Thus we have this modified form of the parallel postulate : In hyperbolic geometry , given any line R , and point P not on R , there are exactly two lines through P which are asymptotic to R , and infinitely many lines through P ultraparallel to R . The differences between these types of lines can also be looked at in the following way : the distance between asymptotic lines shrinks toward zero in one direction and grows without bound in the other ; the distance between ultraparallel lines ( eventually ) increases in both directions .
Thus we have this modified form of the parallel postulate : In hyperbolic geometry , given any line l , and point P not on l , there are exactly two lines through P which are asymptotic to l , and infinitely many lines through P ultraparallel to l . The differences between these types of lines can also be looked at in the following way : the distance between asymptotic lines run to zero in one direction and grows without bound in the other ; the distance between ultraparallel lines increases in both directions .