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# -*- coding: utf-8 -*- 

 

u'''Simplify or linearize a path. 

 

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

produces different simplified results in (very) different run times for 

the same path of C{LatLon} points. 

 

Function L{simplify1} eliminates points based on edge lengths shorter 

than a given tolerance. 

 

The functions L{simplifyRDP} and L{simplifyRDPm} use the original, 

respectively modified Ramer-Douglas-Peucker (RDP) algorithm, iteratively 

finding the point farthest from each path edge. The difference is that 

function L{simplifyRDP} exhaustively searches the most distant point in 

each iteration, while modified L{simplifyRDPm} stops at the first point 

exceeding the distance tolerance. 

 

Function L{simplifyRW} use the Reumann-Witkam method, sliding a "pipe" 

over each path edge, removing all subsequent points within, closer than 

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

 

Functions L{simplifyVW} and L{simplifyVWm} are based on the original, 

respectively modified Visvalingam-Whyatt (VW) method using the area of 

the triangle formed by three neigboring points. Function L{simplifyVW} 

removes only a single point per iteration, while modified L{simplifyVWm} 

eliminates in each iteration all points with a triangular area not 

exceeding the tolerance. 

 

Functions L{simplifyRDP}, L{simplifyRDPm} and L{simplifyRW} provide 

keyword argument I{shortest} to select the computation of the distance 

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

to the path edge or path end points, if C{False} use the perpendicular 

distance to the extended path edge line. 

 

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

radius in meter. Other units can be choosen, provided that the radius 

and tolerance are always specified in the same units. 

 

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

list of simplified point I{indices} instead of the simplified points. 

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

 

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

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

distance approximation. 

 

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

possibly other values, use class L{Numpy2LatLon} to wrap the C{NumPy} 

array into I{on-the-fly-LatLon} points. Pass the L{Numpy2LatLon} 

instance to any I{simplify} function and the returned result will be 

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

the original C{NumPy} array. Use keyword argument C{B{indices}=True} 

to return a list of array row indices inlieu of the simplified array 

subset. 

 

See: 

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

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

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

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

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

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

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

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

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

- U{https://PyPI.org/project/rdp} 

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

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

''' 

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

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

 

# from pygeodesy.basics import len2 # from .fmath 

from pygeodesy.constants import EPS, R_M, _1_0 

from pygeodesy.errors import _AttributeError, _ValueError 

from pygeodesy.fmath import len2, sqrt0 

from pygeodesy.formy import equirectangular_ 

from pygeodesy.interns import _small_, _too_ 

from pygeodesy.iters import isNumpy2, isTuple2 

# from pygeodesy.lazily import _ALL_LAZY # from .units 

from pygeodesy.units import _ALL_LAZY, _1mm 

 

from math import degrees, radians 

 

__all__ = _ALL_LAZY.simplify 

__version__ = '22.09.12' 

 

 

# try: 

# from collections import namedtuple 

# _T2 = namedtuple('_T2', 'ix, h2') 

# except ImportError: 

# class _T2(object): 

# ... 

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

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

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

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

# be suppressed with command line option --stdlib 

class _T2(object): 

'''(INTERNAL) VW 2-tuple (index, area). 

''' 

# __slots__ are no longer space savers, see 

# the comments at the class .points.LatLon_ 

# __slots__ = ('ix', 'h2') 

 

def __init__(self, ix, h2): 

self.ix = ix 

self.h2 = h2 

 

 

class _Sy(object): 

'''(INTERNAL) Simplify state. 

''' 

d2i = None # d2iP or d2iS 

d2yxse = () 

eps = EPS # system epsilon 

indices = False 

n = 0 

options = {} 

pts = [] 

radius = R_M # mean earth radius 

r = {} # RDP indices or VW 2-tuples 

s2 = EPS # tolerance squared 

s2e = EPS # sentinel 

subset = None # isNumpy2 or isTuple2 

 

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

indices, **options): 

'''New C{Simplify} state. 

''' 

n, self.pts = len2(points) 

if n > 0: 

self.n = n 

self.r = {0: True, n-1: True} # dict to avoid duplicates 

 

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

self.subset = points.subset 

 

if indices: 

self.indices = True 

 

if radius: 

self.radius = float(radius) 

if self.radius < self.eps: 

raise _ValueError(radius=radius, txt=_too_(_small_)) 

 

if options: 

self.options = options 

 

# tolerance converted to degrees squared 

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

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

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

self.s2e = self.s2 + 1 # sentinel 

 

# compute either the shortest or perpendicular distance 

self.d2i = self.d2iS if shortest else self.d2iP # PYCHOK false 

 

def d21(self, s, e): 

'''Set path edge or line thru points[s] to -[e]. 

