Topics for 60 Minutes' Presentation:

The lectures will commence using the following schedule (50 minutes lectures):

Topics

• Kinematics of linkages http://en.wikipedia.org/wiki/Linkage_(mechanical)). There is a direct link between kinematics of linkages and algebraic constraints. A 4-bars planar linkage (http://en.wikipedia.org/wiki/Four-bar_linkage) can be formulated using 3 algebraic constraints in 4 unknown and solved for (using spline techniques for example). In this talk, you will introduce the topic of kinematics of linkages and show how one can map the motion into algebraic constraints. The major research and development abilities will be discussed and applications will be presented. Other relevant interesting links are:
  • http://en.wikipedia.org/wiki/Chebyshev_linkage
  • http://en.wikipedia.org/wiki/Cognate_linkage
  • http://en.wikipedia.org/wiki/Chebyshev's_Lambda_Mechanism
  • http://en.wikipedia.org/wiki/Klann_linkage
  • http://en.wikipedia.org/wiki/Jansen%27s_linkage
  This topic could be fully understood immediately. One recent relevant reference to start from is Boris van Sosin and Gershon Elber. ``Solving Piecewise Polynomial Constraint Systems with Decomposition and a Subdivision-Based Solver.'' Computer Aided Design, Vol 78, pp 36-47, September 2016.

• Point inclusion inside curves and surfaces. Reference:
  • "A New Approach to Point Membership Classification in B-rep Solids", by F. Klein, Proceedings of the 13th IMA International Conference on Mathematics of Surfaces XIII, 2009.
  This topic could be fully understood once we study surfaces.

• Tensor Product Volumes and Volumetric representations. This topic could be handled once tensor product surfaces are discussed in class. References:
  • "Geometric Modeling with Splines, An Introduction" by Cohen, Riesenfeld, Elber: Chapter 21.
  • ``A B-spline based Framework for Volumetric Object Modeling.'' by Fady Massarwi and Gershon Elber. Computer Aided Design, Vol 78, pp 36-47, September 2016.
  • ``Fabricating Functionally Graded Material Objects Using Trimmed Trivariate Volumetric Representations.`` by Ben Ezair and Gershon Elber. Proceedings of SMI'2017 Fabrication and Sculpting Event (FASE), Berkeley, CA, USA, June 2017.

• Blending surfaces. This topic would require reading several papers as well as the appropriate section in Hoschek's book. This topic could be fully understood once we start to discuss surfaces. Papers:
  • "A Blending Model for Parametrically Defined Geometric Objects", by A. P. Bien and F. Cheng. Symposium on Solid Modeling Foundation and CAD/CAM Applications, 1991, pp. 339-347.
  • "Constant Radius Blending in Surface Modeling" by B. K. Choi and S. Y. Ju. Computer Aided Design, Vol. 21, No. 4, pp 213-220, May 1989.
  • "Fillet and Surface Intersections Defined by Rolling Balls" by R. K. Klass and B. Kuhn. Computer Aided Geometric Design 9, pp 185-193, 1992.
  • "Parametric Blending using Fanout Surfaces", by P. Koparkar. Symposium on Solid Modeling Foundation and CAD/CAM Applications, 1991, pp. 317-327.
  • "Fundamentals of Computer Aided Geometric Design", by J Hoschek and D Lasser, Chapter 14 "Blending Methods", pp 572-600.

• Introduction to iso-geometric analysis (IGA) - analysis in spline spaces. A good start will be "Isogeometric Analysis: Toward Integration of CAD and FEA". 1st Edition, by J. Austin Cottrell, Thomas J.R. Hughes, Yuri Bazilevs. John Wiley & Sons, 2009. Knowledge in FEA is reuired.

• Blossoms. A more general view of the recursive algorithms of Bezier. This topic could be fully understood once we study Bezier curves. References:
  • "Blossoms are Polar Forms", by Lyle Ramshaw. Computer Aided Geometric Design Vol 6, pp 323-358, 1989.
  • "Curves and Surfaces for Computer Aided Geometric Design", by G Farin. Third edition.
  • "NURB curves and surfaces from Projective Geometry to Practical Use", by G. Farin.
  • "Fundamentals of Computer Aided Geometric Design", by J Hoschek and D Lasser.
  The treatment of one more advanced topic such as composition or refinement would be necessary.
  This topic could be fully understood once we study Bezier curves.

• Bezier general subdivision (Chapter 16 in text book). Consider subdivision of arbitrary order Bezier curves (surfaces).
  This topic could be fully understood once we study Bezier curves. Some will have to wait to tensor products.

• Analysis of the topology of implicit curves. References:
  • "Efficient and exact manipulation of algebraic points and curves", by Keyser, J., Culver, T., Manocha, D., and Krishnan, S. Computer-Aided Design 32, 11 (15 September), pp 649--662, 2000.
  • "Guaranteed consistency of surface intersections and trimmed surfaces using a coupled topology resolution and domain decomposition scheme", by J.~Hass, R.~T. Farouki, C.~Y. Han, X.~Song, and T.~W. Sederberg. Computational Mathematics, 2005.
    See http://mae.ucdavis.edu/~farouki/index.html
  This topic could be fully understood once we study tensor product surfaces.

• Solving multivariate polynomial/rational equations using splines. References:
  • "Computation of the solutions of nonlinear polynomial systems", by E. C. Sherbrooke and N. M. patrikalakis. Computer Aided Geometric Design, Vol 10, No 5, pp 379-405, 1993
  • "Geometric Constraint Solver using Multivariate Rational Spline Functions", by G. Elber and M. S. Kim. The Sixth ACM/IEEE Symposium on Solid Modeling and Applications, Ann Arbor, Michigan, pp 1-10, June 2001.
  • "Subdivision methods for solving polynomial equations", by B. Mourrain and J. P. Pavone, Technical Report 5658, Inria, Sophia-Antipolis, 2005.
  This topic could be fully understood once we study tensor product surfaces.