References

The algorithms and mathematical objects in PaNTr come from a well-established literature. This page collects the works cited throughout the documentation, followed by encyclopedic background reading for the core concepts.

Bibliography

[1]

R. Kraft. Adaptive and linearly independent multilevel B-splines. In A. Le Méhauté, C. Rabut, and L. L. Schumaker, editors, Surface Fitting and Multiresolution Methods, pages 209–218. Vanderbilt University Press, Nashville, 1997.

[2]

A.-V. Vuong, C. Giannelli, B. Jüttler, and B. Simeon. A hierarchical approach to adaptive local refinement in isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 200(49–52):3554–3567, 2011. doi:10.1016/j.cma.2011.09.004.

[3]

Carlotta Giannelli, Bert Jüttler, and Hendrik Speleers. THB-splines: the truncated basis for hierarchical splines. Computer Aided Geometric Design, 29(7):485–498, 2012. doi:10.1016/j.cagd.2012.03.025.

[4]

T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194(39–41):4135–4195, 2005. doi:10.1016/j.cma.2004.10.008.

[5]

Les Piegl and Wayne Tiller. The NURBS Book. Springer, Berlin, Heidelberg, 2 edition, 1997. ISBN 978-3-540-61545-3. doi:10.1007/978-3-642-59223-2.

[6]

Carl de Boor. A Practical Guide to Splines. Volume 27 of Applied Mathematical Sciences. Springer, New York, revised edition, 2001. ISBN 978-0-387-95366-3.

[7]

Michael J. Borden, Michael A. Scott, John A. Evans, and Thomas J. R. Hughes. Isogeometric finite element data structures based on Bézier extraction of NURBS. International Journal for Numerical Methods in Engineering, 87(1–5):15–47, 2011. doi:10.1002/nme.2968.

[8]

Michael A. Scott, Michael J. Borden, Clemens V. Verhoosel, Thomas W. Sederberg, and Thomas J. R. Hughes. Isogeometric finite element data structures based on Bézier extraction of T-splines. International Journal for Numerical Methods in Engineering, 88(2):126–156, 2011. doi:10.1002/nme.3167.

[9]

Steven A. Coons. Surfaces for computer-aided design of space forms. Technical Report MAC-TR-41, Massachusetts Institute of Technology, Project MAC, 1967. URL: https://publications.csail.mit.edu/lcs/pubs/pdf/MIT-LCS-TR-041.pdf.

[10]

Gerald Farin. Curves and Surfaces for CAGD: A Practical Guide. Morgan Kaufmann, San Francisco, 5 edition, 2002. ISBN 978-1-55860-737-8.

[11]

Davide D'Angella, Stefan Kollmannsberger, Ernst Rank, and Alessandro Reali. Multi-level Bézier extraction for hierarchical local refinement of isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 328:147–174, 2018. doi:10.1016/j.cma.2017.08.017.

[12]

Eduardo M. Garau and Rafael Vázquez. Algorithms for the implementation of adaptive isogeometric methods using hierarchical B-splines. Applied Numerical Mathematics, 123:58–87, 2018. doi:10.1016/j.apnum.2017.08.006.

[13]

Hendrik Speleers and Carla Manni. Effortless quasi-interpolation in hierarchical spaces. Numerische Mathematik, 132(1):155–184, 2016. doi:10.1007/s00211-015-0711-z.

[14]

Cem Yuksel. High-performance polynomial root finding for graphics. Proceedings of the ACM on Computer Graphics and Interactive Techniques, 5(3):1–15, 2022. doi:10.1145/3543865.

[15]

T. W. Sederberg and T. Nishita. Curve intersection using Bézier clipping. Computer-Aided Design, 22(9):538–549, 1990. doi:10.1016/0010-4485(90)90039-F.

[16]

Gene H. Golub and John H. Welsch. Calculation of Gauss quadrature rules. Mathematics of Computation, 23(106):221–230, 1969. doi:10.1090/S0025-5718-69-99647-1.

[17]

George Karypis and Vipin Kumar. A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM Journal on Scientific Computing, 20(1):359–392, 1998. doi:10.1137/S1064827595287997.

Background reading

Encyclopedic introductions to the underlying mathematics, useful as a gentle on-ramp before the primary literature above:

• B-spline — basis functions, knot vectors, and the Cox–de Boor recurrence.

• Non-uniform rational B-spline (NURBS) — rational spline geometry.

• Bézier curve and Bernstein polynomial — the Bézier toolkit and its polynomial basis.

• De Boor’s algorithm — stable evaluation of B-spline curves.

• De Casteljau’s algorithm — stable evaluation and subdivision of Bézier curves.

• Isogeometric analysis — the CAD/analysis bridge PaNTr targets.

• Gaussian quadrature — the Gauss–Legendre rules behind pantr.quad.

• Coons patch — transfinite interpolation used by the constructive-geometry surfaces in pantr.cad.