CINPACT-splines: A Class of \({C^{{\infty }}}\) Curves with Compact Support

  • Adam Runions
  • Faramarz SamavatiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9213)


Recently, Runions and Samavati [7] proposed Partion of Unity Parametrics (PUPs), a generalization of NURBS which replaces B-spline basis functions with arbitrary Weight-Functions (WFs) while preserving affine invariance. A key problem identified by Runions and Samavati was the identification of classes of weight-functions which are well-suited to geometric modeling. In this paper, we propose a class of WF based on bump-functions, which arise in the study of smooth, non-analytic manifolds. These give rise to a class of \(C^{\infty }\) curves with compact-support, which we call CINPACT splines. The WFs are similar in form to B-spline basis functions, and are parameterized by a degree-like shape parameter. We examine the approximating and interpolating curves created using the proposed class of WF. Furthermore, we propose and demonstrate a method to specify the tangents and higher order derivatives of the curve at control points for CINPACT and PUPs curves.


Parametric curves Interpolating curves Approximating curves PUPs B-spline NURBS Bump functions 


  1. 1.
    Bartels, R., Beatty, J., Barsky, B.: An Introduction to Splines for Use in Computer Graphics and Geometric Modeling. Morgan Kaufmann, Los Altos (1987)zbMATHGoogle Scholar
  2. 2.
    Catmull, E., Rom, R.: A class of local interpolating splines. In: Barnhill, R.E., Riesenfeld, R.F. (eds.) Computer Aided Geometric Design, pp. 317–326. Academic Press, New York (1974)Google Scholar
  3. 3.
    Farin, G.: Curves and Surfaces for CAGD. Morgan Kauffmann, San Francisco (2002)Google Scholar
  4. 4.
    Knuth, D.: The METAFONT Book. Addison-Wesley, Boston (1995)Google Scholar
  5. 5.
    Lee, J.: Introduction to Smooth Manifolds. Springer, New York (2000)Google Scholar
  6. 6.
    Parent, R.: Computer Animation. Morgan Kauffmann, San Francisco (2012)Google Scholar
  7. 7.
    Runions, A., Samavati, F.: Partition of unity parametrics: a framework for meta-modeling. Vis. Comput. 27, 495–505 (2011)zbMATHCrossRefGoogle Scholar
  8. 8.
    Sederberg, T., Zheng, J., Bakenov, A., Nasri, A.: T-splines and T-NURCCs. ACM Trans. Graph. 22, 477–484 (2003)zbMATHCrossRefGoogle Scholar
  9. 9.
    Wang, Q., Hua, W., Guiqing, L., Bao, H.: Generalized NURBS curves and surfaces. In: Geometric Modeling and Processing, pp. 365–368 (2004)Google Scholar
  10. 10.
    Zhang, R.-J., Ma, W.: An efficient scheme for curve and surface construction based on a set of interpolatory basis functions. ACM Trans. Graph. 30, 10:1–10:11 (2011)CrossRefGoogle Scholar
  11. 11.
    Zhang, R.-J.: Curve and surface reconstruction based on a set of improved interpolatory basis functions. Comput. Aided Des. 44, 749–756 (2012)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of CalgaryAlbertaCanada

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