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1. Show that the P 0 from our geometrical construction (shown below) is actually the inverse of P with
respect to the circle with center C and radius r.
2. Let O = (0, 0) be the origin and consider the inversion IO,5 .
(a) Compute IO,5 (−2, 3) and IO,5 (6, 8)
(b) Where does IO,5 send the line y = 2x?
(c) Where does IO,5 send the line x + y = 5?
(d) Where does IO,5 send the circle centered at O with radius 3?
3. Find an inversion which maps the circle x2 + y 2 = 16 onto the straight line x = 2.
4. Suppose we are given a rectangle with height a and width b. Show that there is a circle going through
all four vertices of the rectangle. What is the radius of that circle?
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Math 115 Homework Assignment #6
In problems 1 and 2 below, you must use the definition of the hyperbolic length of a path γ :
Z p 2
dx + dy 2
hyperbolic length of γ =
y
γ
1. Suppose y2 ≥ y1 > 0, and let x0 be an arbitrary real number. Show that the hyperbolic length of the
vertical line segment joining (x0 , y1 ) to (x0 , y2 ) is ln(y2 /y1 ).
2. Compute the hyperbolic length of the path which runs along the line y = 3x + 1 from (0, 1) to (1, 4).
3. Let A = (5, 3) and B = (1, 1).
(a) What is the midpoint of the segment AB ?
(b) What is the slope of AB ?
(c) What is the equation of the perpendicular bisector of AB ?
(d) Where does the perpendicular bisector of AB intersect the x-axis? Call this point C = (c, 0).
(e) What is the (Euclidean) distance r from C to A and B ?
(f) Compute the hyperbolic distance h(A, B).
4. Find a horizontal line segment such that the hyperbolic length agrees with the Euclidean length. Is this
possible with a vertical line segment? Try to give an example or explain why not.
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Math 115 Homework Assignment #7
1. Show that a straight geodesic in the upper half-plane H can be extended as a geodesic arbitrarily far in
either direction. You need to prove that the limit of the hyperbolic distance between two points with the
same x-coordinate goes to infinity when we move the points further and further away from one another.
2. Find the hyperbolic center and hyperbolic radius of the Euclidean circle with Euclidean center (5, 4)
3. Suppose you have a hyperbolic triangle whose interior angles are all equal. Are the all sides necessarily
equal? Explain why or give an example for why not. Hint: revisit Euclid’s Proposition 6 in Book I.
4. Show that the hyperbolic circumference of a hyperbolic circle with hyperbolic radius R is 2π sinh(R)
where sinh(x) = (ex − e−x )/2 is the hyperbolic sine function. Use the formulas for the hyperbolic radius
R and the hyperbolic circumference we had in terms of the Euclidean radius r.
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