1

2# -*- coding: utf-8 -*-

3

4 u'''Simplify or linearize a path of C{LatLon} points.

5

6 Each of the 4 I{simplify} functions is based on a different algorithm and

7 produces different, simplified results in (very) different run times for the

8 same path:

9

10 - Function L{simplify1} eliminates points with edge lengths shorter than

11 the given tolerance.

12

13 - Function L{simplifyRDP} implements the I{Ramer-Douglas-Peucker} (RDP)

14 algorithm, iteratively finding the point farthest from each path edge.

15 Original RDP exhaustively searches the most distant point in each iteration,

16 I{modified} RDP stops at the first point exceeding the distance tolerance.

17

18 - Function L{simplifyRW} uses the I{Reumann-Witkam} (RW) method, sliding a

19 "pipe" over each path edge, removing all subsequent points within the pipe

20 radius, up to the first point outside the pipe.

21

22 - Function L{simplifyVW} provides the I{Visvalingam-Whyatt} (VW) method

23 using the area of the triangle formed by three neigboring points. Original

24 VW removes only a single point per iteration, I{modified} VW eliminates all

25 points with a triangular area not exceeding the tolerance in each iteration.

26

27 Keyword argument I{shortest} of functions L{simplifyRDP} and L{simplifyRW}

28 specifies of the distance between a point and a path edge. If C{True}, use

29 the I{shortest} distance to the path edge or edge points, otherwise use the

30 I{perpendicular} distance to the (extended) edge through both points.

31

32 Keyword argument B{C{radius}} of all fuctions is set to the mean earth

33 radius in C{meter}, conventionally. Other units may be used, provided

34 that radius and tolerance are specified in the same units.

35

36 Use keyword argument C{B{indices}=True} in any function to return a list

37 of I{indices} of simplified point instead of the simplified points with

38 the first and last index are always the first and last original index.

39

40 Finally, any additional keyword arguments B{C{options}} to all functions

41 are passed thru to function L{pygeodesy.equirectangular4} to specify the

42 distance approximation.

43

44 To process C{NumPy} arrays containing rows of lat-, longitude and possibly

45 other values, use class L{Numpy2LatLon} to wrap the C{NumPy} array into

46 I{on-the-fly-LatLon} points. Pass the L{Numpy2LatLon} instance to any

47 I{simplify} function and the returned result will be a C{NumPy} array

48 containing the simplified subset, a partial copy of the original C{NumPy}

49 array. Use keyword argument C{B{indices}=True} to return a list of array

50 row indices inlieu of the simplified array subset.

51

52 See:

53 - U{https://Bost.Ocks.org/mike/simplify}

54 - U{https://WikiPedia.org/wiki/Ramer-Douglas-Peucker_algorithm}

55 - U{https://www.ScienceDirect.com/science/article/pii/S0098300402000092}

56 - U{https://hydra.Hull.ac.UK/resources/hull:8338}

57 - U{https://psimpl.SourceForge.net/reumann-witkam.html}

58 - U{https://www.CS.UBC.Ca/cgi-bin/tr/1992/TR-92-07.pdf}

59 - U{https://GitHub.com/FlorianWilhelm/gps_data_with_python}

60 - U{https://www.BDCC.co.UK/Gmaps/GDouglasPeuker.js}

61 - U{https://GitHub.com/mourner/simplify-js}

62 - U{https://GitHub.com/OmarEstrella/simplify.py}

63 - U{https://PyPI.org/project/rdp}

64 - U{https://PyPI.org/project/visvalingam}

65 - U{https://PyPI.org/project/simplification}

66'''

67# make sure int/int division yields float quotient, see .basics

68 from __future__ import division as _; del _ # PYCHOK semicolon

69

70# from pygeodesy.basics import len2 # from .fmath

71 from pygeodesy.constants import EPS, R_M, _1_0

72 from pygeodesy.errors import _AttributeError, _ValueError, _xkwds_pop2

73 from pygeodesy.fmath import fdot_, len2, sqrt0

74 from pygeodesy.formy import equirectangular4

75 from pygeodesy.interns import _small_, _too_

76 from pygeodesy.iters import isNumpy2, isTuple2

77# from pygeodesy.lazily import _ALL_LAZY # from .units

78 from pygeodesy.units import _ALL_LAZY, _1mm, Radius_

79

80 from math import degrees, fabs, radians

81

82__all__ = _ALL_LAZY.simplify

83__version__ = '24.12.02'

84

85

86# try:

87# from collections import namedtuple

88# _T2 = namedtuple('_T2', 'ix, h2')

89# except ImportError:

90# class _T2(object):

91# ...

