Geometry Continuity and epsilon-geometry continuity

In differential geometry, Riemann (1826 1866), continuities play a very important kernel role. G-Continuity could be defined as the smoothness properties of a curve or a surface that are more than its order of differentiability. To day, scientists try to find a kind of continuities, which are the intuitive intrinsic properties of curves and surfaces, and the orders of the continuities are independent of the parameterization. In order to make through the bottleneck, we have developed the theories of epsilon-geometry continuities to accommodate the representation and the rounding errors of float-point arithmetic, and designed new geometric modelling operators under the constraints of epsilon-geometry continuities. Since representation and rounding errors of real numbers by floating-point numbers are ubiquitous, we have developed the epsilon-geometry continuity theories and algorithms with error tolerances to match both the features of the floating-point arithmetic and the requirements of the engineering design. Thus, we proposed the theories and algorithms to manipulate the transition between sharp and rounded features. We have provided theories and algorithms for the $ϵ-G^2$ B-spline surfaces interpolating the specified four groups of boundary derivative curves in the B-spline form. We bound all kinds of the discontinuities by the invariant tolerances, and classify the compatibility problems. Then, we proposed the algorithms for continuity-preserving re-parameterization, knot-insertion and local control-point tuning to solve the compatibility problems, and achieve the $ϵ-G^2$ continuities.

Geometry beautification

Although geometric uncertainties are often related to robustness and tolerance, there are a number of extra issues well worth deeper investigations. Geometric arrangements are full of special cases. The most notable ones are: cases of touch, overlapping, containment, etc.; cases of parallelism, perpendicularity, coincidence, etc.; axes of symmetrical data, data clustering, dense or sparse data, etc.; cases of degeneracy, discontinuity, inconsistencies, etc.; problems with cracks, excess material, lack of detail, etc. In just about any code that deals with geometry, the number of special cases is significantly larger than the general ones. Data explosion is the result of careless selection of the methods, e.g. parameter space-based sampling, and improper implementation, e.g. recursive algorithms. Some of the relevant issues are: sampling: over sampling, sampling in incorrect places, etc., procedural definitions, e.g. lofting a large set of curves or merging surfaces may result in an explosion of control points. Our contributions in the last years proposed elegant solutions to deal with these problems.

Shape generation

As an alternative solution to NURBS, we proposed a canonical form of the curved-knot B-spline surface called the regular curved-knot B-spline. The curved knot vector of one parametric coordinate is defined by a group of blending functions that depend on the other coordinate. So the knot vectors of two opposite boundaries can be different. That property makes it possible to represent a smooth transition between two B-spline boundaries with different knots and continuities, since knots determine the continuity. The regular form guarantees the simplicity in storage, evaluation and construction algorithms. It therefore provides the curved-knot B-spline with practicability in geometric modeling systems. The applications of surface bridging and transition illustrate its suitability for blending sharp and rounded features. Compared with B-splines and T-splines, it not only increases the surface quality, but also reduces the complexity of the surface construction.

We mention here papers published in the best international reviews in CAD i.e. Computer-Aided Design (Elsevier) and Computer Aided Geometry Design (Elsevier) in the last four years.

[1] Kan-Le Shi, Jun-Hai Yong, Jia-Guang Sun, Jean-Claude Paul. Epsilon-G2 B-spline surface interpolation. Computer Aided Geometric Design 2011. 28(6): 368-381. (2010 SCI IF: 0.859; SCI (IDS): 822SS; EI Compendex: 20113514270430)

[2] Kan-Le Shi, Sen Zhang, Hui Zhang, Jun-Hai Yong, Jia-Guang Sun, Jean-Claude Paul. G2 B-spline interpolation to a closed mesh. Computer-Aided Design 2011. 43(2): 145-160. (2010 SCI IF: 1.542; SCI: 718GT; EI Compendex: 20110213580729)

[3] Kan-Le Shi, Jun-Hai Yong, Jia-Guang Sun, Jean-Claude Paul. Gn blending multiple surfaces in polar coordinates. Computer-Aided Design 2010. 42(6): 479-494. (2010 SCI IF: 1.542; SCI: 602YI; EI Compendex: 20101712878906; Inspec: 11282424)

