# Coverage for pygeodesy/formy.py : 97%

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 | # -*- coding: utf-8 -*- u'''Formulary of basic geodesy functions and approximations. @newfield example: Example, Examples ''' from pygeodesy.basics import EPS, R_M, LimitError, \ _limiterrors, len2, map1 from pygeodesy.fmath import fsum_, hypot, hypot2 from pygeodesy.lazily import _ALL_LAZY from pygeodesy.named import Distance4Tuple, LatLon2Tuple, PhiLam2Tuple, \ Points2Tuple, Vector3Tuple from pygeodesy.streprs import fstr from pygeodesy.utily import PI, PI2, PI_2, degrees2m, \ degrees90, degrees180, degrees360, \ isNumpy2, isTuple2, sincos2, unroll180, unrollPI, \ wrap90, wrap180, wrapPI, wrapPI_2 from math import atan2, cos, degrees, radians, sin, sqrt # pow # all public contants, classes and functions __all__ = _ALL_LAZY.formy __version__ = '20.03.19' def _scaled(lat1, lat2): # degrees # scale delta lon by cos(mean of lats) m = (lat1 + lat2) * 0.5 return cos(radians(m)) if abs(m) < 90 else 0 def _scaler(rad1, rad2): # radians, imported by heights.HeighIDW2 # scale delta lon by cos(mean of lats) m = (rad1 + rad2) * 0.5 return cos(m) if abs(m) < PI_2 else 0 def antipode(lat, lon): '''Return the antipode, the point diametrically opposite to a given point in C{degrees}. @arg lat: Latitude (C{degrees}). @arg lon: Longitude (C{degrees}). @return: A L{LatLon2Tuple}C{(lat, lon)}. @see: U{Geosphere<https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}. ''' return LatLon2Tuple(-wrap90(lat), wrap180(lon + 180)) def antipode_(phi, lam): '''Return the antipode, the point diametrically opposite to a given point in C{radians}. @arg phi: Latitude (C{radians}). @arg lam: Longitude (C{radians}). @return: A L{PhiLam2Tuple}C{(phi, lam)}. @see: U{Geosphere<https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}. ''' return PhiLam2Tuple(-wrapPI_2(phi), wrapPI(lam + PI)) def bearing(lat1, lon1, lat2, lon2, **options): '''Compute the initial or final bearing (forward or reverse azimuth) between a (spherical) start and end point. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg options: Optional keyword arguments for function L{bearing_}. @return: Initial or final bearing (compass C{degrees360}) or zero if start and end point coincide. ''' ab4 = map1(radians, lat1, lon1, lat2, lon2) return degrees(bearing_(*ab4, **options)) def bearing_(a1, b1, a2, b2, final=False, wrap=False): '''Compute the initial or final bearing (forward or reverse azimuth) between a (spherical) start and end point. @arg a1: Start latitude (C{radians}). @arg b1: Start longitude (C{radians}). @arg a2: End latitude (C{radians}). @arg b2: End longitude (C{radians}). @kwarg final: Return final bearing if C{True}, initial otherwise (C{bool}). @kwarg wrap: Wrap and L{unrollPI} longitudes (C{bool}). @return: Initial or final bearing (compass C{radiansPI2}) or zero if start and end point coincide. ''' if final: a1, b1, a2, b2 = a2, b2, a1, b1 r = PI + PI2 else: r = PI2 db, _ = unrollPI(b1, b2, wrap=wrap) sa1, ca1, sa2, ca2, sdb, cdb = sincos2(a1, a2, db) # see <https://MathForum.org/library/drmath/view/55417.html> x = ca1 * sa2 - sa1 * ca2 * cdb y = sdb * ca2 return (atan2(y, x) + r) % PI2 # wrapPI2 def compassAngle(lat1, lon1, lat2, lon2, adjust=True, wrap=False): '''Return the angle from North for the direction vector M{(lon2 - lon1, lat2 - lat1)} between two points. Suitable only for short, non-near-polar vectors up to a few hundred Km or Miles. Use function L{bearing} for longer vectors. @arg lat1: From latitude (C{degrees}). @arg lon1: From longitude (C{degrees}). @arg lat2: To latitude (C{degrees}). @arg lon2: