Abstract
Studying convolutions of hypersurfaces (especially of curves and surfaces) has become an active research area in recent years. The main characterization from the point of view of convolutions is their convolution degree, which is an affine invariant associated to a hypersurface describing the complexity of the shape with respect to the operation of convolution. Extending the results from [1], we will focus on the two simplest classes of planar algebraic curves with respect to the operation of convolution, namely on the curves with the convolution degree one (so called LN curves) and two. We will present an algebraic analysis of these curves, provide their decomposition, and study their properties.
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References
Vršek, J., Lávička, M.: On convolution of algebraic curves. Journal of Symbolic Computation 45(6), 657–676 (2010)
Farouki, R.: Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable. Springer, Heidelberg (2008)
Farouki, R., Sakkalis, T.: Pythagorean hodographs. IBM Journal of Research and Development 34(5), 736–752 (1990)
Farouki, R., Sakkalis, T.: Pythagorean-hodograph space curves. Adv. Comput. Math. 2, 41–66 (1994)
Kosinka, J., Jüttler, B.: G 1 Hermite interpolation by Minkowski Pythagorean hodograph cubics. Computer Aided Geometric Design 23, 401–418 (2006)
Kosinka, J., Jüttler, B.: MOS surfaces: Medial surface transforms with rational domain boundaries. In: Martin, R., Sabin, M.A., Winkler, J.R. (eds.) Mathematics of Surfaces 2007. LNCS, vol. 4647, pp. 245–262. Springer, Heidelberg (2007)
Kosinka, J., Lávička, M.: On rational Minkowski Pythagorean hodograph curves. Computer Aided Geometric Design 27(7), 514–524 (2010)
Peternell, M., Pottmann, H.: A Laguerre geometric approach to rational offsets. Computer Aided Geometric Design 15, 223–249 (1998)
Pottmann, H., Peternell, M.: Applications of Laguerre geometry in CAGD. Computer Aided Geometric Design 15, 165–186 (1998)
Jüttler, B.: Triangular Bézier surface patches with linear normal vector field. In: Cripps, R. (ed.) The Mathematics of Surfaces VIII. Information Geometers, pp. 431–446 (1998)
Peternell, M., Manhart, F.: The convolution of a paraboloid and a parametrized surface. Journal for Geometry and Graphics 7(2), 157–171 (2003)
Sampoli, M.L., Peternell, M., Jüttler, B.: Rational surfaces with linear normals and their convolutions with rational surfaces. Computer Aided Geometric Design 23(2), 179–192 (2006)
Lávička, M., Bastl, B.: Rational hypersurfaces with rational convolutions. Computer Aided Geometric Design 24(7), 410–426 (2007)
Lee, I.K., Kim, M.S., Elber, G.: Polynomial/rational approximation of Minkowski sum boundary curves. Graphical Models and Image Processing 60(2), 136–165 (1998)
Gravesen, J., Jüttler, B., Šír, Z.: On rationally supported surfaces. Computer Aided Geometric Design 25, 320–331 (2008)
Šír, Z., Gravesen, J., Jüttler, B.: Curves and surfaces represented by polynomial support functions. Theoretical Computer Science 392(1-3), 141–157 (2008)
Arrondo, E., Sendra, J., Sendra, J.R.: Parametric generalized offsets to hypersurfaces. Journal of Symbolic Computation 23, 267–285 (1997)
Arrondo, E., Sendra, J., Sendra, J.R.: Genus formula for generalized offset curves. Journal of Pure and Applied Algebra 136, 199–209 (1999)
Sendra, J.R., Sendra, J.: Algebraic analysis of offsets to hypersurfaces. Mathematische Zeitschrift 237, 697–719 (2000)
Šír, Z., Bastl, B., Lávička, M.: Hermite interpolation by hypocycloids and epicycloids with rational offsets. Computer Aided Geometric Design 27, 405–417 (2010)
Jüttler, B.: Hermite interpolation by Pythagorean hodograph curves of degree seven. Math. Comp. 70, 1089–1111 (2001)
Jüttler, B., Sampoli, M.: Hermite interpolation by piecewise polynomial surfaces with rational offsets. Computer Aided Geometric Design 17, 361–385 (2000)
Meek, D.S., Walton, D.J.: Geometric Hermite interpolation with Tschirnhausen cubics. J. Comput. Appl. Math. 81(2), 299–309 (1997)
Brieskorn, E., Knörer, H.: Plane algebraic curves. Birkhaüser, Basel (1986)
Cox, D.A., Little, J., O’Shea, D.: Using algebraic geometry, 2nd edn. Springer, Heidelberg (2005)
Fulton, W.: Algebraic Curves. Benjamin, New York (1969)
Kim, M.S., Elber, G.: Problem reduction to parameter space. In: Proceedings of the 9th IMA Conference on the Mathematics of Surfaces, pp. 82–98. Springer, Heidelberg (2000)
Walker, R.: Algebraic Curves. Princeton University Press, Princeton (1950)
Lávička, M., Bastl, B., Šír, Z.: Reparameterization of curves and surfaces with respect to their convolution. In: Dæhlen, M., Floater, M., Lyche, T., Merrien, J.-L., Mørken, K., Schumaker, L.L. (eds.) MMCS 2008. LNCS, vol. 5862, pp. 285–298. Springer, Heidelberg (2010)
Cohen, H., Frey, G. (eds.): Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman & Hall/CRC (2005)
Hartshorne, R.: Algebraic Geometry. Springer, Heidelberg (1977)
Moon, H.: Minkowski Pythagorean hodographs. Computer Aided Geometric Design 16, 739–753 (1999)
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Vršek, J., Lávička, M. (2012). Algebraic Curves of Low Convolution Degree. In: Boissonnat, JD., et al. Curves and Surfaces. Curves and Surfaces 2010. Lecture Notes in Computer Science, vol 6920. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27413-8_45
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DOI: https://doi.org/10.1007/978-3-642-27413-8_45
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