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 ?1.3.0?
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::angleBetween vector1 vector2
44 ::math::geometry::inproduct vector1 vector2
46 ::math::geometry::areaParallellogram vector1 vector2
48 ::math::geometry::calculateDistanceToLine P line
50 ::math::geometry::calculateDistanceToLineSegment P linesegment
52 ::math::geometry::calculateDistanceToPolyline P polyline
54 ::math::geometry::calculateDistanceToPolygon P polygon
56 ::math::geometry::findClosestPointOnLine P line
58 ::math::geometry::findClosestPointOnLineSegment P linesegment
60 ::math::geometry::findClosestPointOnPolyline P polyline
62 ::math::geometry::lengthOfPolyline polyline
64 ::math::geometry::movePointInDirection P direction dist
66 ::math::geometry::lineSegmentsIntersect linesegment1 linesegment2
68 ::math::geometry::findLineSegmentIntersection linesegment1 linesegment2
70 ::math::geometry::findLineIntersection line1 line2
72 ::math::geometry::polylinesIntersect polyline1 polyline2
74 ::math::geometry::polylinesBoundingIntersect polyline1 polyline2 granu‐
76 larity
77 ::math::geometry::intervalsOverlap y1 y2 y3 y4 strict
79 ::math::geometry::rectanglesOverlap P1 P2 Q1 Q2 strict
81 ::math::geometry::bbox polyline
83 ::math::geometry::pointInsidePolygon P polyline
85 ::math::geometry::pointInsidePolygonAlt P polyline
87 ::math::geometry::rectangleInsidePolygon P1 P2 polyline
89 ::math::geometry::areaPolygon polygon
91 ::math::geometry::translate vector polyline
93 ::math::geometry::rotate angle polyline
95 ::math::geometry::reflect angle polyline
97 ::math::geometry::degToRad angle
99 ::math::geometry::radToDeg angle
101 ::math::geometry::circle centre radius
103 ::math::geometry::circleTwoPoints point1 point2
105 ::math::geometry::pointInsideCircle point circle
107 ::math::geometry::lineIntersectsCircle line circle
109 ::math::geometry::lineSegmentIntersectsCircle segment circle
111 ::math::geometry::intersectionLineWithCircle line circle
113 ::math::geometry::intersectionCircleWithCircle circle1 circle2
115 ::math::geometry::tangentLinesToCircle point circle
117______________________________________________________________________________
119
121 The math::geometry package is a collection of functions for computa‐
122 tions and manipulations on two-dimensional geometrical objects, such as
123 points, lines and polygons.
124 The geometrical objects are implemented as plain lists of coordinates.
126 For instance a line is defined by a list of four numbers, the x- and y-
127 coordinate of a first point and the x- and y-coordinates of a second
128 point on the line.
129 The various types of object are recognised by the number of coordinate
131 pairs and the context in which they are used: a list of four elements
132 can be regarded as an infinite line, a finite line segment but also as
133 a polyline of one segment and a point set of two points.
134 Currently the following types of objects are distinguished:
136 • point - a list of two coordinates representing the x- and y-co‐
138 ordinates respectively.
139 • line - a list of four coordinates, interpreted as the x- and y-
141 coordinates of two distinct points on the line.
142 • line segment - a list of four coordinates, interpreted as the x-
144 and y-coordinates of the first and the last points on the line
145 segment.
146 • polyline - a list of an even number of coordinates, interpreted
148 as the x- and y-coordinates of an ordered set of points.
149 • polygon - like a polyline, but the implicit assumption is that
151 the polyline is closed (if the first and last points do not co‐
152 incide, the missing segment is automatically added).
153 • point set - again a list of an even number of coordinates, but
155 the points are regarded without any ordering.
156 • circle - a list of three numbers, the first two are the coordi‐
158 nates of the centre and the third is the radius.
159
161 The package defines the following public procedures:
162 ::math::geometry::+ point1 point2
164 Compute the sum of the two vectors given as points and return
165 it. The result is a vector as well.
166 ::math::geometry::- point1 point2
168 Compute the difference (point1 - point2) of the two vectors
169 given as points and return it. The result is a vector as well.
170 ::math::geometry::p x y
172 Construct a point from its coordinates and return it as the re‐
173 sult of the command.
174 ::math::geometry::distance point1 point2
176 Compute the distance between the two points and return it as the
177 result of the command. This is in essence the same as
178 math::geometry::length [math::geomtry::- point1 point2]
181 ::math::geometry::length point
184 Compute the length of the vector and return it as the result of
185 the command.