To longitude (C{degrees}). @kwarg adjust: Adjust the longitudinal delta by the cosine of the mean latitude (C{bool}). @kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). @return: Compass angle from North (C{degrees360}). @note: Courtesy Martin Schultz. @see: U{Local, flat earth approximation <https://www.EdWilliams.org/avform.htm#flat>}. ''' d_lon, _ = unroll180(lon1, lon2, wrap=wrap) if adjust: # scale delta lon d_lon *= _scaled(lat1, lat2) return degrees360(atan2(d_lon, lat2 - lat1)) def equirectangular(lat1, lon1, lat2, lon2, radius=R_M, **options): '''Compute the distance between two points using the U{Equirectangular Approximation / Projection <https://www.Movable-Type.co.UK/scripts/latlong.html>}. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg radius: Mean earth radius (C{meter}). @kwarg options: Optional keyword arguments for function L{equirectangular_}. @return: Distance (C{meter}, same units as B{C{radius}}). @see: Function L{equirectangular_} for more details, the available B{C{options}}, errors, restrictions and other, approximate or accurate distance functions. ''' _, dy, dx, _ = equirectangular_(lat1, lon1, lat2, lon2, **options) # PYCHOK Distance4Tuple return degrees2m(hypot(dx, dy), radius=radius) def equirectangular_(lat1, lon1, lat2, lon2, adjust=True, limit=45, wrap=False): '''Compute the distance between two points using the U{Equirectangular Approximation / Projection <https://www.Movable-Type.co.UK/scripts/latlong.html>}. This approximation is valid for short distance of several hundred Km or Miles, see the B{C{limit}} keyword argument and the L{LimitError}. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg adjust: Adjust the wrapped, unrolled longitudinal delta by the cosine of the mean latitude (C{bool}). @kwarg limit: Optional limit for lat- and longitudinal deltas (C{degrees}) or C{None} or C{0} for unlimited. @kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). @return: A L{Distance4Tuple}C{(distance2, delta_lat, delta_lon, unroll_lon2)}. @raise LimitError: If the lat- and/or longitudinal delta exceeds the B{C{-limit..+limit}} range and L{limiterrors} set to C{True}. @see: U{Local, flat earth approximation <https://www.EdWilliams.org/avform.htm#flat>}, functions L{equirectangular}, L{euclidean}, L{haversine} and L{vincentys} and methods L{Ellipsoid.distance2}, C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. ''' d_lat = lat2 - lat1 d_lon, ulon2 = unroll180(lon1, lon2, wrap=wrap) if limit and _limiterrors \ and max(abs(d_lat), abs(d_lon)) > limit > 0: t = fstr((lat1, lon1, lat2, lon2), prec=4) raise LimitError('%s(%s, limit=%s) delta exceeds limit' % ('equirectangular_', t, fstr(limit, prec=2))) if adjust: # scale delta lon d_lon *= _scaled(lat1, lat2) d2 = hypot2(d_lat, d_lon) # degrees squared! return Distance4Tuple(d2, d_lat, d_lon, ulon2 - lon2) def euclidean(lat1, lon1, lat2, lon2, radius=R_M, adjust=True, wrap=False): '''Approximate the C{Euclidian} distance between two (spherical) points. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg radius: Mean earth radius (C{meter}). @kwarg adjust: Adjust the longitudinal delta by the cosine of the mean latitude (C{bool}). @kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). @return: Distance (C{meter}, same units as B{C{radius}}). @see: U{Distance between two (spherical) points <https://www.EdWilliams.org/avform.htm#Dist>}, functions L{equirectangular}, L{haversine} and L{vincentys} and methods L{Ellipsoid.distance2}, C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. ''' r = float(radius) if r: d, _ = unroll180(lon1, lon2, wrap=wrap) r *= euclidean_(radians(lat2), radians(lat1), radians(d), adjust=adjust) return r def euclidean_(a2, a1, b21, adjust=True): '''Approximate the I{angular} C{Euclidean} distance between two (spherical) points. @arg a2: End latitude (C{radians}). @arg a1: Start latitude (C{radians}). @arg b21: Longitudinal delta, M{end-start} (C{radians}). @kwarg adjust: Adjust the longitudinal delta by the cosine of the mean latitude (C{bool}). @return: Angular distance (C{radians}). @see: Functions L{euclidean}, L{equirectangular_}, L{haversine_} and L{vincentys_}. ''' a, b = abs(a2 - a1), abs(b21) if adjust: b *= _scaler(a2, a1) if a < b: a, b = b, a return a + b * 0.5 # 0.4142135623731 def haversine(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): '''Compute the distance between two (spherical) points using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} formula. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg radius: Mean earth radius (C{meter}). @kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). @return: Distance (C{meter}, same units as B{C{radius}}). @see: U{Distance between two (spherical) points <https://www.EdWilliams.org/avform.htm#Dist>}, functions L{equirectangular}, L{euclidean} and L{vincentys} and methods L{Ellipsoid.distance2}, C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. @note: See note under L{vincentys_}. ''' r = float(radius) if r: d, _ = unroll180(lon1, lon2, wrap=wrap) r *= haversine_(radians(lat2), radians(lat1), radians(d)) return r def haversine_(a2, a1, b21): '''Compute the I{angular} distance between two (spherical) points using the U{Haversine<https://www.Movable-Type.co.UK/scripts/latlong.html>} formula. @arg a2: End latitude (C{radians}). @arg a1: Start latitude (C{radians}). @arg b21: Longitudinal delta, M{end-start} (C{radians}). @return: Angular distance (C{radians}). @see: Functions L{haversine}, L{equirectangular_}, L{euclidean_} and L{vincentys_}. @note: See note under L{vincentys_}. ''' def _hsin(rad): return sin(rad * 0.5)**2 h = _hsin(a2 - a1) + cos(a1) * cos(a2) * _hsin(b21) # haversine try: r = atan2(sqrt(h), sqrt(1 - h)) * 2 # == asin(sqrt(h)) * 2 except ValueError: r = 0 if h < 0.5 else PI return r def heightOf(angle, distance, radius=R_M): '''Determine the height above the (spherical) earth after traveling along a straight line at a given tilt. @arg angle: Tilt angle above horizontal (C{degrees}). @arg distance: Distance along the line (C{meter} or same units as B{C{radius}}). @kwarg radius: Optional mean earth radius (C{meter}). @return: Height (C{meter}, same units as B{C{distance}} and B{C{radius}}). @raise ValueError: Invalid B{C{angle}}, B{C{distance}} or B{C{radius}}. @see: U{MultiDop geog_lib.GeogBeamHt<https://GitHub.com/NASA/MultiDop>} (U{Shapiro et al. 2009, JTECH <https://Journals.AMetSoc.org/doi/abs/10.1175/2009JTECHA1256.1>} and U{Potvin et al. 2012, JTECH <https://Journals.AMetSoc.org/doi/abs/10.1175/JTECH-D-11-00019.1>}). ''' d, r = distance, radius if d > r: d, r = r, d if d > EPS: d = d / float(r) s = sin(radians(angle)) s = fsum_(1, 2 * s * d, d**2) if s > 0: return r * sqrt(s) - float(radius) raise ValueError('%s%r' % (heightOf.__name__, (angle, distance, radius))) def horizon(height, radius=R_M, refraction=False): '''Determine the distance to the horizon from a given altitude above the (spherical) earth. @arg height: Altitude (C{meter} or same units as B{C{radius}}). @kwarg radius: Optional mean earth radius (C{meter}). @kwarg refraction: Consider atmospheric refraction (C{bool}). @return: Distance (C{meter}, same units as B{C{height}} and B{C{radius}}). @raise ValueError: Invalid B{C{height}} or B{C{radius}}. @see: U{Distance to horizon<https://www.EdWilliams.org/avform.htm#Horizon>}. ''' if min(height, radius) < 0: raise ValueError('%s%r' % (horizon.