Coverage for pyrdnap/rdnap2018.py: 91%

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

4u'''Main classes L{RDNAP2018v1} and L{RDNAP2018v2} follow C{variant 1} respectively C{variant 2}

5of the specification U{RDNAPTRANS(tm)2018<https://formulieren.kadaster.nl/aanvragen_rdnaptrans>}.

6Each provide a C{forward} method to convert geodetic lat-/longitudes and heights to C{RD}

7coodinates and C{NAP} heights and a C{reverse} method for converting the other way.

9The C{forward} and C{reverse} results of L{RDNAP2018v1} meet the C{RDNAPTRANS(tm)2018_v220627}

10self-validation requirements of C{0.000000010 degrees} and C{0.0010 meter} for tests inside

11the C{RD} region, see B{C{Note below}}. Class L{RDNAP2018v2} does not and is not required to.

13The original C{RDNAPTRANS(tm)2018_v220627} grid files for both variants are I{not included}

14in C{PyPRDNAP} and C{pyrdnap} due to the size of those files. Instead, the grid files for each

15variant I{include only} the C{lat_corr_}, C{lon_corr_} and C{_NAP_quasi_geoid_height_...} columns,

16extracted from the original grid files with leading and trailing zeros removed and formatted as

17row-order matrices.

19@note: L{RDNAP2018v1}, C{PyRDNAP} and C{pyrdnap} have B{not been formally validated} and are

20 B{not certified} to carry the trademark name C{RDNAPTRANS(tm)}.

21'''

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

23from __future__ import division as _; del _ # noqa: E702 ;

25from pyrdnap.rd0 import _RD, _RD0 as A0, RDNAP7Tuple

26from pyrdnap.v_grids import RDNAPError, _V_grid, _v_gridz_import

27from pyrdnap.__pygeodesy import (_0_0, _0_5, _1_0, _2_0,

28 _isNAN, _earth_datum,

29 _ALL_DOCS, _ALL_OTHER, _FOR_DOCS,

30 _NamedBase, notOverloaded)

31from pygeodesy import (EPS0, EPS1, NAN, PI_2, PI, PI2, # "consterns"

32 Datum, Datums, Ellipsoid, # datums, ellipsoids

33 property_RO, property_ROnce, # props

34 Lamd, Phid, # units

35 sincos2, sincos2d) # utily

37from math import asin, atan, copysign, degrees, exp, \

38 fabs, floor, hypot, radians, sin, sqrt

40__all__ = ()

41__version__ = '26.05.14'

43_TOL_D = 1e-9 # degrees 2.3.3f+

44_TOL_M = 1e-6 # meter

45_TOL_R = radians(_TOL_D) # 2e-11

46_TRIPS = 16 # 5..6 sufficient

49class _RDNAPbase(_NamedBase):

50 '''(INTERNAL) L{RDNAP2018v1}C{/-v2} base class.

51 '''

52 _datum = None # forward, v1 reverse Datum, lazily

53 _EETRS = None # forward, v1 reverse Ellipsoid, lazily

54 _raiser = False

56 def __init__(self, a_ellipsoid=None, f=None, raiser=False, **name):

57 '''New C{RDNAP2018v1} or C{-v2} instance.

59 @kwarg a_ellipsoid: An ellipsoid (L{Ellipsoid}) or the ellipsoid's equatorial

60 radius (C{scalar}, conventionally in C{meter}), see B{C{f}}

61 or a datum (L{Datum}). Default C{Datums.GRS80}.

62 @kwarg f: The flattening of the ellipsoid (C{scalar}) if B{C{a_ellipsoid}} is

63 specified as C{scalar}, ignored otherwise.

64 @kwarg raiser: If C{True} raise an L{RDNAPError} for lat-/longitudes outside

65 the C{RD} region (C{bool}).

66 @kwarg name: Optional name (C{str}).

68 @raise RDNAPError: Ellipsoid (or datum) is not oblate (i.e. is spherical or

69 prolate) or the datum's C{transform} is not C{unity}.