92# namedtuple (and .named._NamedTuple) can not be

93# used because (a) values can not be updated and

94# (b) it produces PyChecker warning "<string>:28:

95# self is not first method argument" which can't

96# be suppressed with command line option --stdlib

97 class _T2(object):

98 '''(INTERNAL) VW 2-tuple (index, area2).

99 '''

100 # __slots__ are no longer space savers, see

101 # the comments at the class .points.LatLon_

102 # __slots__ = ('ix', 'h2')

103

104 def __init__(self, ix, h2):

105 self.ix = ix

106 self.h2 = h2

107

108

109 class _Sy(object):

110 '''(INTERNAL) Simplify state.

111 '''

112 d2yxse5 = () # 5-tuple

113 eps = EPS # system epsilon

114 indices = False

115 ixs = set() # set(indices)

116 n = 0

117 options = {}

118 pts = []

119 radius = R_M # mean earth radius

120 s2 = EPS # tolerance squared

121 s2e = EPS # VW sentinel

122 shortest = False # i.e. perpendicular

123 subset = None # isNumpy2 or isTuple2

124 t2s = [] # list(_T2s)

125

126 def __init__(self, points, tolerance, radius, shortest,

127 indices, **options):

128 '''New C{Simplify} state.

129 '''

130 n, self.pts = len2(points)

131 if n > 0:

132 self.n = n

133 self.ixs = set((0, n-1))

134

135 if radius is not R_M:

136 self.radius = Radius_(radius, low=self.eps)

137 # tolerance converted to degrees squared

138 self.s2 = s2 = degrees(tolerance / self.radius)**2

139 if min(s2, tolerance) < self.eps:

140 raise _ValueError(tolerance=tolerance, txt=_too_(_small_))

141 self.s2e = s2 + _1_0 # VW sentinel

142 # assert self.s2e > s2

143

144 if indices:

145 self.indices = True

146 if options:

147 _, self.options = _xkwds_pop2(options, modified=None)

148 if shortest:

149 self.shortest = True

150 if isNumpy2(points) or isTuple2(points): # NOT self.pts

151 self.subset = points.subset

152

153 def d21(self, s, e):

154 '''Set path edge or line thru (points[s], -[e]).

155 '''

156 d21, y21, x21, _ = self.d2yxu4(s, e)

157 self.d2yxse5 = d21, y21, x21, s, e

158 return d21 > self.eps

159

160 def d2i2(self, m, n, modified):

161 '''Find the tallest distance among all points[m..n]

162 to (points[s], -[e]) exceeding the tolerance.

163 '''

164 _, _, _, s, _ = self.d2yxse5

165 t2, t = self.s2, 0 # tallest

166 for i in range(m, n):

167 d2, _, _, _ = self.d2yxu4(s, i)

168 if d2 > t2:

169 t2, t = d2, i

170 if modified and d2 > self.eps:

171 break

172 return t2, t

173

174 def d2ix2(self, m, n, modified):

175 '''Find the tallest I{perpendicular B{or} shortest} distance

176 among all points[m..n] to the path edge or line through

177 (points[s], -[e]) exceeding the tolerance.

178 '''

179 h = not self.shortest

180 # point (x, y) on axis rotated by angle a ccw:

181 # x' = y * sin(a) + x * cos(a)

182 # y' = y * cos(a) - x * sin(a)

183 #

184 # distance along (w) and perpendicular to (h)

185 # a line from the origin to point (dx, dy):

186 # w = (y * dy + x * dx) / hypot(dx, dy)

187 # h = (y * dx - x * dy) / hypot(dx, dy)

188 d21, y21, x21, s, e = self.d2yxse5

189 t2, t = self.s2, 0 # tallest

190 for i in range(m, n):

191 # distance points[s] to -[i], ...

192 d2, y01, x01, _ = self.d2yxu4(s, i)

193 if d2 > self.eps:

194 if h: # perpendicular distance

195 d2 = fdot_(y01, x21, -x01, y21)**2 / d21

196 else:

197 w = fdot_(y01, y21, x01, x21)

198 if w > 0:

199 if w < d21: # ... perpendicular ...

