A 4-hyperboloid has negative curvature, with R^2 = x^2 + y^2 + z^2 - w^2 2x(d x)/(d w) + 2y(d y)/(d w) + 2z(d z)/(d w) - 2w = 0. Since r congruent xx^^ + yy^^ + zz^^, it follows that d w = (x d x + y d y + z d z)/w = (r·d r)/sqrt(r^2 - R^2).