Abstract
In this chapter the implicit surface method based on RBF is explained and demonstrated. The classical method that consists of the definition of a scalar valued function in the 3D space that is null at on-surface points and positive (negative) at off-surface points that are positioned outside (inside) the surface is firstly introduced. The projection onto the surface is achieved by computing the gradient of the implicit function. The second one is a novel method that consists of the creation of a projection field (vector valued RBF) defined on the surface and at off-surface points that is zero at on-surface points whereas it is equal to the vector moving to the surface at off-surface points. The effectiveness of both methods is demonstrated with practical examples.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Carr JC, Beatson R, Cherri J, Mitchell T, Fright W, McCallum B (2001) Reconstruction and representation of 3D objects with radial basis functions. In: Proceedings of the 28th annual conference on Computer graphics and interactive techniques, Los Angeles, CA, p 67–76
Carr JC, Beatson RK, McCallum BC, Fright WR, McLennan TJ, Mitchell TJ (2003) Smooth surface reconstruction from noisy range data. In: First international conference on computer graphics and interactive techniques, p 119
Hartmann EA (1998) Marching method for the triangulation of surfaces. Vis Comput 14(3):95–108
Jin X, Sun H, Peng Q (2003) Subdivision interpolating implicit surfaces. Comput Graph 27:763–772. https://doi.org/10.1016/S0097-8493(03)00149-3
Karkanis T, Stewart J (2011) Curvature dependent triangulation of implicit surfaces. IEEE Comput Graphics Appl 21(2):60–9
Savchenko VV, Pasko AA, Okunev OG, Kunni TL (1995) Function representation of solids reconstructed from scattered surface points and contours. Comput Graph Forum 14(4):181–188
Tobor I, Reuter P, Schlick C (2006) Reconstructing multi-scale variational partition of unity implicit surfaces with attributes. Graph Models 68(1):25–41
Turk G, O’Brien JF (2002) Modeling with implicit surfaces that interpolate. ACM Trans Graph 21(4):855–873
Turk G, O’Brien JF (1999) Shape transformation using variational implicit functions. In: SIGGRAPH ‘99 Proceedings of the 26th annual conference on Computer graphics and interactive techniques, ACM Press/Addison-Wesley Publishing Co. New York, NY, USA, pp 335–342, ISBN:0-201-48560-5, https://doi.org/10.1145/311535.311580
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Biancolini, M.E. (2017). RBF Implicit Representation of Geometrical Entities. In: Fast Radial Basis Functions for Engineering Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-75011-8_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-75011-8_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-75009-5
Online ISBN: 978-3-319-75011-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)