200 d2 = fdot_(y01, x21, -x01, y21)**2 / d21

201 else: # ... or points[e] to -[i]

202 d2, _, _, _ = self.d2yxu4(e, i)

203 if d2 > t2:

204 t2, t = d2, i

205 if modified:

206 break

207 return t2, t

208

209 def d2yxu4(self, i, j):

210 '''Return the distance I{squared}, the deltas and the

211 (longitudinal) unrollment between (points[i], -[j]).

212 '''

213 p1, p2 = self.pts[i], self.pts[j]

214 return equirectangular4(p1.lat, p1.lon,

215 p2.lat, p2.lon, **self.options)

216

217 def h2t(self, i1, i2, i3):

218 '''Compute (double) the triangle area, points[i2] is

219 the top and edge (points[i1], -[i3]) is the base

220 of the triangle.

221 '''

222 d21, y21, x21 , _ = self.d2yxu4(i1, i3)

223 if d21 > self.eps:

224 d01, y01, x01, _ = self.d2yxu4(i1, i2)

225 if d01 > self.eps:

226 h2 = fabs(fdot_(y01, x21, -x01, y21))

227 # triangle height h = h2 / sqrt(d21) and

228 # the area = h * sqrt(d21) / 2 == h2 / 2

229 return h2 # double triangle area

230 return 0

231

232 def rdp(self, modified):

233 '''Ramer-Douglas-Peucker (RDP) simplification of a

234 path of C{LatLon} points.

235

236 @arg modified: Use I{modified} RDP (C{bool}).

237 '''

238 r, n = self.ixs, self.n

239 if n > 1:

240 s2, se = self.s2, [(0, n-1)]

241 while se:

242 s, e = se.pop()

243 s1 = s + 1

244 if e > s1:

245 if self.d21(s, e): # points[] to edge [s, e]

246 d2, i = self.d2ix2(s1, e, modified)

247 else: # points[] to point [s]

248 d2, i = self.d2i2( s1, e, modified)

249 if d2 > s2 and i > 0:

250 se.append((i, e))

251 if not modified:

252 se.append((s, i))

253 r.add(i)

254 r.add(s)

255 return self.result(r)

256

257 def result(self, r):

258 '''Return the simplified points or indices.

259 '''

260 r = sorted(r)

261 if self.indices:

262 return list(r)

263 elif self.subset:

264 return self.subset(r)

265 else:

266 return list(self.pts[i] for i in r)

267

268 def rw(self):

269 '''Reumann-Witkam simplification.

270 '''

271 r, n = self.ixs, self.n

272 if n > 1:

273 s, e, s2 = 0, 1, self.s2

274 while s < e < n:

275 if self.d21(s, e):

276 d2, i = self.d2ix2(e + 1, n, True)

277 r.add(s)

278 if d2 > s2 and i > 0:

279 r.add(i)

280 s = e = i

281 else:

282 # r.add(n - 1)

283 break

284 e += 1

285 return self.result(r)

286

287 def sy1(self):

288 '''Basic simplification.

289 '''

290 r, n = self.ixs, self.n

291 if n > 1:

292 s2, i = self.s2, 0

293 for j in range(1, n):

294 d2, _, _, _ = self.d2yxu4(i, j)

295 if d2 > s2:

296 r.add(j)

297 i = j

298 return self.result(r)

299

300 def vwn(self):

301 '''Initialize VW as list of 2-tuples _T2(ix, h2) where

302 ix is the points[] index and h2 is the triangular

303 area I{(times 2)} of that point.

304 '''

305 self.t2s = t = []

306 n, T2 = self.n, _T2

307 if n > 2:

308 _h2t = self.h2t

309 t[:] = [T2(i, _h2t(i-1, i, i+1)) for i in range(1, n - 1)]

310 if n > 1:

311 t.append(T2(n - 1, self.s2e))

312 if n > 0:

313 t.insert(0, T2(0, self.s2e))

314 return len(t)

315

316 def vwr(self, attr):

317 '''Return the VW results, optionally including the

318 triangular area (in C{meter}) as attribute C{attr}

319 to each simplified point.