186 ::math::geometry::s* factor point
188 Scale the vector by the factor and return it as the result of
189 the command. This is a vector as well.
190 ::math::geometry::direction angle
192 Given the angle in degrees this command computes and returns the
193 unit vector pointing into this direction. The vector for angle
194 == 0 points to the right (up), and for angle == 90 up (north).
195 ::math::geometry::h length
197 Returns a horizontal vector on the X-axis of the specified
198 length. Positive lengths point to the right (east).
199 ::math::geometry::v length
201 Returns a vertical vector on the Y-axis of the specified length.
202 Positive lengths point down (south).
203 ::math::geometry::between point1 point2 s
205 Compute the point which is at relative distance s between the
206 two points and return it as the result of the command. A rela‐
207 tive distance of 0 returns point1, the distance 1 returns
208 point2. Distances < 0 or > 1 extrapolate along the line between
209 the two point.
210 ::math::geometry::octant point
212 Compute the octant of the circle the point is in and return it
213 as the result of the command. The possible results are
214 [1] east
216 [2] northeast
218 [3] north
220 [4] northwest
222 [5] west
224 [6] southwest
226 [7] south
228 [8] southeast
230 Each octant is the arc of the circle +/- 22.5 degrees from the
232 cardinal direction the octant is named for.
233 ::math::geometry::rect nw se
235 Construct a rectangle from its northwest and southeast corners
236 and return it as the result of the command.
237 ::math::geometry::nwse rect
239 Extract the northwest and southeast corners of the rectangle and
240 return them as the result of the command (a 2-element list con‐
241 taining the points, in the named order).
242 ::math::geometry::angle line
244 Calculate the angle from the positive x-axis to a given line (in
245 two dimensions only).
246 list line
248 Coordinates of the line
249 ::math::geometry::angleBetween vector1 vector2
251 Calculate the angle between two vectors (in degrees)
252 list vector1
254 First vector
255 list vector2
257 Second vector
258 ::math::geometry::inproduct vector1 vector2
260 Calculate the inner product of two vectors
261 list vector1
263 First vector
264 list vector2
266 Second vector
267 ::math::geometry::areaParallellogram vector1 vector2
269 Calculate the area of the parallellogram with the two vectors as
270 its sides
271 list vector1
273 First vector
274 list vector2
276 Second vector
277 ::math::geometry::calculateDistanceToLine P line
280 Calculate the distance of point P to the (infinite) line and re‐
281 turn the result
282 list P List of two numbers, the coordinates of the point
284 list line
286 List of four numbers, the coordinates of two points on
287 the line
288 ::math::geometry::calculateDistanceToLineSegment P linesegment
291 Calculate the distance of point P to the (finite) line segment
292 and return the result.
293 list P List of two numbers, the coordinates of the point
295 list linesegment
297 List of four numbers, the coordinates of the first and
298 last points of the line segment
299 ::math::geometry::calculateDistanceToPolyline P polyline
303 Calculate the distance of point P to the polyline and return the
304 result. Note that a polyline needs not to be closed.
305 list P List of two numbers, the coordinates of the point
307 list polyline
309 List of numbers, the coordinates of the vertices of the
310 polyline
311 ::math::geometry::calculateDistanceToPolygon P polygon
314 Calculate the distance of point P to the polygon and return the
315 result. If the list of coordinates is not closed (first and last
316 points differ), it is automatically closed.
317 list P List of two numbers, the coordinates of the point
319 list polygon
321 List of numbers, the coordinates of the vertices of the
322 polygon
323 ::math::geometry::findClosestPointOnLine P line
326 Return the point on a line which is closest to a given point.
327 list P List of two numbers, the coordinates of the point
329 list line
331 List of four numbers, the coordinates of two points on
332 the line
333 ::math::geometry::findClosestPointOnLineSegment P linesegment
336 Return the point on a line segment which is closest to a given
337 point.
338 list P List of two numbers, the coordinates of the point
340 list linesegment
342 List of four numbers, the first and last points on the
343 line segment
344 ::math::geometry::findClosestPointOnPolyline P polyline
347 Return the point on a polyline which is closest to a given
348 point.
349 list P List of two numbers, the coordinates of the point
351 list polyline
353 List of numbers, the vertices of the polyline
354 ::math::geometry::lengthOfPolyline polyline
357 Return the length of the polyline (note: it not regarded as a
358 polygon)
359 list polyline
361 List of numbers, the vertices of the polyline
362 ::math::geometry::movePointInDirection P direction dist
365 Move a point over a given distance in a given direction and re‐
366 turn the new coordinates (in two dimensions only).