__name__, (height, radius))) if refraction: d2 = 2.415750694528 * height * radius # 2.0 / 0.8279 else: d2 = height * fsum_(radius, radius, height) return sqrt(d2) def isantipode(lat1, lon1, lat2, lon2, eps=EPS): '''Check whether two points are antipodal, on diametrically opposite sides of the earth. @arg lat1: Latitude of one point (C{degrees}). @arg lon1: Longitude of one point (C{degrees}). @arg lat2: Latitude of the other point (C{degrees}). @arg lon2: Longitude of the other point (C{degrees}). @kwarg eps: Tolerance for near-equality (C{degrees}). @return: C{True} if points are antipodal within the B{C{eps}} tolerance, C{False} otherwise. @see: U{Geosphere<https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}. ''' return abs(wrap90(lat1) + wrap90(lat2)) < eps and \ abs(abs(wrap180(lon1) - wrap180(lon2)) % 360 - 180) < eps def isantipode_(phi1, lam1, phi2, lam2, eps=EPS): '''Check whether two points are antipodal, on diametrically opposite sides of the earth. @arg phi1: Latitude of one point (C{radians}). @arg lam1: Longitude of one point (C{radians}). @arg phi2: Latitude of the other point (C{radians}). @arg lam2: Longitude of the other point (C{radians}). @kwarg eps: Tolerance for near-equality (C{radians}). @return: C{True} if points are antipodal within the B{C{eps}} tolerance, C{False} otherwise. @see: U{Geosphere<https://CRAN.R-Project.org/web/packages/geosphere/geosphere.pdf>}. ''' return abs(wrapPI_2(phi1) + wrapPI_2(phi2)) < eps and \ abs(abs(wrapPI(lam1) - wrapPI(lam2)) % PI2 - PI) < eps def latlon2n_xyz(lat, lon): '''Convert lat-, longitude to C{n-vector} (normal to the earth's surface) X, Y and Z components. @arg lat: Latitude (C{degrees}). @arg lon: Longitude (C{degrees}). @return: A L{Vector3Tuple}C{(x, y, z)}. @see: Function L{philam2n_xyz}. @note: These are C{n-vector} x, y and z components, I{NOT} (geocentric) ECEF x, y and z coordinates! ''' return philam2n_xyz(radians(lat), radians(lon)) def n_xyz2latlon(x, y, z): '''Convert C{n-vector} components to lat- and longitude in C{degrees}. @arg x: X component (C{scalar}). @arg y: Y component (C{scalar}). @arg z: Z component (C{scalar}). @return: A L{LatLon2Tuple}C{(lat, lon)} in C{degrees}. @see: Function L{n_xyz2philam}. ''' a, b = n_xyz2philam(x, y, z) # PYCHOK PhiLam2Tuple return LatLon2Tuple(degrees90(a), degrees180(b)) def n_xyz2philam(x, y, z): '''Convert C{n-vector} components to lat- and longitude in C{radians}. @arg x: X component (C{scalar}). @arg y: Y component (C{scalar}). @arg z: Z component (C{scalar}). @return: A L{PhiLam2Tuple}C{(phi, lam)} in C{radian}. @see: Function L{n_xyz2latlon}. ''' return PhiLam2Tuple(atan2(z, hypot(x, y)), atan2(y, x)) def philam2n_xyz(phi, lam): '''Convert lat-, longitude to C{n-vector} (normal to the earth's surface) X, Y and Z components. @arg phi: Latitude (C{radians}). @arg lam: Longitude (C{radians}). @return: A L{Vector3Tuple}C{(x, y, z)}. @see: Function L{latlon2n_xyz}. @note: These are C{n-vector} x, y and z components, I{NOT} (geocentric) ECEF x, y and z coordinates! ''' # Kenneth Gade eqn 3, but using right-handed # vector x -> 0°E,0°N, y -> 90°E,0°N, z -> 90°N sa, ca, sb, cb = sincos2(phi, lam) return Vector3Tuple(ca * cb, ca * sb, sa) def points2(points, closed=True, base=None, Error=ValueError): '''Check a path or polygon represented by points. @arg points: The path or polygon points (C{LatLon}[]) @kwarg closed: Optionally, consider the polygon closed, ignoring any duplicate or closing final