1math::geometry(n) Tcl Math Library math::geometry(n)
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6
8 math::geometry - Geometrical computations
9
11 package require Tcl ?8.5?
12 package require math::geometry ?1.2.3?
14 ::math::geometry::+ point1 point2
16 ::math::geometry::- point1 point2
18 ::math::geometry::p x y
20 ::math::geometry::distance point1 point2
22 ::math::geometry::length point
24 ::math::geometry::s* factor point
26 ::math::geometry::direction angle
28 ::math::geometry::h length
30 ::math::geometry::v length
32 ::math::geometry::between point1 point2 s
34 ::math::geometry::octant point
36 ::math::geometry::rect nw se
38 ::math::geometry::nwse rect
40 ::math::geometry::angle line
42 ::math::geometry::calculateDistanceToLine P line
44 ::math::geometry::calculateDistanceToLineSegment P linesegment
46 ::math::geometry::calculateDistanceToPolyline P polyline
48 ::math::geometry::calculateDistanceToPolygon P polygon
50 ::math::geometry::findClosestPointOnLine P line
52 ::math::geometry::findClosestPointOnLineSegment P linesegment
54 ::math::geometry::findClosestPointOnPolyline P polyline
56 ::math::geometry::lengthOfPolyline polyline
58 ::math::geometry::movePointInDirection P direction dist
60 ::math::geometry::lineSegmentsIntersect linesegment1 linesegment2
62 ::math::geometry::findLineSegmentIntersection linesegment1 linesegment2
64 ::math::geometry::findLineIntersection line1 line2
66 ::math::geometry::polylinesIntersect polyline1 polyline2
68 ::math::geometry::polylinesBoundingIntersect polyline1 polyline2 granu‐
70 larity
71 ::math::geometry::intervalsOverlap y1 y2 y3 y4 strict
73 ::math::geometry::rectanglesOverlap P1 P2 Q1 Q2 strict
75 ::math::geometry::bbox polyline
77 ::math::geometry::pointInsidePolygon P polyline
79 ::math::geometry::pointInsidePolygonAlt P polyline
81 ::math::geometry::rectangleInsidePolygon P1 P2 polyline
83 ::math::geometry::areaPolygon polygon
85 ::math::geometry::translate vector polyline
87 ::math::geometry::rotate angle polyline
89 ::math::geometry::reflect angle polyline
91 ::math::geometry::degToRad angle
93 ::math::geometry::radToDeg angle
95______________________________________________________________________________
97
99 The math::geometry package is a collection of functions for computa‐
100 tions and manipulations on two-dimensional geometrical objects, such as
101 points, lines and polygons.
102 The geometrical objects are implemented as plain lists of coordinates.
104 For instance a line is defined by a list of four numbers, the x- and y-
105 coordinate of a first point and the x- and y-coordinates of a second
106 point on the line.
107 The various types of object are recognised by the number of coordinate
109 pairs and the context in which they are used: a list of four elements
110 can be regarded as an infinite line, a finite line segment but also as
111 a polyline of one segment and a point set of two points.
112 Currently the following types of objects are distinguished:
114 · point - a list of two coordinates representing the x- and y-
116 coordinates respectively.
117 · line - a list of four coordinates, interpreted as the x- and y-
119 coordinates of two distinct points on the line.
120 · line segment - a list of four coordinates, interpreted as the x-
122 and y-coordinates of the first and the last points on the line
123 segment.
124 · polyline - a list of an even number of coordinates, interpreted
126 as the x- and y-coordinates of an ordered set of points.
127 · polygon - like a polyline, but the implicit assumption is that
129 the polyline is closed (if the first and last points do not
130 coincide, the missing segment is automatically added).
131 · point set - again a list of an even number of coordinates, but
133 the points are regarded without any ordering.
134
136 The package defines the following public procedures:
137 ::math::geometry::+ point1 point2
139 Compute the sum of the two vectors given as points and return
140 it. The result is a vector as well.
141 ::math::geometry::- point1 point2
143 Compute the difference (point1 - point2) of the two vectors
144 given as points and return it. The result is a vector as well.
145 ::math::geometry::p x y
147 Construct a point from its coordinates and return it as the
148 result of the command.
149 ::math::geometry::distance point1 point2
151 Compute the distance between the two points and return it as the
152 result of the command. This is in essence the same as
153 math::geometry::length [math::geomtry::- point1 point2]
156 ::math::geometry::length point
159 Compute the length of the vector and return it as the result of
160 the command.
161 ::math::geometry::s* factor point
163 Scale the vector by the factor and return it as the result of
164 the command. This is a vector as well.
165 ::math::geometry::direction angle
167 Given the angle in degrees this command computes and returns the
168 unit vector pointing into this direction. The vector for angle
169 == 0 points to the right (up), and for angle == 90 up (north).
170 ::math::geometry::h length
172 Returns a horizontal vector on the X-axis of the specified
173 length. Positive lengths point to the right (east).
