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1from fontTools.pens.basePen import BasePen
2from functools import partial
3from itertools import count
4import sympy as sp
5import sys
6
7n = 3 # Max Bezier degree; 3 for cubic, 2 for quadratic
8
9t, x, y = sp.symbols('t x y', real=True)
10c = sp.symbols('c', real=False) # Complex representation instead of x/y
11
12X = tuple(sp.symbols('x:%d'%(n+1), real=True))
13Y = tuple(sp.symbols('y:%d'%(n+1), real=True))
14P = tuple(zip(*(sp.symbols('p:%d[%s]'%(n+1,w), real=True) for w in '01')))
15C = tuple(sp.symbols('c:%d'%(n+1), real=False))
16
17# Cubic Bernstein basis functions
18BinomialCoefficient = [(1, 0)]
19for i in range(1, n+1):
20	last = BinomialCoefficient[-1]
21	this = tuple(last[j-1]+last[j] for j in range(len(last)))+(0,)
22	BinomialCoefficient.append(this)
23BinomialCoefficient = tuple(tuple(item[:-1]) for item in BinomialCoefficient)
24del last, this
25
26BernsteinPolynomial = tuple(
27	tuple(c * t**i * (1-t)**(n-i) for i,c in enumerate(coeffs))
28	for n,coeffs in enumerate(BinomialCoefficient))
29
30BezierCurve = tuple(
31	tuple(sum(P[i][j]*bernstein for i,bernstein in enumerate(bernsteins))
32		for j in range(2))
33	for n,bernsteins in enumerate(BernsteinPolynomial))
34BezierCurveC = tuple(
35	sum(C[i]*bernstein for i,bernstein in enumerate(bernsteins))
36	for n,bernsteins in enumerate(BernsteinPolynomial))
37
38
39def green(f, curveXY):
40	f = -sp.integrate(sp.sympify(f), y)
41	f = f.subs({x:curveXY[0], y:curveXY[1]})
42	f = sp.integrate(f * sp.diff(curveXY[0], t), (t, 0, 1))
43	return f
44
45
46class _BezierFuncsLazy(dict):
47
48	def __init__(self, symfunc):
49		self._symfunc = symfunc
50		self._bezfuncs = {}
51
52	def __missing__(self, i):
53		args = ['p%d'%d for d in range(i+1)]
54		f = green(self._symfunc, BezierCurve[i])
55		f = sp.gcd_terms(f.collect(sum(P,()))) # Optimize
56		return sp.lambdify(args, f)
57
58class GreenPen(BasePen):
59
60	_BezierFuncs = {}
61
62	@classmethod
63	def _getGreenBezierFuncs(celf, func):
64		funcstr = str(func)
65		if not funcstr in celf._BezierFuncs:
66			celf._BezierFuncs[funcstr] = _BezierFuncsLazy(func)
67		return celf._BezierFuncs[funcstr]
68
69	def __init__(self, func, glyphset=None):
70		BasePen.__init__(self, glyphset)
71		self._funcs = self._getGreenBezierFuncs(func)
72		self.value = 0
73
74	def _moveTo(self, p0):
75		self.__startPoint = p0
76
77	def _closePath(self):
78		p0 = self._getCurrentPoint()
79		if p0 != self.__startPoint:
80			self._lineTo(self.__startPoint)
81
82	def _endPath(self):
83		p0 = self._getCurrentPoint()
84		if p0 != self.__startPoint:
85			# Green theorem is not defined on open contours.
86			raise NotImplementedError
87
88	def _lineTo(self, p1):
89		p0 = self._getCurrentPoint()
90		self.value += self._funcs[1](p0, p1)
91
92	def _qCurveToOne(self, p1, p2):
93		p0 = self._getCurrentPoint()
94		self.value += self._funcs[2](p0, p1, p2)
95
96	def _curveToOne(self, p1, p2, p3):
97		p0 = self._getCurrentPoint()
98		self.value += self._funcs[3](p0, p1, p2, p3)
99
100# Sample pens.
101# Do not use this in real code.
102# Use fontTools.pens.momentsPen.MomentsPen instead.
103AreaPen = partial(GreenPen, func=1)
104MomentXPen = partial(GreenPen, func=x)
105MomentYPen = partial(GreenPen, func=y)
106MomentXXPen = partial(GreenPen, func=x*x)
107MomentYYPen = partial(GreenPen, func=y*y)
108MomentXYPen = partial(GreenPen, func=x*y)
109
110
111def printGreenPen(penName, funcs, file=sys.stdout):
112
113	print(
114'''from fontTools.pens.basePen import BasePen
115
116class %s(BasePen):
117
118	def __init__(self, glyphset=None):
119		BasePen.__init__(self, glyphset)
120'''%penName, file=file)
121	for name,f in funcs:
122		print('		self.%s = 0' % name, file=file)
123	print('''
124	def _moveTo(self, p0):
125		self.__startPoint = p0
126
127	def _closePath(self):
128		p0 = self._getCurrentPoint()
129		if p0 != self.__startPoint:
130			self._lineTo(self.__startPoint)
131
132	def _endPath(self):
133		p0 = self._getCurrentPoint()
134		if p0 != self.__startPoint:
135			# Green theorem is not defined on open contours.
136			raise NotImplementedError
137''', end='', file=file)
138
139	for n in (1, 2, 3):
140
141		if n == 1:
142			print('''
143	def _lineTo(self, p1):
144		x0,y0 = self._getCurrentPoint()
145		x1,y1 = p1
146''', file=file)
147		elif n == 2:
148			print('''
149	def _qCurveToOne(self, p1, p2):
150		x0,y0 = self._getCurrentPoint()
151		x1,y1 = p1
152		x2,y2 = p2
153''', file=file)
154		elif n == 3:
155			print('''
156	def _curveToOne(self, p1, p2, p3):
157		x0,y0 = self._getCurrentPoint()
158		x1,y1 = p1
159		x2,y2 = p2
160		x3,y3 = p3
161''', file=file)
162		subs = {P[i][j]: [X, Y][j][i] for i in range(n+1) for j in range(2)}
163		greens = [green(f, BezierCurve[n]) for name,f in funcs]
164		greens = [sp.gcd_terms(f.collect(sum(P,()))) for f in greens] # Optimize
165		greens = [f.subs(subs) for f in greens] # Convert to p to x/y
166		defs, exprs = sp.cse(greens,
167				     optimizations='basic',
168				     symbols=(sp.Symbol('r%d'%i) for i in count()))
169		for name,value in defs:
170			print('		%s = %s' % (name, value), file=file)
171		print(file=file)
172		for name,value in zip([f[0] for f in funcs], exprs):
173			print('		self.%s += %s' % (name, value), file=file)
174
175	print('''
176if __name__ == '__main__':
177	from fontTools.misc.symfont import x, y, printGreenPen
178	printGreenPen('%s', ['''%penName, file=file)
179	for name,f in funcs:
180		print("		      ('%s', %s)," % (name, str(f)), file=file)
181	print('		     ])', file=file)
182
183
184if __name__ == '__main__':
185	pen = AreaPen()
186	pen.moveTo((100,100))
187	pen.lineTo((100,200))
188	pen.lineTo((200,200))
189	pen.curveTo((200,250),(300,300),(250,350))
190	pen.lineTo((200,100))
191	pen.closePath()
192	print(pen.value)
193