The Relationship Between Hyperbola and Catenary

When the points a, b, c, and d are projected back down to the asymptote they mark off lengths, a"=1, b"=(2^{1}+1/2^{1})/2; c"=(2^{2}+1/2^{2})/2; d"=(2^{3}+1/2^{3})/2