''' 

d21, y21, x21, _ = self.d2yxu(s, e) 

self.d2yxse = d21, y21, x21, s, e 

return d21 > self.eps 

 

def d2ih(self, n, m, brk): 

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

to points[s] exceeding the tolerance. 

''' 

_, _, _, s, _ = self.d2yxse 

eps, d2yxu = self.eps, self.d2yxu 

t2, t = self.s2, 0 # tallest 

for i in range(n, m): 

d2, _, _, _ = d2yxu(s, i) 

if d2 > t2: 

t2, t = d2, i 

if brk and d2 > eps: 

break 

return t2, t 

 

def d2iP(self, n, m, brk): 

'''Find the tallest perpendicular distance among all 

points[n..m] to the path edge or line thru points[s] 

to -[e] exceeding the tolerance. 

''' 

d21, y21, x21, s, _ = self.d2yxse 

eps, d2yxu = self.eps, self.d2yxu 

t2, t = self.s2, 0 # tallest 

for i in range(n, m): 

d2, y01, x01, _ = d2yxu(s, i) 

if d2 > eps: 

# perpendicular distance squared 

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

if d2 > t2: 

t2, t = d2, i 

if brk: 

break 

return t2, t 

 

def d2iS(self, n, m, brk): 

'''Find the tallest shortest distance among all 

points[n..m] to the path edge or line thru 

points[s] to -[e] exceeding the tolerance. 

''' 

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

# x' = y * sin(a) + x * cos(a) 

# y' = y * cos(a) - x * sin(a) 

# 

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

# a line thru point (dx, dy) and the origin: 

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

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

 

d21, y21, x21, s, e = self.d2yxse 

eps, d2yxu = self.eps, self.d2yxu 

t2, t = self.s2, 0 # tallest 

for i in range(n, m): 

# distance points[i] to -[s] 

d2, y01, x01, _ = d2yxu(s, i) 

if d2 > eps: 

w = y01 * y21 + x01 * x21 

if w > 0: 

if w < d21: 

# perpendicular distance squared 

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

else: # distance points[i] to -[e] 

d2, _, _, _ = d2yxu(e, i) 

if d2 > t2: 

t2, t = d2, i 

if brk: 

break 

return t2, t 

 

def d2yxu(self, i, j): 

'''Return distance squared, points[i] to -[j] deltas and 

longitudinal unrollment. 

''' 

p1 = self.pts[i] 

p2 = self.pts[j] 

return equirectangular_(p1.lat, p1.lon, 

p2.lat, p2.lon, **self.options) 

 

def h2t(self, i1, i0, i2): 

'''Compute the Visvalingam-Whyatt triangular area, 

points[i0] is the top and points[i1] to -[i2] 

form the base of the triangle. 

''' 

d21, y21, x21 , _= self.d2yxu(i1, i2) 

if d21 > self.eps: 

d01, y01, x01, _ = self.d2yxu(i1, i0) 

if d01 > self.eps: 

h2 = abs(y01 * x21 - x01 * y21) 

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

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

return h2 # double triangle area 

return 0 

 

def points(self, r): 

'''Return the list of simplified points or indices. 

''' 

r = sorted(r.keys()) 

if self.indices: 

return list(r) 

elif self.subset: 

return self.subset(r) 

else: 

return [self.pts[i] for i in r] 

 

def rdp(self, modified): 

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

path of C{LatLon} points. 

 

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

''' 

n, r = self.n, self.r 

if n > 1: 

s2, d21 = self.s2, self.d21 

d2i, d2ih = self.d2i, self.d2ih 

 

se = [(0, n-1)] 

while se: 

s, e = se.pop() 

if e > (s + 1): 

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

d2, i = d2i(s+1, e, modified) 

else: # points[] to point [s] 

d2, i = d2ih(s+1, e, modified) 

if d2 > s2 and i > 0: 

r[s] = r[i] = True 

se.append((i, e)) 

if not modified: 

se.append((s, i)) 

else: 

r[s] = True 

 

return self.points(r) 

 

def rm1(self, m, tol): 

'''Eliminate one Visvalingam-Whyatt point and recomputes 

the trangular area of both neighboring points, but 

removes those too unless the recomputed area exceeds 

the tolerance. 

