The excentric angle notation in the ellipse is extremely useful, and in part we can replace it by the hyperbolic sine and cosine in connection with the hyperbola.

Take the hyperbola x2/a2/-y2/b2/= 1, then the coordinates of any point on it may be written acoshφ, bsinhφ, for cosh2φ–sinh2φ=1. The objection to its use in all cases is that the hyperbolic cosine of an angle is always positive, so that (acoshφ, bsinhφ) can only represent any point on the branch on the positive side of the axis of y, for any point on the other branch we must take its coordinates as (–acoshφ, bsinhφ).