174 ::math::geometry::v length
176 Returns a vertical vector on the Y-axis of the specified length.
177 Positive lengths point down (south).
178 ::math::geometry::between point1 point2 s
180 Compute the point which is at relative distance s between the
181 two points and return it as the result of the command. A rela‐
182 tive distance of 0 returns point1, the distance 1 returns
183 point2. Distances < 0 or > 1 extrapolate along the line between
184 the two point.
185 ::math::geometry::octant point
187 Compute the octant of the circle the point is in and return it
188 as the result of the command. The possible results are
189 [1] east
191 [2] northeast
193 [3] north
195 [4] northwest
197 [5] west
199 [6] southwest
201 [7] south
203 [8] southeast
205 Each octant is the arc of the circle +/- 22.5 degrees from the
207 cardinal direction the octant is named for.
208 ::math::geometry::rect nw se
210 Construct a rectangle from its northwest and southeast corners
211 and return it as the result of the command.
212 ::math::geometry::nwse rect
214 Extract the northwest and southeast corners of the rectangle and
215 return them as the result of the command (a 2-element list con‐
216 taining the points, in the named order).
217 ::math::geometry::angle line
219 Calculate the angle from the positive x-axis to a given line (in
220 two dimensions only).
221 list line
223 Coordinates of the line
224 ::math::geometry::calculateDistanceToLine P line
227 Calculate the distance of point P to the (infinite) line and
228 return the result
229 list P List of two numbers, the coordinates of the point
231 list line
233 List of four numbers, the coordinates of two points on
234 the line
235 ::math::geometry::calculateDistanceToLineSegment P linesegment
238 Calculate the distance of point P to the (finite) line segment
239 and return the result.
240 list P List of two numbers, the coordinates of the point
242 list linesegment
244 List of four numbers, the coordinates of the first and
245 last points of the line segment
246 ::math::geometry::calculateDistanceToPolyline P polyline
250 Calculate the distance of point P to the polyline and return the
251 result. Note that a polyline needs not to be closed.
252 list P List of two numbers, the coordinates of the point
254 list polyline
256 List of numbers, the coordinates of the vertices of the
257 polyline
258 ::math::geometry::calculateDistanceToPolygon P polygon
261 Calculate the distance of point P to the polygon and return the
262 result. If the list of coordinates is not closed (first and last
263 points differ), it is automatically closed.
264 list P List of two numbers, the coordinates of the point
266 list polygon
268 List of numbers, the coordinates of the vertices of the
269 polygon
270 ::math::geometry::findClosestPointOnLine P line
273 Return the point on a line which is closest to a given point.
274 list P List of two numbers, the coordinates of the point
276 list line
278 List of four numbers, the coordinates of two points on
279 the line
280 ::math::geometry::findClosestPointOnLineSegment P linesegment
283 Return the point on a line segment which is closest to a given
284 point.
285 list P List of two numbers, the coordinates of the point
287 list linesegment
289 List of four numbers, the first and last points on the
290 line segment
291 ::math::geometry::findClosestPointOnPolyline P polyline
294 Return the point on a polyline which is closest to a given
295 point.
296 list P List of two numbers, the coordinates of the point
298 list polyline
300 List of numbers, the vertices of the polyline
301 ::math::geometry::lengthOfPolyline polyline
304 Return the length of the polyline (note: it not regarded as a
305 polygon)
306 list polyline
308 List of numbers, the vertices of the polyline
309 ::math::geometry::movePointInDirection P direction dist
312 Move a point over a given distance in a given direction and
313 return the new coordinates (in two dimensions only).
314 list P Coordinates of the point to be moved
316 double direction
318 Direction (in degrees; 0 is to the right, 90 upwards)
319 list dist
321 Distance over which to move the point
322 ::math::geometry::lineSegmentsIntersect linesegment1 linesegment2
325 Check if two line segments intersect or coincide. Returns 1 if
326 that is the case, 0 otherwise (in two dimensions only). If an
327 endpoint of one segment lies on the other segment (or is very
328 close to the segment), they are considered to intersect
329 list linesegment1
331 First line segment
332 list linesegment2
334 Second line segment
335 ::math::geometry::findLineSegmentIntersection linesegment1 linesegment2
338 Find the intersection point of two line segments. Return the
339 coordinates or the keywords "coincident" or "none" if the line
340 segments coincide or have no points in common (in two dimensions
341 only).
342 list linesegment1
344 First line segment
345 list linesegment2
347 Second line segment
348 ::math::geometry::findLineIntersection line1 line2
351 Find the intersection point of two (infinite) lines. Return the
352 coordinates or the keywords "coincident" or "none" if the lines
353 coincide or have no points in common (in two dimensions only).
354 list line1
356 First line
357 list line2
359 Second line
360 See section References for details on the algorithm and math
362 behind it.
363 ::math::geometry::polylinesIntersect polyline1 polyline2
366 Check if two polylines intersect or not (in two dimensions
367 only).