367 list P Coordinates of the point to be moved
369 double direction
371 Direction (in degrees; 0 is to the right, 90 upwards)
372 list dist
374 Distance over which to move the point
375 ::math::geometry::lineSegmentsIntersect linesegment1 linesegment2
378 Check if two line segments intersect or coincide. Returns 1 if
379 that is the case, 0 otherwise (in two dimensions only). If an
380 endpoint of one segment lies on the other segment (or is very
381 close to the segment), they are considered to intersect
382 list linesegment1
384 First line segment
385 list linesegment2
387 Second line segment
388 ::math::geometry::findLineSegmentIntersection linesegment1 linesegment2
391 Find the intersection point of two line segments. Return the co‐
392 ordinates or the keywords "coincident" or "none" if the line
393 segments coincide or have no points in common (in two dimensions
394 only).
395 list linesegment1
397 First line segment
398 list linesegment2
400 Second line segment
401 ::math::geometry::findLineIntersection line1 line2
404 Find the intersection point of two (infinite) lines. Return the
405 coordinates or the keywords "coincident" or "none" if the lines
406 coincide or have no points in common (in two dimensions only).
407 list line1
409 First line
410 list line2
412 Second line
413 See section References for details on the algorithm and math be‐
415 hind it.
416 ::math::geometry::polylinesIntersect polyline1 polyline2
419 Check if two polylines intersect or not (in two dimensions
420 only).
421 list polyline1
423 First polyline
424 list polyline2
426 Second polyline
427 ::math::geometry::polylinesBoundingIntersect polyline1 polyline2 granu‐
430 larity
431 Check whether two polylines intersect, but reduce the correct‐
432 ness of the result to the given granularity. Use this for
433 faster, but weaker, intersection checking.
434 How it works:
436 Each polyline is split into a number of smaller polylines, con‐
438 sisting of granularity points each. If a pair of those smaller
439 lines' bounding boxes intersect, then this procedure returns 1,
440 otherwise it returns 0.
441 list polyline1
443 First polyline
444 list polyline2
446 Second polyline
447 int granularity
449 Number of points in each part (<=1 means check every
450 edge)
451 ::math::geometry::intervalsOverlap y1 y2 y3 y4 strict
454 Check if two intervals overlap.
455 double y1,y2
457 Begin and end of first interval
458 double y3,y4
460 Begin and end of second interval
461 logical strict
463 Check for strict or non-strict overlap
464 ::math::geometry::rectanglesOverlap P1 P2 Q1 Q2 strict
467 Check if two rectangles overlap.
468 list P1
470 upper-left corner of the first rectangle
471 list P2
473 lower-right corner of the first rectangle
474 list Q1
476 upper-left corner of the second rectangle
477 list Q2
479 lower-right corner of the second rectangle
480 list strict
482 choosing strict or non-strict interpretation
483 ::math::geometry::bbox polyline
486 Calculate the bounding box of a polyline. Returns a list of four
487 coordinates: the upper-left and the lower-right corner of the
488 box.
489 list polyline
491 The polyline to be examined
492 ::math::geometry::pointInsidePolygon P polyline
495 Determine if a point is completely inside a polygon. If the
496 point touches the polygon, then the point is not completely in‐
497 side the polygon.
498 list P Coordinates of the point
500 list polyline
502 The polyline to be examined
503 ::math::geometry::pointInsidePolygonAlt P polyline
506 Determine if a point is completely inside a polygon. If the
507 point touches the polygon, then the point is not completely in‐
508 side the polygon. Note: this alternative procedure uses the so-
509 called winding number to determine this. It handles self-inter‐
510 secting polygons in a "natural" way.
511 list P Coordinates of the point
513 list polyline
515 The polyline to be examined
516 ::math::geometry::rectangleInsidePolygon P1 P2 polyline
519 Determine if a rectangle is completely inside a polygon. If
520 polygon touches the rectangle, then the rectangle is not com‐
521 plete inside the polygon.
