%%%%%%%%%%%%%%%%%%%%% Schwartz LFT Triple Constructor %%%%%%%%%%%%%%%%%%%%%
%
% LFTSCHWARTZTRIPLE
% Usage: [A,B,C] = LFTSCHWARTZTRIPLE(p,q,r)
% Inputs:  p,q,r  - real numbers with 1/p +1/q +1/r < 1 
%
% Output: Three matrices  A,B,C such that 
%         A has order p, B has order q, and C has order r as rotations at the corners of the standard Schwartz triangle
%         and     ABC = I
               
function [A,B,C] =  LFTschwartztriple(p,q,r)             
T = schwartz(p,q,r);
z1 = T(2);
z2 = T(3);
z0 = euccentre(z1,z2);
P = [exp(pi*i/p), 0; 0, exp(-pi*i/p)];
Q = [1,0; 0,1];
R = [z0,-1;1,-z0'];  
R = R/sqrt(det(R));
A = P*conj(Q);
B = Q*conj(R);
C = R*conj(P);