## Properties of generalized offset curves and surfaces.(English)Zbl 1439.68025

Summary: This paper proposes a definition of generalized offsets for curves and surfaces, which have the variable offset distance and direction, by using the local coordinate system. Based on this definition, some analytic properties and theorems of generalized offsets are put forward. The regularity and the topological property of generalized offsets are simply given by representing the generalized offset as the standard offset. Some examples are provided as well to show the applications of generalized offsets. The conclusions in this paper can be taken as the foundation for further study on extending the standard offset.

### MSC:

 68U07 Computer science aspects of computer-aided design 53A04 Curves in Euclidean and related spaces 65D17 Computer-aided design (modeling of curves and surfaces)
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### References:

 [1] Shen, H.; Fu, J.; Chen, Z.; Fan, Y., Generation of o set surface for tool path in nc machining through level set methods, International Journal of Advanced Manufacturing Technology, 46, 9-12, 1043-1047 (2009) [2] Krasauskas, R.; Peternell, M., Rational offset surfaces and their modeling applications, Nonlinear Computational Geometry, 151, 109-135 (2010) · Zbl 1190.53005 [3] Brechner, E., General tool offset curves and surfaces, Geometry Processing For Design and Manufacturing (1992), Philadelphia, Pa, USA: SIAM, Philadelphia, Pa, USA · Zbl 0767.53006 [4] Pottmann, H., General offset surfaces, Neural Parallel and Scientific Computations, 5, 55-80 (1997) [5] Arrondo, E.; Sendra, J.; Sendra, J. R., Genus formula for generalized offset curves, Journal of Pure and Applied Algebra, 136, 3, 199-209 (1999) · Zbl 0944.14014 [6] Lin, Q.; Rokne, J. G., Variable-radius offset curves and surfaces, Mathematical and Computer Modelling, 26, 7, 97-108 (1997) · Zbl 0890.68127 [7] Sendra, J. R.; Sendra, J., Algebraic analysis of offsets to hypersurfaces, Mathematische Zeitschrift, 234, 4, 697-719 (2000) · Zbl 0996.14027 [8] Georgiev, G. H., Rational generalized offsets of rational surfaces, Mathematical Problems in Engineering, 2012 (2012) · Zbl 1264.53012 [9] Barnhill, R. E., General tool offset curves and surfaces, Geometry Processing For Design and Manufacturing (1992), Philadelphia, Pa, USA: SIAM, Philadelphia, Pa, USA [10] Anton, F.; Emiris, I.; Mourrain, B.; Teillaud, M., The offset to an algebraic curve and an application to conics, Proceedings of the International Conference on Computational Science and Its Applications (ICCSA ’05) [11] Farouki, R. T.; Neff, C. A., Algebraic properties of plane offset curves, Computer Aided Geometric Design, 7, 1-4, 101-127 (1990) · Zbl 0724.65008 [12] Farouki, R. T.; Neff, C. A., Analytic properties of plane offset curves, Computer Aided Geometric Design, 7, 1-4, 83-99 (1990) · Zbl 0718.53003 [13] Klok, F., Two moving coordinate frames for sweeping along a 3D trajectory, Computer Aided Geometric Design, 3, 3, 217-229 (1986) · Zbl 0631.65145 [14] Kreyszig, E., Differential Geometry (1959), University of Toronto Press · Zbl 0088.13901 [15] Encheva, R. P.; Georgiev, G. H., Similar frenet curves, Results in Mathematics, 55, 3, 359-372 (2009) · Zbl 1180.53004 [16] Bancho, T. F.; Gaffney, T.; McCrory, C., Cusps of Gauss Mappings Pitman. Cusps of Gauss Mappings Pitman, Research Notes in Mathematics (1982), London, UK: Pitman, London, UK [17] Nutbourne, A. W.; Martin, R. R., Differential Geometry Applied To Curve and Surface Design, 1: Foundations (1988), New York, NY, USA: West Sussex, Chichester, UK; E. Horwood, New York, NY, USA [18] Piegl, L. A.; Tiller, W., Computing offsets of NURBS curves and surfaces, CAD Computer Aided Design, 31, 2, 147-156 (1999) · Zbl 1053.68749
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