70 '''

71 if a_ellipsoid is f is None:

72 self._datum = Datums.GRS80

73 else:

74 _earth_datum(a_ellipsoid, f, **name) # sets self._datum

75 E = self._datum.ellipsoid

76 if not E.isOblate:

77 raise RDNAPError('not oblate: %r' % (E,))

78 self._EETRS = E

79 if raiser:

80 T = self._datum.transform

81 if not T.isunity:

82 raise RDNAPError('not unity: %r' % (T,))

83 self._raiser = True

84 if name:

85 self.name = name

87 def forward(self, lat, lon, height=0, raiser=None, name='forward'):

88 '''Convert GRS80 (ETRS98) geodetic C{(B{lat}, B{lon})} and B{C{height}}

89 to local C{(RDx, RDy)} coordinates and C{NAPh} quasi-geoid-height.

91 @arg lat: Latitude (C{degrees} geodetic).

92 @arg lon: Longitude (C{degrees} geodetic).

93 @kwarg height: Height, optional (C{meter} above geoid) or C{NAN}

94 to ignore C{NAPh} interpolation.

95 @kwarg raiser: If C{True} raise an L{RDNAPError} if B{C{lat}} or

96 B{C{lon}} is outside the C{RD} region (C{bool}),

97 if C{False} don't, overriding property C{raiser}.

98 @kwarg name: Optional name (C{str}).

100 @return: An L{RDNAP7Tuple}C{(RDx, RDy, NAPh, lat, lon, height, datum)}

101 with local C{RDx}, C{RDy} coordinates and C{NAPh} height.

102 '''

103 lat0, lon0 = \

104 lat_, lon_ = self._forwardXform(lat, lon, raiser)

105 for _ in range(_TRIPS): # 2.3.3a-f, 1..2

106 latc, lonc = self._rdlatlon2(lat_, lon_, lat0, lon0)

107 if fabs(latc - lat_) < _TOL_D and \

108 fabs(lonc - lon_) < _TOL_D:

109 break

110 lat_, lon_ = latc, lonc

111

112 phiClamC = _ellipsoidal2spherical(latc, lonc)

113 RDx, RDy = _spherical2oblique(*phiClamC)

114 NAPh = NAN if _isNAN(height) else (height - self.rdNAPh(lat, lon)) # 2.5.2

115 return RDNAP7Tuple(RDx, RDy, NAPh, lat, lon, height, self.forwardDatum, name=name)

116

117 @property_RO

118 def forwardDatum(self):

119 '''Get the C{forward} datum (L{Datum}, default GRS80).

120 '''

121 return self._datum

122

123 def _inside2(self, lat, lon, raiser):

124 # default and variant 2: no datum Xform

125 if (raiser or (raiser is None and self._raiser)) and \

126 not _RD.isinside(lat, lon):

127 raise self._outsidError(lat, lon)

128 return lat, lon

129

130 _forwardXform = _inside2 # no datum Xform

131

132 def isinside(self, lat, lon, eps=0):

133 '''Is C{(B{lat}, B{lon})} inside the C{RD} region (C{bool})?

134

135 @kwarg eps: Over-/undersize the C{RD} region (C{degrees}).

136 '''

137 return _RD.isinside(lat, lon, eps)

138

139 def _outsidError(self, *lat_lon):

140 # format an RDNAPError for C{lat_lon} outside C{RD} region

141 return RDNAPError('%r outside %s' % (lat_lon, self.region))

142

143 @property_RO

144 def _rdgrid(self):

145 raise notOverloaded(self)

146

147 def _rdlatlon2(self, lat, lon, lat0=None, lon0=None): # 2.3.2

148 # return the RD-corrected C{(lat, lon)}

149 if _RD.isinside(lat, lon):

150 c_f_N_f6 = _RD.c_f_N_f6(lat, lon)

151 lat_corr = _bilinear(self._rdgrid._lat_corr, *c_f_N_f6)

152 lon_corr = _bilinear(self._rdgrid._lon_corr, *c_f_N_f6)

153

154 if lat0 is lon0 is None: # reverse

155 lat += lat_corr

156 lon += lon_corr

157 else: # forward

158 lat = lat0 - lat_corr

159 lon = lon0 - lon_corr

160 return lat, lon # NAN, NAN?

