Abstract
The construction of offset curves is an important problem encountered often in design processes and in interrogation of geometric models. In this paper the problem of construction of offsets of curves lying on the same parametric surface is addressed. A novel algorithm is introduced, whose main feature is the use of geodesic paths to determine points of the offset. The offset is then approximated in the underlying surface parameter space by B-splines interpolating data points obtained by traveling a known distance along the geodesics departing from corresponding points of the progenitor in a direction perpendicular to the latter. A comprehensive error checking scheme has been devised allowing adaptive improvement of the approximation of the offset. The applicability of the algorithm is demonstrated by number of numerical examples.
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References
Blum H.; Nagel R.N. (1978) Shape description using weighted symmetric axis features. Pattern Recogn. 10(3), 167–180
Patrikalakis, N.M.; Gursoy, H.N. (1988) Skeletons in Shape Feature Recognition for Automated Analysis. Technical Report, MIT, Department of Ocean Engineering, Design Laboratory, No. 88-4; also submitted for publication, June
Hansmann W. (1987) Interactive design and geometric description of smooth transitions between curved surfaces. Computers in Offshore and Arctic Engineering, Sixth International Symposium on Offshore Mechanics and Arctic Engineering, Houston, Texas. ASME, New York, pp. 19–26
Klass, R. (1983) An offset spline approximation for plane cubic splines. Comput. Aided Des. 15(5), September, 297–299
Tiller, W.; Hanson, E.G. (1984) Offsets of two-dimensional profiles. IEEE Comput. Graph. Applic. 4(9), September, 36–46
Hoschek, J. (1988) Spline approximation of offset curves. Comput. Aided Geom. Des. 5(1), June, 33–40
Farouki, R.T. (1986) The approximation of non-degenerate offset surfaces. Comput. Aided Geom. Des. 3, 15–43
Patrikalakis, N.M.; Prakash, P.V. (1987) Free-form plate modeling using offset surfaces. J Offshore Mech Arctic Eng. ASME Transactions 110(3), 287–294; also Computers in Offshore and Arctic Engineering, Sixth International Symposium on Offshore Mechanics and Arctic Engineering, Houston, Texas. ASME, New York, March, pp. 37–44
De Boor, C. (1978) A Practical Guide to Splines. New York, Springer-Verlag
Prakash, P.V. (1988) Computation of Surface-Surface Intersections for Geometric Modeling, PhD dissertation. Cambridge, MA: Massachusetts Institute of Technology, May
Prakash, P.V., Patrikalakis, N.M. (1988) Algebraic and rational polynomial parametric surface intersections. Submitted for publication, July
do Carmo P.M. (1976) Differential Geometry of Curves and Surfaces. Englewood Cliffs, NJ: Prentice-Hall
Munchmeyer F.C.; Haw, R. (1982) Applications of differential geometry to ship design. Computer Applications in the Automation of Shipyard Operation and Ship Design IV: Amsterdam; North Holland, pp. 183–188
Beck, J.M.; Farouki, R.T., Hinds, J.K. (1986) Surface analysis methods. IEEE Comput. Graph. Applic. 6(12), December, pp. 18–36
Hildebrand, F.B. (1976) Advanced Calculus for Applications. Englewood Cliffs, NJ. Prentice-Hall
Hall, G.; Watt, J.M. (eds.). (1976) Modern Numerical Methods for Ordinary Differential Equations. Oxford, England: Clarendon Press
NAG (1985) Numerical Algorithms Group FORTRAN Library. Oxford, England: NAG
Gordon, W.J.; Riesenfeld, R.F. (1974) Bernstein-Bezier methods for the computer-aided design of free-form curves and surfaces. J. ACM 21(2), 293–310
Hartley, P.J.; Judd, C.J. (1980) Parametrization and shape of B-spline curves for CAD. Comput. Aided Des. 12(5), September, 235–238
Gill, P.E.; Murray, W. (1973) Safeguard Steplength Algorithms for Optimization Using Descent Methods. National Physical Laboratory Report NAC 37
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Patrikalakis, N.M., Bardis, L. Offsets of curves on rational B-spline surfaces. Engineering with Computers 5, 39–46 (1989). https://doi.org/10.1007/BF01201996
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DOI: https://doi.org/10.1007/BF01201996