320 '''

321 pts, t = self.pts, self.t2s

322

323 # double check the minimal triangular area

324 assert min(t2.h2 for t2 in t) > self.s2 > 0

325

326 if attr: # return each trangular area (actually

327 # the sqrt of double the triangular area)

328 # converted back from degrees to meter

329 if isNumpy2(pts):

330 raise _AttributeError(attr=attr)

331 t[0].h2 = t[-1].h2 = 0 # zap sentinels

332 m = radians(_1_0) * self.radius # meter

333 for t2 in t: # convert back to meter

334 setattr(pts[t2.ix], attr, sqrt0(t2.h2) * m)

335

336 n = len(t) # double check for duplicates

337 r = set(t2.ix for t2 in t)

338 assert len(r) == n

339 return self.result(r)

340

341 def vwrm(self):

342 '''Keep removing the VW point with the smallest triangular

343 area until that area exceeds the tolerance.

344 '''

345 s2, t = self.s2, self.t2s

346 while len(t) > 2:

347 m2, m = t[1].h2, 1

348 for i in range(2, len(t) - 1):

349 h2 = t[i].h2

350 if h2 < m2:

351 m2, m = h2, i

352 if m2 > s2:

353 break

354 self.vwrm1(m, 0)

355

356 def vwrm1(self, m, tol):

357 '''Eliminate VW point[m], keep recomputing the trangular

358 area of both neighboring points and removing those

359 too until the recomputed area exceeds C{tol}.

360 '''

361 t, _h2t = self.t2s, self.h2t

362 t.pop(m)

363 for n in (m, m - 1): # neighbors

364 while 0 < n < (len(t) - 1):

365 h2 = _h2t(t[n-1].ix, t[n].ix, t[n+1].ix)

366 if h2 > tol:

367 t[n].h2 = h2

368 break # while

369 t.pop(n)

370

371 def vwrm2(self, tol):

372 '''Eliminate all VW points with a triangular area not

373 exceeding C{tol}.

374 '''

375 t = self.t2s

376 m = len(t) - 1

377 while m > 1:

378 m -= 1

379 if t[m].h2 <= tol:

380 self.vwrm1(m, tol)

381 m = min(m, len(t) - 1)

382

383

384 def simplify1(points, distance=_1mm, radius=R_M, indices=False, **options):

385 '''Basic simplification of a path of C{LatLon} points by eliminating

386 any points closer together than the given I{distance} tolerance.

387

388 @arg points: Iterable with the path points (C{LatLon}[]).

389 @kwarg distance: Tolerance (C{meter}, same units as B{C{radius}}).

390 @kwarg radius: Mean earth radius (C{meter}, conventionally).

391 @kwarg indices: If C{True}, return B{C{points}} indices instead

392 of the simplified points (C{bool}).

393 @kwarg options: Optional keyword arguments passed thru to function

394 L{pygeodesy.equirectangular4}.

395

396 @return: Simplified points (C{LatLon}[]) or B{C{points}} indices.

397

398 @raise LimitError: Lat- and/or longitudinal delta exceeds the B{C{limit}},

399 see function L{pygeodesy.equirectangular4}.

400

401 @raise ValueError: Tolerance B{C{distance}} or B{C{radius}} too small.

402 '''

403 S = _Sy(points, distance, radius, True, indices, **options)

404 return S.sy1()

405

406

407 def simplifyRDP(points, distance=_1mm, radius=R_M, shortest=False,

408 indices=False, modified=False, **options):

409 '''I{Ramer-Douglas-Peucker} (RDP) simplification of a path of C{LatLon}

410 points by eliminating any points too close together or closer to an

411 edge than the given I{distance} tolerance.

412

413 @arg points: Iterable with the path points (C{LatLon}[]).

414 @kwarg distance: Tolerance (C{meter}, same units as B{C{radius}}).

415 @kwarg radius: Mean earth radius (C{meter}, conventionally).

416 @kwarg shortest: If C{True}, use the I{shortest} otherwise the

417 I{perpendicular} distance (C{bool}).

418 @kwarg indices: If C{True}, return B{C{points}} indices instead

419 of the simplified points (C{bool}).

420 @kwarg modified: If C{True}, use the C{modified RDP} method (C{bool}),

421 see the B{note}.

422 @kwarg options: Optional keyword arguments passed thru to function

423 L{pygeodesy.equirectangular4}.

424

425 @return: Simplified points (C{LatLon}[]) or B{C{points}} indices.

426

427 @raise LimitError: Lat- and/or longitudinal delta exceeds the B{C{limit}},

428 see function L{pygeodesy.equirectangular4}.

429

430 @raise ValueError: Tolerance B{C{distance}} or B{C{radius}} too small.