161

162 def rdNAPh(self, lat, lon, raiser=False): # 2.5.1 and 3.5

163 '''Interpolate the C{NAPh} quasi-geoid-height I{within}

164 the C{RD} region.

165

166 @arg lat: Latitude (C{degrees} GRS80 (ETRS89), geodetic).

167 @arg lon: Longitude (C{degrees} GRS80 (ETRS89), geodetic).

168 @kwarg raiser: If C{True} raise an L{RDNAPError} if B{C{lat}} or

169 B{C{lon}} is outside the C{RD} region (C{bool}),

170 otherwise don't and return C{NAN}.

171

172 @return: C{NAPh} (C{meter}) or C{NAN} if C{B{raiser} is False}

173 and B{C{lat}} or B{C{lon}} is outside the C{RD} region.

174 '''

175 if _RD.isinside(lat, lon):

176 c_f_N_f6 = _RD.c_f_N_f6(lat, lon)

177 return _bilinear(self._rdgrid._NAP_h, *c_f_N_f6)

178 elif raiser:

179 raise self._outsidError(lat, lon)

180 return NAN # c0 2.5.1e+

181

182 @property_RO

183 def region(self):

184 '''Get the C{RD} region as L{RDregion4Tuple}C{(S, W, N, E)}, all C{GRS80 (ETRS89) degrees}.

185 '''

186 return _RD.region

187

188 def _reverse(self, RDx, RDy, NAPh, raiser=None, name='reverse'):

189 '''(INTERNAL) Convert local C{(B{RDx}, B{RDy})} coordinates and

190 B{C{NAPh}} quasi-geoid-height to GRS80 (ETRS89) or Bessel1841

191 (RD-Bessel) geodetic C{lat}, C{lon} and C{height}.

192 '''

193 philCamC = _oblique2spherical(RDx, RDy)

194 lat, lon = _spherical2ellipsoidal(*philCamC)

195

196 lat, lon = self._rdlatlon2(lat, lon)

197 lat, lon = self._reverseXform(lat, lon, raiser)

198 h = NAN if _isNAN(NAPh) else (NAPh + self.rdNAPh(lat, lon))

199 return RDNAP7Tuple(RDx, RDy, NAPh, lat, lon, h,

200 self.reverseDatum, name=name)

201

202 @property_RO

203 def reverseDatum(self):

204 '''Get the C{reverse} datum (L{Datum}), GRS80 or Bessel1841.

205 '''

206 return {1: self._datum,

207 2: A0.D0}.get(self.variant)

208

209 _reverseXform = _inside2 # no datum Xform

210

211 def toStr(self, prec=9, **unused): # PYCHOK signature

212 '''Return this C{RDNAP2018#v} instance as a string.

213

214 @kwarg prec: Precision, number of decimal digits (0..9).

215

216 @return: This C{RDNAP2018#v} (C{str}).

217 '''

218 return self.attrs('name', 'variant', 'forwardDatum', prec=prec) # _ellipsoid_, _name__

219

220 @property_RO

221 def variant(self):

222 raise None

223

224

225class RDNAP2018v1(_RDNAPbase):

226 '''Transformer implementing RD NAP 2018 C{variant 1}.

227

228 @note: Method L{RDNAP2018v1.reverse} returns B{not GRS80 (ETRS89)}

229 geodetic lat- and longitudes.

230 '''

231 if _FOR_DOCS:

232 __init__ = _RDNAPbase.__init__

233 forward = _RDNAPbase.forward

234

235 def _forwardXform(self, lat, lon, raiser):

236 # transform C{(lat, lon)} from GRS80 (ETRS89) to RD-Bessel datum

237 x, y, z = _geodetic2cartesian(lat, lon, self._EETRS, A0.H0_ETRS)

238 x, y, z = _RD._xETRS2RD.transform(x, y, z)

239 lat, lon = _cartesian2geodetic(x, y, z, A0.E0)

240 return self._inside2(lat, lon, raiser)

241

242 @property_ROnce

243 def _rdgrid(self):

244 try:

245 from pyrdnap import v1grid

246 except (AttributeError, ImportError, RDNAPError):

247 v1grid = _v_gridz_import(self.variant)

248 return v1grid

249

250 def reverse(self, RDx, RDy, NAPh=0, **raiser_name): # RDNAP to GRS80 (ETRS89)

251 '''Convert a local C{(B{RDx}, B{RDy})} point and B{C{NAPh}} height to

252 B{GRS80 (ETRS89)} geodetic C{(lat, lon, height)}.

253

254 @arg RDx: Local C{RD} X (C{meter}, conventionally).