''' 

h2t, r = self.h2t, self.r 

 

r.pop(m) 

for n in (m, m - 1): 

while 0 < n < (len(r) - 1): 

h2 = h2t(r[n-1].ix, r[n].ix, r[n+1].ix) 

if h2 > tol: 

r[n].h2 = h2 

break # while 

else: 

r.pop(n) 

 

def rm2(self, tol): 

'''Eliminate all Visvalingam-Whyatt points with a 

triangular area not exceeding the tolerance. 

''' 

r, rm1 = self.r, self.rm1 

 

i = len(r) - 1 

while i > 1: 

i -= 1 

if r[i].h2 <= tol: 

rm1(i, tol) 

i = min(i, len(r) - 1) 

 

def vwn(self): 

'''Initialize Visvalingam-Whyatt as list of 2-tuples 

_T2(ix, h2) where ix is the points[] index and h2 

is the triangular area I{(times 2)} of that point. 

''' 

n, h2t, s2e = self.n, self.h2t, self.s2e 

 

if n > 2: 

self.r = [_T2(0, s2e)] 

self.r.extend(_T2(i, h2t(i-1, i, i+1)) for i in range(1, n-1)) 

self.r.append(_T2(n-1, s2e)) 

elif n > 0: 

self.r = [_T2(i, s2e) for i in range(0, n)] # PYCHOK false 

else: 

self.r = [] 

 

return len(self.r) 

 

def vwr(self, attr): 

'''Return the Visvalingam-Whyatt results, optionally 

including the triangular area (in meters) as 

attribute attr to each simplified point. 

''' 

pts, r = self.pts, self.r 

 

# double check the minimal triangular area 

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

 

if attr: # return the trangular area (actually 

# the sqrt of double the triangular area) 

# converted back from degrees to meter 

if isNumpy2(pts): 

raise _AttributeError(attr=attr) 

m = radians(_1_0) * self.radius 

r[0].h2 = r[-1].h2 = 0 # zap sentinels 

for t2 in r: # convert back to meter 

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

 

# double check for duplicates 

n = len(r) 

r = dict((t2.ix, True) for t2 in r) 

assert len(r) == n 

return self.points(r) 

 

 

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

'''Basic simplification of a path of C{LatLon} points. 

 

Eliminates any points closer together than the given distance 

tolerance. 

 

@arg points: Path points (C{LatLon}[]). 

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

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

@kwarg indices: Optionally return the simplified point indices 

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

@kwarg options: Optional keyword arguments passed thru to 

function L{pygeodesy.equirectangular_}. 

 

@return: Simplified points (C{LatLon}[]). 

 

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

see function L{pygeodesy.equirectangular_}. 

 

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

''' 

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

 

n, r = S.n, S.r 

if n > 1: 

s2, d2yxu = S.s2, S.d2yxu 

 

i = 0 

for j in range(1, n): 

d2, _, _, _= d2yxu(i, j) 

if d2 > s2: 

r[j] = True 

i = j 

 

return S.points(r) 

 

 

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

indices=False, **options): 

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

C{LatLon} points. 

 

Eliminates any points too close together or closer to an 

edge than the given distance tolerance. 

 

This C{RDP} method exhaustively searches for the point with 

the largest distance, resulting in worst-case complexity 

M{O(n**2)} where M{n} is the number of points. 

 

@arg points: Path points (C{LatLon}[]). 

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

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

@kwarg shortest: Optional, shortest or perpendicular distance 

(C{bool}). 

@kwarg indices: Optionally return the simplified point indices 

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

@kwarg options: Optional keyword arguments passed thru to 

function L{pygeodesy.equirectangular_}. 

 

@return: Simplified points (C{LatLon}[]). 

 

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

see function L{pygeodesy.equirectangular_}. 

 

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

''' 

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

 

return S.rdp(False) 

 

 

def simplifyRDPm(points, distance=_1mm, radius=R_M, shortest=False, 

indices=False, **options): 

'''Modified Ramer-Douglas-Peucker (RDPm) simplification of a path 

of C{LatLon} points. 

 

Eliminates any points too close together or closer to an edge 

than the given distance tolerance. 

