I need some help finding the Poincare distance
I have tried to do this problem, but I know my answer is wrong!
Please Help me:
Let Gamma be the interior of the unit disk centered at the origin in the xy-plane, (x^2+y^2 <1). Find the poincare distance between the following pairs of points.
(0,0) and (1/2,0).
Find A’
· TU is the line x= ½
· T= (½ , (√3)/2)
· U=(½ , - (√3)/2)
· Find the equation of the line tangent to gamma at T (Perpendicular to radius of T)
o The slope of point T is √3
o Slope of the perpendicular to the radius is -1/√3
o Equation of the line: y = -1/√3 x + 2/√3
· A’
o 0= -1/√3 x + 2/√3
o X=2
o A’ =(2,0)
Find midpoint of AA’
· ( 5/4 , 0)
Equation of the perpendicular bisector
· X= 5/4
Midpoint of BA
· (¼ , 0)
Intersection of midpoint and perpendicular bisector
· (5/4, 0)
Find the radius (r)
· r = ¾
· delta( the other circle)
o y² + (x- 5/4)² = 9/16
So P and Q are the simultaneous Solutions of
· x² + y² =1
· y² + (x- 5/4)² = 9/16
Then we find P and Q and then the cross ratio, but my problem is that the circle delta creates doesn’t include B, so I feel I did something wrong along the way. I feel my A’ is wrong. I really don’t know Please help me.
Thanks