431

432 @note: The original C{RDP} method exhaustively searches for the point

433 with the largest distance (resulting in complexity M{O(n**2)}

434 with M{n} is the number of points). The B{C{modified}} C{RDP}

435 method stops at the first point farther than the B{C{distance}}

436 tolerance, significantly reducing the run time (but producing

437 results different from the original C{RDP} method).

438 '''

439 S = _Sy(points, distance, radius, shortest, indices, **options)

440 return S.rdp(bool(modified))

441

442

443 def simplifyRW(points, pipe=_1mm, radius=R_M, shortest=False,

444 indices=False, **options):

445 '''I{Reumann-Witkam} (RW) simplification of a path of C{LatLon} points

446 by eliminating any points too close together or within the given

447 I{pipe} tolerance along an edge.

448

449 @arg points: Iterable with the path points (C{LatLon}[]).

450 @kwarg pipe: Pipe radius, half-width (C{meter}, same units as

451 B{C{radius}}).

452 @kwarg radius: Mean earth radius (C{meter}, conventionally).

453 @kwarg shortest: If C{True}, use the I{shortest} otherwise the

454 I{perpendicular} distance (C{bool}).

455 @kwarg indices: If C{True}, return B{C{points}} indices instead

456 of the simplified points (C{bool}).

457 @kwarg options: Optional keyword arguments passed thru to function

458 L{pygeodesy.equirectangular4}.

459

460 @return: Simplified points (C{LatLon}[]) or B{C{points}} indices.

461

462 @raise LimitError: Lat- and/or longitudinal delta exceeds the B{C{limit}},

463 see function L{pygeodesy.equirectangular4}.

464

465 @raise ValueError: Tolerance B{C{pipe}} or B{C{radius}} too small.

466 '''

467 S = _Sy(points, pipe, radius, shortest, indices, **options)

468 return S.rw()

469

470

471 def simplifyVW(points, area=_1mm, radius=R_M, indices=False,

472 attr=None, modified=False, **options):

473 '''I{Visvalingam-Whyatt} (VW) simplification of a path of C{LatLon}

474 points by eliminating any points too close or with a triangular

475 area not exceeding the given I{area} tolerance I{squared}.

476

477 @arg points: Iterable with the path points (C{LatLon}[]).

478 @kwarg area: Tolerance (C{meter}, same units as B{C{radius}}).

479 @kwarg radius: Mean earth radius (C{meter}, conventionally).

480 @kwarg indices: If C{True}, return B{C{points}} indices instead

481 of the simplified points (C{bool}).

482 @kwarg attr: Optional, B{C{points}} attribute to save the area

483 value (C{str}).

484 @kwarg modified: If C{True}, use the C{modified VW} method (C{bool}),

485 see the B{note}.

486 @kwarg options: Optional keyword arguments passed thru to function

487 L{pygeodesy.equirectangular4}.

488

489 @return: Simplified points (C{LatLon}[]) or B{C{points}} indices.

490

491 @raise AttributeError: An B{C{attr}} isinvalid for I{Numpy2} B{C{points}}.

492

493 @raise LimitError: Lat- and/or longitudinal delta exceeds the B{C{limit}},

494 see function L{pygeodesy.equirectangular4}.

495

496 @raise ValueError: Tolerance B{C{area}} or B{C{radius}} too small.

497

498 @note: The original C{VW} method exhaustively searches for the point

499 with the smallest triangular I{area} (resulting in complexity

500 M{O(n**2)} with M{n} the number of points). The B{C{modified}}

501 C{VW} method removes I{all} points with a triangular I{area}

502 below the tolerance in each iteration, significantly reducing

503 the run time (but producing results different from the original

504 C{VW} method).

505 '''

506 S = _Sy(points, area, radius, False, indices, **options)

507 if S.vwn() > 2:

508 if modified:

509 S.vwrm2(S.s2)

510 else:

511 S.vwrm2(0)

512 S.vwrm()

513 return S.vwr(attr)

514

515# **) MIT License

516#

517# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved.

518#

519# Permission is hereby granted, free of charge, to any person obtaining a

520# copy of this software and associated documentation files (the "Software"),

521# to deal in the Software without restriction, including without limitation

522# the rights to use, copy, modify, merge, publish, distribute, sublicense,

523# and/or sell copies of the Software, and to permit persons to whom the

524# Software is furnished to do so, subject to the following conditions:

525#

526# The above copyright notice and this permission notice shall be included

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528#

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530# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

531# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

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