[4] Hai-Chuan Song, Jun-Hai Yong, Yi-Jun Yang, Xiao-Ming Liu. Algorithm for orthogonal projection of parametric curves onto B-spline surfaces. Computer-Aided Design 2011. 43(4): 381-393. (2010 SCI IF: 1.542; SCI (IDS): 743KT; EI Compendex: 20111013734335)

[5] Bernard Anselmetti, Robin Chavanne, Jian-Xin Yang, and Nabil Anwer, Quick GPS: A new CAT system for single-part tolerancing, Computer Aided design, Vol. 42, Issue 9, sept. 2010, pages 768-780.

[6] Yan-Bing Bai, Jun-Hai Yong, Chang-Yuan Liu, Xiao-Ming Liu, Yu Meng. Polyline approach for approximating Hausdorff distance between planar free-form curves. Computer-Aided Design 2011. 43(6): 687-698. (2010 SCI IF: 1.542; SCI: 773US; EI Compendex: 20111913976224)

[7] Xiao-Diao Chen, Weiyin Ma, Gang Xu, Jean-Claude Paul. Computing the Hausdorff distance between two B-spline curves. Computer-Aided Design 2010. 42(12): 1197-1206. (2010 SCI IF: 1.542; SCI IDS: 677BA; Inspec: 11589392; EI Compendex: 20104213304195)

[8] Chao Wang, Yu Shen Liu, Min Liu, Jun-Hai Yong, Jean-Claude Paul. Robust Shape Normalization of 3D articulated volumetric models. Computer-Aided Design 2012. 44: 1253-1258.

[9] Wen-Ke Wang, Hui Zhang, Xiao-Ming Liu, Jean-Claude Paul. Conditions for coincidence of two cubic Bézier curves. Journal of Computational and Applied Mathematics 2011. 235(17): 5198–5202. (2010 SCI IF: 1.030; SCI: 801IR; EI Compendex: 20112814135958; Inspec: 12599381)

[10] WuJun Che, XiaoPeng Zhang, Yi-Kuan Zhang, Jean-Claud Paul, Ridge extraction of a smooth 2-manifold surface based on vector field, Computer Aided Geometric Design, Vol. 28, Issue 4, May 2011.

[11] Yamei Wen, Hui Zhang, Jiaguang Sun, Jean-Claude Paul, A new method for identifying and validating features from 2D sectional views, Computer-Aided Design 43 (2011) 677–686.

[12] Xiao-Diao Chen, Weiyin Ma, Jean-Claude Paul. Cubic B-spline curve approximation by curve unclamping. Computer-Aided Design 2010. 42(6): 523-534. (2010 SCI IF: 1.542; SCI IDS: 602YI; Inspec: 11282428; EI Compendex: 20101712878910)

[13] Dong-Ming Yan, Wenping Wang, Bruno Lévy, Yang Liu. Efficient computation of clipped Voronoi diagram for mesh generation. Computer-Aided Design. (2010 SCI IF: 1.542; EI Compendex: IP51671352)

[14] Xiao-Diao Chen, Weiyin Ma: Geometric point interpolation method in R3 space with tangent directional constraint. Computer-Aided Design 44 (2012) 1217–1228

[15] Dong-Ming Yan, Wenping Wang, Yang Liu, Zhouwang Yang. Variational mesh segmentation via quadric surface fitting. Computer-Aided Design 2012. 44(11): 1072-1082. (2010 SCI IF: 1.542)

[16] Kai-Mo Hu, Bin Wang, Jun-Hai Yong, Jean-Claude Paul. Relaxed lightweight assembly retrieval using vector space model. Computer-Aided Design 2013. 45(3): 739-750.

[17] Yong Liu, Hai-Chuan Song, Jun-Hai Yong. Calculating Jacobian coefficients of primitive constraints with respect to Euler parameters. International Journal of Advanced Manufacturing Technology. Accepted.

[18] Yong Liu, Hai-Chuan Song, Jun-Hai Yong. Solving under-constrained assembly problems incrementally using a kinematic method. Computer-Aided Design 2013 to appear.