%%%%%%%%%%%%%%%%%%%%% Schwartz Triangle Constructor %%%%%%%%%%%%%%%%%%%%%
%
% SCHWARTZ
% Usage: T = SCHWARTZ(p,q,r)
% Inputs:  p,q,r  - real numbers with 1/p +1/q +1/r < 1 
%
% Output: matrix of numbers [ 0, s, t*exp(pi*I/p)] such that the 
%         angles at the vertices are given by
%       
%               vertex         hyperbolic angle   
%
%                  0                pi/p             
%                  s                pi/q
%             t*exp(pi*I/p)         pi/r            
%          
%              
function T = schwartz(p,q,r)             

   I = sqrt(-1);
   a = pi/p;   
   b = pi/q;                   
   c = pi/r;   
   d = pi-a-b-c;                  % this is positive

% the relationship of the all these quantities is:
%
%                        Hyperbolic                 Euclidean 
%      vertex         hyperbolic angle   euclidean angles  opposite side
%
%         0                pi/p           a                2*r*sin(d/2)       
%         s                pi/q           b+d/2                 t
%    t*exp(pi*I/p)         pi/r           c+d/2                 s
%          
%    where r is the radius of the circle defining the geodesic 

%  r,s,t satisfy  (Law of Sines, Law of Cosines)
%
%  2*r*sin(d/2)/sin(a) = s/sin(c+d/2) = t/sin(d+d/2)  (Law of Sines, above triangle)
%  1 = s^2 +2*r*s*sin(b)   triangle [ 0, s, centre of circle]
%  1 = t^2 +2*r*t*sin(c)   triangle [ 0, t*exp(pi*I/p), centre of circle]


% the equations above may be solved for r,s,t:
   
   w = 2*sin(d/2)/sin(a);
   u = sin(c+d/2);
   v = sin(b+d/2);
   r = 1/sqrt(w^2*u^2 + 2*u*w*sin(b));
   s = u*w*r;
   t = v*w*r ;
   
   T = [ 0, s, t*exp(pi*I/p)];