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3-space adjoint analytic analytic curve analytic functions apply approaches zero asymptotic average catenoid chain circle coefficients computation concyclic points congruence conjugate consider constant continuous continuous functions coordinates corresponding cubic denote derivatives determined differential equation double points E. H. Moore energy envelope equal expression factor finite formula function geodesic geometric given gives Green's function harmonic harmonic function helicoid Hence heptads identical Illustrative equation imaginary roots inequalities infinite number integer integral curve intersection invariant Lagrange's Identity limit linear locus method motion multiples negative nine-point circle obtained orthogonal pair particle plane positive prime problem projective proof quintic rational integral region represented respect satisfied shadow rails solution substitution sufficient condition surface tacnodes tangent Theorem theory tion transformation tube values vanish variables velocity