def construct_center_polygon(n: int, k: int, quasiregular: bool) -> Polygon:
# Initialize P as the center polygon in an n-k regular or quasiregular tiling.
# Let ABC be a triangle in a regular (n,k0-tiling, where
# A is the center of an n-gon (also center of the disk),
# B is a vertex of the n-gon, and
# C is the midpoint of a side of the n-gon adjacent to B.
angle_a = pi / n
angle_b = pi / k
angle_c = pi / 2
# For a regular tiling, we need to compute the distance s from A to B.
sin_a = sin(angle_a)
sin_b = sin(angle_b)
s = sin(angle_c - angle_b - angle_a) / sqrt(Decimal(1) - sin_b * sin_b - sin_a * sin_a)
# But for a quasiregular tiling, we need the distance s from A to C.
if quasiregular:
s = (s * s + Decimal(1)) / (Decimal(2) * s * cos(angle_a))
s = s - sqrt(s * s - Decimal(1))
# Now determine the coordinates of the n vertices of the n-gon.
# They're all at distance s from the center of the Poincare disk.
poly = [Point(s, s) for _ in range(n)]
for i, pt in enumerate(poly):
pt.x *= cos(Decimal(3 + 2 * i) * angle_a)
pt.y *= sin(Decimal(3 + 2 * i) * angle_a)
return poly
def p_expression_function(t):
'expression : FUNCTION LPAREN expression RPAREN'
if t[1] == 'sqrt':
t[0] = t[3].sqrt()
elif t[1] == 'sin':
t[0] = sin(t[3])
elif t[1] == 'cos':
t[0] = cos(t[3])
elif t[1] == 'ln':
t[0] = t[3].ln()
elif t[1] == 'log':
t[0] = t[3].log10()
elif t[1][:4] == 'log_':
try:
b = Decimal(t[1][4:])
t[0] = t[3].log10() / b.log10()
except decimal.InvalidOperation:
print "Cannot evaluate %s%s%s%s. " \
"Setting it to 0." % (t[1], t[2], t[3], t[4])
t[0] = Decimal(0)
def p_expression_function(t):
'expression : FUNCTION LPAREN expression RPAREN'
if t[1] == 'sqrt':
t[0] = t[3].sqrt()
elif t[1] == 'sin':
t[0] = sin(t[3])
elif t[1] == 'cos':
t[0] = cos(t[3])
elif t[1] == 'ln':
t[0] = t[3].ln()
elif t[1] == 'log':
t[0] = t[3].log10()
elif t[1][:4] == 'log_':
try:
b = Decimal(t[1][4:])
t[0] = t[3].log10() / b.log10()
except decimal.InvalidOperation:
print "Cannot evaluate %s%s%s%s. " \
"Setting it to 0." % (t[1], t[2], t[3], t[4])
t[0] = Decimal(0)
def _eval_power(b, e):
"""
b is Real but not equal to rationals, integers, 0.5, oo, -oo, nan
e is symbolic object but not equal to 0, 1
(-p) ** r -> exp(r * log(-p)) -> exp(r * (log(p) + I*Pi)) ->
-> p ** r * (sin(Pi*r) + cos(Pi*r) * I)
"""
if isinstance(e, Number):
if isinstance(e, Integer):
e = e.p
return Real(decimal_math.pow(b.num, e))
e2 = e._as_decimal()
if b.is_negative and not e.is_integer:
m = decimal_math.pow(-b.num, e2)
a = decimal_math.pi() * e2
s = m * decimal_math.sin(a)
c = m * decimal_math.cos(a)
return Real(s) + Real(c) * S.ImaginaryUnit
return Real(decimal_math.pow(b.num, e2))
return
def _eval_power(b, e):
if e.is_odd: return S.NegativeOne
if e.is_even: return S.One
if isinstance(e, Number):
if isinstance(e, Real):
a = e.num * decimal_math.pi()
s = decimal_math.sin(a)
c = decimal_math.cos(a)
return Real(s) + Real(c) * S.ImaginaryUnit
if isinstance(e, NaN):
return S.NaN
if isinstance(e, (Infinity, NegativeInfinity)):
return S.NaN
if isinstance(e, Half):
return S.ImaginaryUnit
if isinstance(e, Rational):
if e.q == 2:
return S.ImaginaryUnit ** Integer(e.p)
q = int(e)
if q:
q = Integer(q)
return b ** q * b ** (e - q)
return
def x(self, t: Decimal) -> Decimal: return self._x + self._r * cos(t)
def cos(self): return Real(decimal_math.cos(self._as_decimal())) def tan(self): return Real(decimal_math.tan(self._as_decimal()))