255 @arg RDy: Local C{RD} Y (C{meter}, conventionally).

256 @kwarg NAPh: C{NAP} quasi-geoid-height (C{meter}, conventionally)

257 or C{NAN} to ignore C{NAPh} interpolation.

258 @kwarg raiser_name: Like the C{forward} method, C{B{raiser}=None}

259 (C{bool}) and optional C{B{name}='reverse'} (C{str}).

260

261 @return: An L{RDNAP7Tuple}C{(RDx, RDy, NAPh, lat, lon, height, datum)}

262 with geodetic C{lat} and C{lon}, C{height} and C{datum}

263 B{GRS80 (ETRS89)}.

264 '''

265 return self._reverse(RDx, RDy, NAPh, **raiser_name)

266

267 def _reverseXform(self, lat, lon, raiser):

268 # transform C{(lat, lon)} from RD-Bessel to GRS80 (ETRS89) datum

269 x, y, z = _geodetic2cartesian(lat, lon, A0.E0, A0.H0)

270 x, y, z = _RD._xRD2ETRS.transform(x, y, z)

271 lat, lon = _cartesian2geodetic(x, y, z, self._EETRS)

272 return self._inside2(lat, lon, raiser)

273

274 def similarity(self, inverse=False):

275 '''Get the similarity transform (C{Similarity}).

276

277 @kwarg inverse: Use C{True} for the C{reverse} or C{False}

278 for the C{forward} transform (C{bool}).

279 '''

280 return _RD._xRD2ETRS if inverse else _RD._xETRS2RD

281

282 @property_ROnce

283 def variant(self):

284 '''Get this C{RDNAP2018}'s variant (C{int}).

285 '''

286 return 1

287

288

289class RDNAP2018v2(_RDNAPbase):

290 '''Transformer implementing RD NAP 2018 C{variant 2}.

291

292 @note: Method L{RDNAP2018v2.reverse} returns B{Bessel1841 (RD-Bessel)}

293 and B{not GRS80 (ETRS89)} geodetic lat- and longitudes.

294 '''

295 if _FOR_DOCS:

296 __init__ = _RDNAPbase.__init__

297 forward = _RDNAPbase.forward

298

299 @property_ROnce

300 def _rdgrid(self):

301 try:

302 from pyrdnap import v2grid

303 except (AttributeError, ImportError, RDNAPError):

304 v2grid = _v_gridz_import(self.variant)

305 return v2grid

306

307 def reverse(self, RDx, RDy, NAPh=0, **raiser_name): # RDNAP to RD-Bessel

308 '''Convert a local C{(B{RDx}, B{RDy})} point and B{C{NAPh}} height

309 to B{Bessel1841 (RD-Bessel)} geodetic C{(lat, lon, height)}.

310

311 @arg RDx: Local C{RD} X (C{meter}, conventionally).

312 @arg RDy: Local C{RD} Y (C{meter}, conventionally).

313 @kwarg NAPh: C{NAP} quasi-geoid-height (C{meter}, conventionally)

314 or C{NAN} to ignore C{NAPh} interpolation.

315 @kwarg raiser_name: Like the C{forward} method, C{B{raiser}=None}

316 (C{bool}) and optional C{B{name}='reverse'} (C{str}).

317

318 @return: An L{RDNAP7Tuple}C{(RDx, RDy, NAPh, lat, lon, height, datum)}

319 with geodetic C{lat} and C{lon}, C{height} and C{datum}

320 B{Bessel1841 (RD-Bessel)}.

321 '''

322 return self._reverse(RDx, RDy, NAPh, **raiser_name)

323

324 def similarity(self, *unused): # PYCHOK signature

325 '''Get the similarity transform, always C{None}.

