First line segment
Second line segment
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).
First line segment
Second line segment
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).
First line
Second line
See section References for details on the algorithm and math behind it.
Check if two polylines intersect or not (in two dimensions only).
First polyline
Second polyline
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.
First polyline
Second polyline
Number of points in each part (<=1 means check every edge)
Check if two intervals overlap.
Begin and end of first interval
Begin and end of second interval
Check for strict or non-strict overlap
Check if two rectangles overlap.
upper-left corner of the first rectangle
lower-right corner of the first rectangle
upper-left corner of the second rectangle
lower-right corner of the second rectangle
choosing strict or non-strict interpretation
Calculate the bounding box of a polyline. Returns a list of four coordinates: the upper-left and the lower-right corner of the box.
The polyline to be examined
Check if the bounding boxes of two polylines overlap or not.
Arguments:
The first polyline
The second polyline
Whether strict overlap is to checked (1) or if the bounding boxes may touch (0, default)
Check if the point is inside or on the bounding box or not. Arguments:
The bounding box given as a list of x/y coordinates
The point to be checked
Return the third point of the rectangular triangle defined by the two given end points of the hypothenusa. The triangle's side from point A (or B, if the location is given as "b") to the third point is the cathetus length. If the cathetus' length is lower than the length of the hypothenusa, an empty list is returned.
Arguments:
The starting point on hypotenuse
The ending point on hypotenuse
The length of the cathetus of the triangle
The location of the given cathetus, "a" means given cathetus shares point pa (default) "b" means given cathetus shares point pb
Return a line parallel to the given line, with a distance "offset". The orientation is determined by the two points defining the line.
Arguments:
The given line
The distance to the given line
Orientation of the new line with respect to the given line (defaults to "right")
Return a unit vector from the given line or direction, if the line argument is a single point (then a line through the origin is assumed) Arguments:
The line in question (or a single point, implying a line through the origin)
Determine if a point is completely inside a polygon. If the point touches the polygon, then the point is not completely inside the polygon.
Coordinates of the point
The polyline to be examined
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.
Coordinates of the point
The polyline to be examined
Determine if a rectangle is completely inside a polygon. If polygon touches the rectangle, then the rectangle is not complete inside the polygon.
Upper-left corner of the rectangle
Lower-right corner of the rectangle
The polygon in question
Calculate the area of a polygon.
The polygon in question
Translate a polyline over a given vector
Translation vector
The polyline to be translated
Rotate a polyline over a given angle (degrees) around the origin
Angle over which to rotate the polyline (degrees)
The polyline to be rotated
Rotate a polyline around a given point p and return the new polyline.
Arguments:
The point of rotation
The angle over which to rotate the polyline (degrees)
The polyline to be rotated
Reflect a polyline in a line through the origin at a given angle (degrees) to the x-axis
Angle of the line of reflection (degrees)
The polyline to be reflected
Convert from degrees to radians
Angle in degrees
Convert from radians to degrees
Angle in radians
Convenience procedure to create a circle from a point and a radius.
Coordinates of the circle centre
Radius of the circle
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.
First point
Second point
Determine if the given point is inside the circle or on the circumference (1) or outside (0).
Point to be checked
Circle that may or may not contain the point
Determine if the given line intersects the circle or touches it (1) or does not (0).
Line to be checked
Circle that may or may not be intersected
Determine if the given line segment intersects the circle or touches it (1) or does not (0).
Line segment to be checked
Circle that may or may not be intersected
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.)
Line to be checked
Circle that may or may not be intersected
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.)
First circle
Second 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.)
Point in question
Circle to which the tangent lines are to be determined
Return the first point or all points where the two polylines intersect. If the number of points in the polylines is large, you can use the granularity to get an approximate answer faster.
Arguments:
The first polyline
The second polyline
Whether to return only the first (default) or to return all intersection points ("all")
The number of points that will be skipped plus 1 in the search for intersection points (1 or smaller means an exact answer is returned)
Return the first point or all points where the polyline intersects the circle. If the number of points in the polyline is large, you can use the granularity to get an approximate answer faster.
Arguments:
The polyline that may intersect the circle
The circle in question
Whether to return only the first (default) or to return all intersection points ("all")
The number of points that will be skipped plus 1 in the search for intersection points (1 or smaller means an exact answer is returned)
Return the part of the first polyline from the origin up to the first intersection with the second. If the number of points in the polyline is large, you can use the granularity to get an approximate answer faster.
Arguments:
The first polyline (from which a part is to be returned)
The second polyline
The number of points that will be skipped plus 1 in the search for intersection points (1 or smaller means an exact answer is returned)
Return the part of the first polyline from the last intersection point with the second to the end. If the number of points in the polyline is large, you can use the granularity to get an approximate answer faster.
Arguments:
The first polyline (from which a part is to be returned)
The second polyline
The number of points that will be skipped plus 1 in the search for intersection points (1 or smaller means an exact answer is returned)
Split the poyline into a set of polylines where each separate polyline holds "numberVertex" vertices between the two end points.
Arguments:
The polyline to be split up
The number of "internal" vertices
Split up each segment of a polyline into a number of smaller segments and return the result.
Arguments:
The polyline to be refined
The number of subsegments to be created
Remove duplicate neighbouring vertices and return the result.
Arguments:
The polyline to be cleaned up
The coordinate system used by the package is the ordinary cartesian system, where the positive x-axis is directed to the right and the positive y-axis is directed upwards. Angles and directions are defined with respect to the positive x-axis in a counter-clockwise direction, so that an angle of 90 degrees is the direction of the positive y-axis. Note that the Tk canvas coordinates differ from this, as there the origin is located in the upper left corner of the window. Up to and including version 1.3, the direction and octant procedures of this package used this convention inconsistently.
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
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