522 list P1
524 Upper-left corner of the rectangle
525 list P2
527 Lower-right corner of the rectangle
528 list polygon
531 The polygon in question
532 ::math::geometry::areaPolygon polygon
535 Calculate the area of a polygon.
536 list polygon
538 The polygon in question
539 ::math::geometry::translate vector polyline
542 Translate a polyline over a given vector
543 list vector
545 Translation vector
546 list polyline
548 The polyline to be translated
549 ::math::geometry::rotate angle polyline
552 Rotate a polyline over a given angle (degrees) around the origin
553 list angle
555 Angle over which to rotate the polyline (degrees)
556 list polyline
558 The polyline to be rotated
559 ::math::geometry::reflect angle polyline
562 Reflect a polyline in a line through the origin at a given angle
563 (degrees) to the x-axis
564 list angle
566 Angle of the line of reflection (degrees)
567 list polyline
569 The polyline to be reflected
570 ::math::geometry::degToRad angle
573 Convert from degrees to radians
574 list angle
576 Angle in degrees
577 ::math::geometry::radToDeg angle
580 Convert from radians to degrees
581 list angle
583 Angle in radians
584 ::math::geometry::circle centre radius
586 Convenience procedure to create a circle from a point and a ra‐
587 dius.
588 list centre
590 Coordinates of the circle centre
591 list radius
593 Radius of the circle
594 ::math::geometry::circleTwoPoints point1 point2
596 Convenience procedure to create a circle from two points on its
597 circumference The centre is the point between the two given
598 points, the radius is half the distance between them.
599 list point1
601 First point
602 list point2
604 Second point
605 ::math::geometry::pointInsideCircle point circle
607 Determine if the given point is inside the circle or on the cir‐
608 cumference (1) or outside (0).
609 list point
611 Point to be checked
612 list circle
614 Circle that may or may not contain the point
615 ::math::geometry::lineIntersectsCircle line circle
617 Determine if the given line intersects the circle or touches it
618 (1) or does not (0).
619 list line
621 Line to be checked
622 list circle
624 Circle that may or may not be intersected
625 ::math::geometry::lineSegmentIntersectsCircle segment circle
627 Determine if the given line segment intersects the circle or
628 touches it (1) or does not (0).
629 list segment
631 Line segment to be checked
632 list circle
634 Circle that may or may not be intersected
635 ::math::geometry::intersectionLineWithCircle line circle
637 Determine the points at which the given line intersects the cir‐
638 cle. There can be zero, one or two points. (If the line touches
639 the circle or is close to it, then one point is returned. An ar‐
640 bitrary margin of 1.0e-10 times the radius is used to determine
641 this situation.)
642 list line
644 Line to be checked
645 list circle
647 Circle that may or may not be intersected
648 ::math::geometry::intersectionCircleWithCircle circle1 circle2
650 Determine the points at which the given two circles intersect.
651 There can be zero, one or two points. (If the two circles touch
652 the circle or are very close, then one point is returned. An ar‐
653 bitrary margin of 1.0e-10 times the mean of the radii of the two
654 circles is used to determine this situation.)
655 list circle1
657 First circle
658 list circle2
660 Second circle
661 ::math::geometry::tangentLinesToCircle point circle
663 Determine the tangent lines from the given point to the circle.
664 There can be zero, one or two lines. (If the point is on the
665 cirucmference or very close to the circle, then one line is re‐
666 turned. An arbitrary margin of 1.0e-10 times the radius of the
667 circle is used to determine this situation.)
668 list point
670 Point in question
671 list circle
673 Circle to which the tangent lines are to be determined
674
676 [1] Polygon Intersection [http:/wiki.tcl.tk/12070]
677 [2] http://en.wikipedia.org/wiki/Line-line_intersection
679 [3] http://local.wasp.uwa.edu.au/~pbourke/geometry/lineline2d/
681
683 This document, and the package it describes, will undoubtedly contain
684 bugs and other problems. Please report such in the category math ::
685 geometry of the Tcllib Trackers [http://core.tcl.tk/tcllib/reportlist].
686 Please also report any ideas for enhancements you may have for either
687 package and/or documentation.
688 When proposing code changes, please provide unified diffs, i.e the out‐
690 put of diff -u.
691 Note further that attachments are strongly preferred over inlined
693 patches. Attachments can be made by going to the Edit form of the
694 ticket immediately after its creation, and then using the left-most
695 button in the secondary navigation bar.
696
698 angle, distance, line, math, plane geometry, point
699
701 Mathematics
702
704 Copyright (c) 2001 by Ideogramic ApS and other parties
705 Copyright (c) 2010 by Andreas Kupries
706 Copyright (c) 2010 by Kevin Kenny
707 Copyright (c) 2018 by Arjen Markus
708tcllib 1.3.0 math::geometry(n)