The auxiliary circle $x^2$ + $y^2$ = $a^2$ is used to define the eccentric angle. For point P = (asec(theta), btan(theta)) on the hyperbola, the corresponding point on the auxiliary circle is not directly above P (unlike the ellipse). Instead, if the ordinate through P meets the auxiliary circle at Q, then angle QOA = theta where O is centre and A is vertex. The construction is: draw PN perpendicular to transverse axis, then the circle at x = ON gives angle theta.