Envelopes of Splines in the Projective Plane Krzysztof Andrzej Krakowski School of Mathematics, Statistics and Computer Science, The University of New England, Australia Abstract In this paper a family of curves—Riemannian cubics—in the unit sphere and the real projective plane is investigated. Riemannian cubics naturally arise as solutions to variational problems in Riemannian spaces. It is remarkable to find that an envelope of lines generated by a Rieman- nian cubic in one space is (nearly) a Riemannian cubic in another space. Keywords: Riemannian cubic polynomials; real projective plane; envelopes. 1 Introduction and Statement of Results The research outlined in this paper was motivated by an application of Rieman- nian cubics to reconstruct a shape from ray-based information. Given a family of ordered tangential lines (rays) to a contour of a convex body in R 2 the con- tour can be approximated by reformulating the problem to the real projective plane (RP (2)). Now, the tangential lines become points in RP (2) that can be interpolated by a spline. The envelope of the family of plane lines generated by the interpolating curve in the projective plane is the curve interpolating the tangential lines. Hence, the envelope can be thought of as a solution to the problem of reconstructing a convex shape from a family of ordered rays. An attractive way to extend the classical theory of splines in Euclidean spaces is through studying curves as solutions to variational problems. Varia- tional properties of curves naturally extend to non-Euclidean spaces. Splines in Riemannian spaces studied in Noakes, Heinzinger & Paden (1989) inspired a number of extensions to the theory in the form of elastic curves (see Camarinha 1996, Brunnett & Crouch 1994, Silva Leite, Camarinha & Crouch 2000), and higher order geometric splines (see Camarinha, Silva Leite & Crouch 1995). Curves derived with variational methods in Riemannian spaces have a num- ber of applications, for example in robotics, motion planning, graphics and computer aided design (Bullo & Murray 1999, Park & Ravani 1997, Kang & Park 1999, Belta & Kumar 2002). Let (M,g) be a Riemannian structure and I ⊂ R be a closed interval. A regular curve is a smooth curve γ : I → M such that ˙ γ (t) = 0, for all t ∈ I . For a regular curve γ : I → M and k ≥ 0 define a functional Φ (k) (γ ) def =  I ‖D k t ˙ γ ‖ 2 dt, (1) 1