def _geodesic_between_two_points_d(x1,y1,x2,y2,z0=I):
r""" Geodesic path between two points represented in the unit disc
by the map w = (z-I)/(z+I)
INPUTS:
- ''(x1,y1)'' -- starting point (0<y1<=infinity)
- ''(x2,y2)'' -- ending point (0<y2<=infinity)
- ''z0'' -- (default I) the point in the upper corresponding
to the point 0 in the disc. I.e. the transform is
w -> (z-I)/(z+I)
OUTPUT:
- ''ca'' -- a polygonal approximation of a circular arc centered
at c and radius r, starting at t0 and ending at t1
EXAMPLES::
sage: l=_geodesic_between_two_points_d(0.1,0.2,0.0,0.5)
"""
pi=RR.pi()
from sage.plot.plot import line
from sage.functions.trig import (cos,sin)
# First compute the points
if(y1<0 or y2<0 ):
raise ValueError,"Need points in the upper half-plane! Got y1=%s, y2=%s" %(y1,y2)
if(y1==infinity):
P1=CC(1 )
else:
P1=CC((x1+I*y1-z0)/(x1+I*y1-z0.conjugate()))
if(y2==infinity):
P2=CC(1 )
else:
P2=CC((x2+I*y2-z0)/(x2+I*y2-z0.conjugate()))
# First find the endpoints of the completed geodesic in D
if(x1==x2):
a=CC((x1-z0)/(x1-z0.conjugate()))
b=CC(1 )
else:
c=RR(y1**2 -y2**2 +x1**2 -x2**2 )/RR(2 *(x1-x2))
r=RR(sqrt(y1**2 +(x1-c)**2 ))
a=c-r
b=c+r
a=CC((a-z0)/(a-z0.conjugate()))
b=CC((b-z0)/(b-z0.conjugate()))
if( abs(a+b) < 1E-10 ): # On a diagonal
return line([[P1.real(),P1.imag()],[P2.real(),P2.imag()]])
th_a=a.argument()
th_b=b.argument()
# Compute the center of the circle in the disc model
if( min(abs(b-1 ),abs(b+1 ))< 1E-10 and min(abs(a-1 ),abs(a+1 ))>1E-10 ):
c=b+I*(1 -b*cos(th_a))/sin(th_a)
elif( min(abs(b-1 ),abs(b+1 ))> 1E-10 and min(abs(a-1 ),abs(a+1 ))<1E-10 ):
c=a+I*(1 -a*cos(th_b))/RR(sin(th_b))
else:
cx=(sin(th_b)-sin(th_a))/sin(th_b-th_a)
c=cx+I*(1 -cx*cos(th_b))/RR(sin(th_b))
# First find the endpoints of the completed geodesic
r=abs(c-a)
t1=CC(P1-c).argument()
t2=CC(P2-c).argument()
#print "t1,t2=",t1,t2
return _circ_arc(t1,t2,c,r)
def _geodesic_between_two_points_d(x1, y1, x2, y2, z0=I):
r""" Geodesic path between two points represented in the unit disc
by the map w = (z-I)/(z+I)
INPUTS:
- ''(x1,y1)'' -- starting point (0<y1<=infinity)
- ''(x2,y2)'' -- ending point (0<y2<=infinity)
- ''z0'' -- (default I) the point in the upper corresponding
to the point 0 in the disc. I.e. the transform is
w -> (z-I)/(z+I)
OUTPUT:
- ''ca'' -- a polygonal approximation of a circular arc centered
at c and radius r, starting at t0 and ending at t1
EXAMPLES::
sage: l=_geodesic_between_two_points_d(0.1,0.2,0.0,0.5)
"""
pi = RR.pi()
from sage.plot.plot import line
from sage.functions.trig import (cos, sin)
# First compute the points
if (y1 < 0 or y2 < 0):
raise ValueError, "Need points in the upper half-plane! Got y1=%s, y2=%s" % (
y1, y2)
if (y1 == infinity):
P1 = CC(1)
else:
P1 = CC((x1 + I * y1 - z0) / (x1 + I * y1 - z0.conjugate()))
if (y2 == infinity):
P2 = CC(1)
else:
P2 = CC((x2 + I * y2 - z0) / (x2 + I * y2 - z0.conjugate()))
# First find the endpoints of the completed geodesic in D
if (x1 == x2):
a = CC((x1 - z0) / (x1 - z0.conjugate()))
b = CC(1)
else:
c = RR(y1**2 - y2**2 + x1**2 - x2**2) / RR(2 * (x1 - x2))
r = RR(sqrt(y1**2 + (x1 - c)**2))
a = c - r
b = c + r
a = CC((a - z0) / (a - z0.conjugate()))
b = CC((b - z0) / (b - z0.conjugate()))
if (abs(a + b) < 1E-10): # On a diagonal
return line([[P1.real(), P1.imag()], [P2.real(), P2.imag()]])
th_a = a.argument()
th_b = b.argument()
# Compute the center of the circle in the disc model
if (min(abs(b - 1), abs(b + 1)) < 1E-10
and min(abs(a - 1), abs(a + 1)) > 1E-10):
c = b + I * (1 - b * cos(th_a)) / sin(th_a)
elif (min(abs(b - 1), abs(b + 1)) > 1E-10
and min(abs(a - 1), abs(a + 1)) < 1E-10):
c = a + I * (1 - a * cos(th_b)) / RR(sin(th_b))
else:
cx = (sin(th_b) - sin(th_a)) / sin(th_b - th_a)
c = cx + I * (1 - cx * cos(th_b)) / RR(sin(th_b))
# First find the endpoints of the completed geodesic
r = abs(c - a)
t1 = CC(P1 - c).argument()
t2 = CC(P2 - c).argument()
#print "t1,t2=",t1,t2
return _circ_arc(t1, t2, c, r)