326 '''

327 return None

328

329 @property_ROnce

330 def variant(self):

331 '''Get this C{RDNAP2018}'s variant (C{int}).

332 '''

333 return 2

334

335

336def _atan3(y, x, x0): # 2.2.3e and 3.1.1i

337 # equiv to math.atan2 iff x0 is y

338 if x > 0:

339 r = atan(y / x)

340 elif x < 0:

341 r = atan(y / x) + copysign(PI, x0)

342 else:

343 r = copysign(PI_2, x0) if x0 else _0_0

344 return r

345

346

347def _atan_exp(w): # 2.4.1c

348 return atan(exp(w)) * _2_0 - PI_2

349

350

351def _bilinear(v_grid, c_latI, f_latI, latN_f, # 2.3.1f and g

352 c_lonI, f_lonI, lonN_f):

353 # interpolate a lat_corr_, lon_corr_ or NAP_...

354 assert isinstance(v_grid, _V_grid)

355 nw = v_grid(c_latI, f_lonI)

356 ne = v_grid(c_latI, c_lonI)

357 sw = v_grid(f_latI, f_lonI)

358 se = v_grid(f_latI, c_lonI)

359 lonN_f1 = _1_0 - lonN_f # == 1 - (lonN - f_lonN)

360 return (nw * lonN_f1 + ne * lonN_f) * latN_f + \

361 (sw * lonN_f1 + se * lonN_f) * (_1_0 - latN_f)

362

363

364def _cartesian2geodetic(x, y, z, E): # 2.2.3 == EcefUPC.reverse?

365 # convert cartesian C{(x, y, z)} to C{E}-geodetic C{(lat, lon)}

366 r = hypot(x, y)

367 if r > _TOL_M:

368 a = E.a * E.e2

369 phi_ = atan(z / r)

370 for _ in range(_TRIPS): # 4..6

371 s = sin(phi_)

372 s *= a / sqrt(_1_0 - s**2 * E.e2)

373 phi = atan((z + s) / r)

374 if fabs(phi - phi_) < _TOL_R:

375 break

376 phi_ = phi

377 else:

378 phi = copysign(PI_2, z)

379 lam = _atan3(y, x, y)

380 return degrees(phi), degrees(lam)

381

382

383def _ellipsoidal2spherical(lat, lon): # 2.4.1

384 # convert geodetic C{(lat, lon)} to spherical C{(𝛷, 𝛬)}

385 phiC = phi = Phid(lat)

386 if PI_2 > phi > -PI_2: # 2.4.1c

387 q = A0.ln_tan(phi) - A0.ln_e_2(phi)

388 w = A0.N0 * q + A0.M0 # 2.4.1b

389 phiC = _atan_exp(w)

390 lamC = (Lamd(lon) - A0.LAM0) * A0.N0 + A0.LAM0C # 2.4.1d

391 return phiC, lamC # -Capital

392

393

394def _eq0(r, r0=_0_0):

395 return fabs(r - r0) < _TOL_R

396

397

398# def _eq0d(d, d0=_0_0):

399# return fabs(d - d0) < _TOL_D

400

401

402def _geodetic2cartesian(lat, lon, E, h0=0): # 2.2.1

403 # convert C{E}-geodetic C{(lat, lon)} to cartesian C{(x, y, z)}

404 y, x = sincos2d(lon)

405 z, c = sincos2d(lat)

406 n = E.a / sqrt(_1_0 - z**2 * E.e2)

407 c *= n + h0

408 x *= c

409 y *= c

410 z *= n * (_1_0 - E.e2) + h0

411 return x, y, z

412

413

414def _ne0(r, r0=_0_0):

415 return fabs(r - r0) > _TOL_R

416

417

418# def _ne0d(d, d0=_0_0):

419# return fabs(d - d0) > _TOL_D

420

421

422def _oblique2spherical(x, y): # 3.1.1

423 # inverse oblique stereographic conformal projection

424 # from 2-D C{(x, y)} to spherical C{(𝛷, 𝛬)}

425 x -= A0.X0

426 y -= A0.Y0

427 r = hypot(x, y)

