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analytic function angle asymptotes axis brilliant curve brilliant points congruence conic conies conjugate consider constant curvature continued fraction convergent series converges uniformly converges uniformly throughout coordinates corresponding cross ratio cyclic cyclic groups defined denote determined differential equation Euclidean Euclidean space finite follows fundamental pencil geodesic circles geodesic curvature geodesic lines geodesic surface geometry given group of order Hence holomorphisms hypothesis imaginary integral invariant involution leave a safe limit line at infinity line element linear complex locus multiply perfect obtained pairs of points parameter perpendicular piles plane points of intersection points of space positive proof proved quadric quadrilateral quantity quartic real line region represented right line roots safe combination satisfy sequence series converges space of constant straight line subgroups tangents term by term theorem tion transformation triangle uniform convergence values vanish variable vertex vertices zero