B{C{points}} (C{bool}). @kwarg base: Optionally, check all B{C{points}} against this base class, if C{None} don't check. @kwarg Error: Exception to raise (C{ValueError}). @return: A L{Points2Tuple}C{(number, points)} with the number of points and the points C{list} or C{tuple}. @raise TypeError: Some B{C{points}} are not B{C{base}}. @raise Error: Insufficient number of B{C{points}}. ''' n, points = len2(points) if closed: # remove duplicate or closing final points while n > 1 and (points[n-1] == points[0] or points[n-1] == points[n-2]): n -= 1 # XXX following line is unneeded if points # are always indexed as ... i in range(n) points = points[:n] # XXX numpy.array slice is a view! if n < (3 if closed else 1): raise Error('too few %s: %s' % ('points', n)) if base and not (isNumpy2(points) or isTuple2(points)): for i in range(n): base.others(points[i], name='%s[%s]' % ('points', i)) return Points2Tuple(n, points) def vincentys(lat1, lon1, lat2, lon2, radius=R_M, wrap=False): '''Compute the distance between two (spherical) points using U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} spherical formula. @arg lat1: Start latitude (C{degrees}). @arg lon1: Start longitude (C{degrees}). @arg lat2: End latitude (C{degrees}). @arg lon2: End longitude (C{degrees}). @kwarg radius: Mean earth radius (C{meter}). @kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). @return: Distance (C{meter}, same units as B{C{radius}}). @see: Functions L{equirectangular}, L{euclidean} and L{haversine} and methods L{Ellipsoid.distance2}, C{LatLon.distanceTo*} and C{LatLon.equirectangularTo}. @note: See note under L{vincentys_}. ''' d, _ = unroll180(lon1, lon2, wrap=wrap) r = vincentys_(radians(lat2), radians(lat1), radians(d)) return r * float(radius) def vincentys_(a2, a1, b21): '''Compute the I{angular} distance between two (spherical) points using U{Vincenty's<https://WikiPedia.org/wiki/Great-circle_distance>} spherical formula. @arg a2: End latitude (C{radians}). @arg a1: Start latitude (C{radians}). @arg b21: Longitudinal delta, M{end-start} (C{radians}). @return: Angular distance (C{radians}). @see: Functions L{vincentys}, L{equirectangular_}, L{euclidean_} and L{haversine_}. @note: Functions L{vincentys_} and L{haversine_} produce equivalent results, but L{vincentys_} is suitable for antipodal points and slightly more expensive than L{haversine_} (M{3 cos, 3 sin, 1 hypot, 1 atan2, 7 mul, 2 add} versus M{2 cos, 2 sin, 2 sqrt, 1 atan2, 5 mul, 1 add}). ''' sa1, ca1, sa2, ca2, sb21, cb21 = sincos2(a1, a2, b21) x = sa1 * sa2 + ca1 * ca2 * cb21 y = ca1 * sa2 - sa1 * ca2 * cb21 y = hypot(ca2 * sb21, y) return atan2(y, x) # **) MIT License # # Copyright (C) 2016-2020 -- mrJean1 at Gmail -- All Rights Reserved. # # Permission is hereby granted, free of charge, to any person obtaining a # copy of this software and associated documentation files (the "Software"), # to deal in the Software without restriction, including without limitation # the rights to use, copy, modify, merge, publish, distribute, sublicense, # and/or sell copies of the Software, and to permit persons to whom the # Software is furnished to do so, subject to the following conditions: # # The above copyright notice and this permission notice shall be included # in all copies or substantial portions of the Software. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS # OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL # THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR # OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, # ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR # OTHER DEALINGS IN THE SOFTWARE.