 

This C{RDPm} method stops at the first point farther than the 

given distance tolerance, significantly reducing the run time 

(but producing results different from the original C{RDP} method). 

 

@arg points: Path points (C{LatLon}[]). 

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

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

@kwarg shortest: Optional, shortest or perpendicular distance 

(C{bool}). 

@kwarg indices: Optionally return the simplified point indices 

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

@kwarg options: Optional keyword arguments passed thru to 

function L{pygeodesy.equirectangular_}. 

 

@return: Simplified points (C{LatLon}[]). 

 

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

see function L{pygeodesy.equirectangular_}. 

 

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

''' 

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

 

return S.rdp(True) 

 

 

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

indices=False, **options): 

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

 

Eliminates any points too close together or within the given 

pipe tolerance along an edge. 

 

@arg points: Path points (C{LatLon}[]). 

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

B{C{radius}}). 

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

@kwarg shortest: Optional, shortest or perpendicular distance 

(C{bool}). 

@kwarg indices: Optionally return the simplified point indices 

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

@kwarg options: Optional keyword arguments passed thru to 

function L{pygeodesy.equirectangular_}. 

 

@return: Simplified points (C{LatLon}[]). 

 

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

see function L{pygeodesy.equirectangular_}. 

 

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

''' 

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

 

n, r = S.n, S.r 

if n > 1: 

s2, d21, d2i = S.s2, S.d21, S.d2i 

 

s, e = 0, 1 

while s < e < n: 

if d21(s, e): 

d2, i = d2i(e + 1, n, True) 

if d2 > s2 and i > 0: 

r[s] = r[i] = True 

s, e = i, i + 1 

else: 

r[s] = True # r[n-1] = True 

break # while loop 

else: # drop points[e] 

e += 1 

 

return S.points(r) 

 

 

def simplifyVW(points, area=_1mm, radius=R_M, attr=None, 

indices=False, **options): 

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

points. 

 

Eliminates any points too close together or with a triangular 

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

 

This C{VW} method exhaustively searches for the single point 

with the smallest triangular area, resulting in worst-case 

complexity M{O(n**2)} where M{n} is the number of points. 

 

@arg points: Path points (C{LatLon}[]). 

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

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

@kwarg attr: Optional, points attribute to save the area value 

(C{str}). 

@kwarg indices: Optionally return the simplified point indices 

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

@kwarg options: Optional keyword arguments passed thru to 

function L{pygeodesy.equirectangular_}. 

 

@return: Simplified points (C{LatLon}[]). 

 

@raise AttributeError: If an B{C{attr}} is specified for I{Numpy2} 

B{C{points}}. 

 

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

see function L{pygeodesy.equirectangular_}. 

 

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

''' 

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

 

if S.vwn() > 2: 

# remove any points too close or 

# with a zero triangular area 

S.rm2(0) 

 

r, s2, s2e = S.r, S.s2, S.s2e 

# keep removing the point with the smallest 

# area until latter exceeds the tolerance 

while len(r) > 2: 

m, m2 = 0, s2e 

for i in range(1, len(r) - 1): 

h2 = r[i].h2 

if h2 < m2: 

m, m2 = i, h2 

if m2 > s2: 

break 

S.rm1(m, 0) 

 

return S.vwr(attr) 

 

 

def simplifyVWm(points, area=_1mm, radius=R_M, attr=None, 

indices=False, **options): 

'''Modified Visvalingam-Whyatt (VWm) simplification of a path of 

C{LatLon} points. 

 

Eliminates any points too close together or with a triangular 

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

 

This C{VWm} method removes all points with a triangular area 

below the tolerance in each iteration, significantly reducing 

the run time (but producing results different from the 

original C{VW} method). 

 

@arg points: Path points (C{LatLon}[]). 

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

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

@kwarg attr: Optional, points attribute to save the area value 

(C{str}). 

@kwarg indices: Optionally return the simplified point indices 

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

@kwarg options: Optional keyword arguments passed thru to 

function L{pygeodesy.equirectangular_}. 

 

@return: Simplified points (C{LatLon}[]). 

 

@raise AttributeError: If an B{C{attr}} is specified for I{Numpy2} 

B{C{points}}. 

 

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

see function L{pygeodesy.equirectangular_}. 

 

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

''' 

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

 

if S.vwn() > 2: 

# remove all points with an area 

# not exceeding the tolerance 

S.rm2(S.s2) 

 

return S.vwr(attr) 

 

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