Copy of Exercise 11.3.3

Metric Axiom 1: If A and B are points, then d(A,B) is greater than or equal to zero and d(A,B)=0 iff A=B. Proof: The distance formula in the Poincare disk model is By nature of the absolute value signs, d(A,B) will always be nonnegative. To show that d(A,B)=0 iff A=B, first assume that d(A,B)=0. We know that only ln(1)=0. Thus, . This can only occur when A=B. Similarly, if we assume that A=B, we can show that . Thus, the second qualification of Metric Axiom 1 is also satisfied. So, Metric Axiom 1 holds for the Poincare distance. Metric Axiom 2: If A and B are points, then d(A,B)=d(B,A). Proof: We know that . Consider, . Thus, Metric Axiom 2 holds for the Poincare distance.