368 list polyline1
370 First polyline
371 list polyline2
373 Second polyline
374 ::math::geometry::polylinesBoundingIntersect polyline1 polyline2 granu‐
377 larity
378 Check whether two polylines intersect, but reduce the correct‐
379 ness of the result to the given granularity. Use this for
380 faster, but weaker, intersection checking.
381 How it works:
383 Each polyline is split into a number of smaller polylines, con‐
385 sisting of granularity points each. If a pair of those smaller
386 lines' bounding boxes intersect, then this procedure returns 1,
387 otherwise it returns 0.
388 list polyline1
390 First polyline
391 list polyline2
393 Second polyline
394 int granularity
396 Number of points in each part (<=1 means check every
397 edge)
398 ::math::geometry::intervalsOverlap y1 y2 y3 y4 strict
401 Check if two intervals overlap.
402 double y1,y2
404 Begin and end of first interval
405 double y3,y4
407 Begin and end of second interval
408 logical strict
410 Check for strict or non-strict overlap
411 ::math::geometry::rectanglesOverlap P1 P2 Q1 Q2 strict
414 Check if two rectangles overlap.
415 list P1
417 upper-left corner of the first rectangle
418 list P2
420 lower-right corner of the first rectangle
421 list Q1
423 upper-left corner of the second rectangle
424 list Q2
426 lower-right corner of the second rectangle
427 list strict
429 choosing strict or non-strict interpretation
430 ::math::geometry::bbox polyline
433 Calculate the bounding box of a polyline. Returns a list of four
434 coordinates: the upper-left and the lower-right corner of the
435 box.
436 list polyline
438 The polyline to be examined
439 ::math::geometry::pointInsidePolygon P polyline
442 Determine if a point is completely inside a polygon. If the
443 point touches the polygon, then the point is not completely
444 inside the polygon.
445 list P Coordinates of the point
447 list polyline
449 The polyline to be examined
450 ::math::geometry::pointInsidePolygonAlt P polyline
453 Determine if a point is completely inside a polygon. If the
454 point touches the polygon, then the point is not completely
455 inside the polygon. Note: this alternative procedure uses the
456 so-called winding number to determine this. It handles self-
457 intersecting polygons in a "natural" way.
458 list P Coordinates of the point
460 list polyline
462 The polyline to be examined
463 ::math::geometry::rectangleInsidePolygon P1 P2 polyline
466 Determine if a rectangle is completely inside a polygon. If
467 polygon touches the rectangle, then the rectangle is not com‐
468 plete inside the polygon.
469 list P1
471 Upper-left corner of the rectangle
472 list P2
474 Lower-right corner of the rectangle
475 list polygon
478 The polygon in question
479 ::math::geometry::areaPolygon polygon
482 Calculate the area of a polygon.
483 list polygon
485 The polygon in question
486 ::math::geometry::translate vector polyline
489 Translate a polyline over a given vector
490 list vector
492 Translation vector
493 list polyline
495 The polyline to be rotated
496 ::math::geometry::rotate angle polyline
499 Rotate a polyline over a given angle (degrees) around the origin
500 list angle
502 Angle over which to rotate the polyline (degrees)
503 list polyline
505 The polyline to be translated
506 ::math::geometry::reflect angle polyline
509 Reflect a polyline in a line through the origin at a given angle
510 (degrees) to the x-axis
511 list angle
513 Angle of the line of reflection (degrees)
514 list polyline
516 The polyline to be reflected
517 ::math::geometry::degToRad angle
520 Convert from degrees to radians
521 list angle
523 Angle in degrees
524 ::math::geometry::radToDeg angle
527 Convert from radians to degrees
528 list angle
530 Angle in radians
531
533 [1] Polygon Intersection [http:/wiki.tcl.tk/12070]
534 [2] http://en.wikipedia.org/wiki/Line-line_intersection
536 [3] http://local.wasp.uwa.edu.au/~pbourke/geometry/lineline2d/
538
540 This document, and the package it describes, will undoubtedly contain
541 bugs and other problems. Please report such in the category math ::
542 geometry of the Tcllib Trackers [http://core.tcl.tk/tcllib/reportlist].
543 Please also report any ideas for enhancements you may have for either
544 package and/or documentation.
545 When proposing code changes, please provide unified diffs, i.e the out‐
547 put of diff -u.
548 Note further that attachments are strongly preferred over inlined
550 patches. Attachments can be made by going to the Edit form of the
551 ticket immediately after its creation, and then using the left-most
552 button in the secondary navigation bar.
553
555 angle, distance, line, math, plane geometry, point
556
558 Mathematics
559
561 Copyright (c) 2001 by Ideogramic ApS and other parties
562 Copyright (c) 2004 by Arjen Markus
563 Copyright (c) 2010 by Andreas Kupries
564 Copyright (c) 2010 by Kevin Kenny
565tcllib 1.2.2 math::geometry(n)