428 if r > _TOL_M: # x and y

429 s0, c0 = A0.sincos2PHI0C

430 sp, cp = sincos2(atan(r / A0.RK2) * _2_0) # psi

431 ca = sp * y / r

432 xN = cp * c0 - ca * s0

433 yN = sp * x / r

434 zN = ca * c0 + cp * s0

435 phiC = asin(zN)

436 else:

437 _, xN = A0.sincos2PHI0C

438 yN = _0_0

439 phiC = A0.PHI0C # asin(sin(PHI0C))

440 lamC = _atan3(yN, xN, x) + A0.LAM0C

441 return phiC, lamC # -Capital

442

443

444def _spherical2ellipsoidal(phiC, lamC): # 3.1.2

445 # inverse Gauss conformal projection from

446 # spherical C{(𝛷, 𝛬)} to geodetic C{(lat, lon)}

447 phi = phiC

448 if PI_2 > phi > -PI_2:

449 q = (A0.ln_tan(phi) - A0.M0) / A0.N0

450# w = A0.ln_tan(phi)

451 for _ in range(_TRIPS): # 3..6

452 phi_ = phi

453 phi = _atan_exp(A0.ln_e_2(phi) + q)

454 if fabs(phi - phi_) < _TOL_R:

455 break

456 lam = (lamC - A0.LAM0C) / A0.N0 + A0.LAM0

457 lam = floor((PI - lam) / PI2) * PI2 + lam

458 return degrees(phi), degrees(lam)

459

460

461def _spherical2oblique(phiC, lamC): # 2.4.2

462 # oblique stereographic conformal projection

463 # from spherical C{(𝛷, 𝛬)} to 2-D C{(x, y)}

464 a = phiC - A0.PHI0C # 𝛷 - 𝛷0

465 b = lamC - A0.LAM0C # 𝛬 - 𝛬0

466 if (_ne0(a) or _ne0(b)) and (_ne0(phiC, -A0.PHI0C) or

467 _ne0(lamC, -A0.LAM0C + PI)):

468 s0, c0 = A0.sincos2PHI0C # sin(𝛷0), cos(𝛷0)

469 s, c = sincos2(phiC) # sin(𝛷), cos(𝛷)

470 sp_22 = sin(a * _0_5)**2 + \

471 sin(b * _0_5)**2 * c * c0 # sin(𝜓/2)**2

472 if EPS0 < sp_22 < EPS1:

473 # r = 2kR * tan(𝜓/2)

474 # q = r / (sin(𝜓/2) * cos(𝜓/2) * 2)

475 # = 2kR * sin(𝜓/2) / (sin(𝜓/2) * cos(𝜓/2)**2 * 2)

476 # = 2kR / (cos(𝜓/2)**2 * 2)

477 # = 2kR / ((1 - sin(𝜓/2)**2) * 2)

478 # = 2kR / (2 - sin(𝜓/2)**2 * 2)

479 t = sp_22 * _2_0 # 0 < t < 2

480 q = A0.RK2 / (_2_0 - t)

481 x = q * (c * sin(b))

482 y = q * (s - s0 + s0 * t) / c0

483 else:

484 x = y = _0_0 # NAN?

485 x += A0.X0

486 y += A0.Y0

487 elif _eq0(phiC, A0.PHI0C) and _eq0(lamC, A0.LAM0C):

488 x = A0.X0 # x0 2.4.2g

489 y = A0.Y0 # y0 2.4.2h

490 else: # if _eq0(phiC, -A0.PHI0C) and _eq0(lamC, A0.LAM0C - PI):

491 x = y = NAN

492# else:

493# raise RDNAPError(str((phiC, lamC)))

494 return x, y

495

496

497__all__ += _ALL_DOCS(_RDNAPbase)

498__all__ += _ALL_OTHER(RDNAP2018v1, RDNAP2018v2, RDNAPError,

499 Datum, Datums, Ellipsoid) # passed along from PyGeodesy

500

501# **) MIT License

502#

504#

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

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

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

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

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

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

511#

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

513# in all copies or substantial portions of the Software.

514#

515# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

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

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

518# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

519# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

520# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

521# OTHER DEALINGS IN THE SOFTWARE.