Geometric conditions for injectivity of 3D Bézier volumes

Xuanyi Zhao; Jinggai Li; Shiqi He; Chungang Zhu; Xuanyi Zhao; Jinggai Li; Shiqi He; Chungang Zhu

doi:10.3934/math.2021694

AIMS Mathematics

2021, Volume 6, Issue 11: 11974-11988. doi: 10.3934/math.2021694

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Research article

Geometric conditions for injectivity of 3D Bézier volumes

1.
School of Science, Dalian Maritime University, Dalian 116026, China
2.
College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China
3.
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Received: 18 April 2021 Accepted: 09 August 2021 Published: 17 August 2021
MSC : 65D05, 65D17

The one-to-one property of injectivity is a crucial concept in computer-aided design, geometry, and graphics. The injectivity of curves (or surfaces or volumes) means that there is no self-intersection in the curves (or surfaces or volumes) and their images or deformation models. Bézier volumes are a special class of Bézier polytope in which the lattice polytope equals $ \Box_{m, n, l}, (m, n, l\in Z) $. Piecewise 3D Bézier volumes have a wide range of applications in deformation models, such as for face mesh deformation. The injectivity of 3D Bézier volumes means that there is no self-intersection. In this paper, we consider the injectivity conditions of 3D Bézier volumes from a geometric point of view. We prove that a 3D Bézier volume is injective for any positive weight if and only if its control points set is compatible. An algorithm for checking the injectivity of 3D Bézier volumes is proposed, and several explicit examples are presented.
- 3D Bézier volumes,
- injectivity,
- toric surfaces,
- deformations,
- self-intersections
Citation: Xuanyi Zhao, Jinggai Li, Shiqi He, Chungang Zhu. Geometric conditions for injectivity of 3D Bézier volumes[J]. AIMS Mathematics, 2021, 6(11): 11974-11988. doi: 10.3934/math.2021694

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Abstract

The one-to-one property of injectivity is a crucial concept in computer-aided design, geometry, and graphics. The injectivity of curves (or surfaces or volumes) means that there is no self-intersection in the curves (or surfaces or volumes) and their images or deformation models. Bézier volumes are a special class of Bézier polytope in which the lattice polytope equals $ \Box_{m, n, l}, (m, n, l\in Z) $. Piecewise 3D Bézier volumes have a wide range of applications in deformation models, such as for face mesh deformation. The injectivity of 3D Bézier volumes means that there is no self-intersection. In this paper, we consider the injectivity conditions of 3D Bézier volumes from a geometric point of view. We prove that a 3D Bézier volume is injective for any positive weight if and only if its control points set is compatible. An algorithm for checking the injectivity of 3D Bézier volumes is proposed, and several explicit examples are presented.

References

[1]	A. Yilmaz, O. Javed, M. Shah, Object tracking: a survey, ACM Comput. Surv., 38 (2006), 1-45. doi: 10.1145/1132952.1132953
[2]	E. Trucco, K. Plakas, Video tracking: a concise survey, IEEE J. Ocean. Eng., 31 (2006), 520-529. doi: 10.1109/JOE.2004.839933
[3]	H. Yang, L. Shao, F. Zheng, L. Wang, Z. Song, Recent advances and trends in visual tracking: a review, Neurocomputing, 74 (2011), 3823-3831. doi: 10.1016/j.neucom.2011.07.024
[4]	H. Tao, T. S. Huang, Bézier volume deformation model for facial animation and video tracking, In: Modelling & motion capture techniques for virtual environments, International Workshop, CAPTECH'98 Geneva, Switzerland, November 26-27, 1998 Proceedings, Berlin, Heidelberg: Springer, 1998,242-253.
[5]	V. Savchenko, A. Pasko, Shape modeling, Encycl. Comput. Sci. Technol., 45 (2002), 311-346.
[6]	R. Krasauskas, Toric surface patches, Adv. Comput. Math., 17 (2002), 89-113. doi: 10.1023/A:1015289823859
[7]	S. Müller, E. Feliu, G. Regensburger, C. Conradi, A. Shiu, A. Dickenstein, Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry, Found. Comput. Math., 16 (2016), 69-97. doi: 10.1007/s10208-014-9239-3
[8]	C. M. Hoffmann, Geometric and solid modeling, San Mateo, California: Morgan Kaufmann Publishers, Inc., 1989.
[9]	G. Xu, B. Mourrain, R. Duvigneau, A. Galligo, Analysis-suitable volume parameterization of multi-block computational domain in isogeometric applications, Comput.-Aided Design, 45 (2013), 395-404. doi: 10.1016/j.cad.2012.10.022
[10]	A. Galligo, J. P. Pavone, Self-intersections of a Bézier bicubic surface, In: Proceedings of the 2005 international symposium on Symbolic and algebraic computation, 2005,148-155.
[11]	L. E. Andersson, T. J. Peters, N. F. Stewart, Self-intersection of composite curves and surfaces, Comput.-Aided Geom. Design, 15 (1998), 507-527. doi: 10.1016/S0167-8396(98)00005-3
[12]	N. M. Patrikalakis, P. V. Prakash, Surface intersections for geometric modeling, J. Mech. Des., 112 (1990), 100-107. doi: 10.1115/1.2912565
[13]	T. N. T. Goodman, K. Unsworth, Injective bivariate maps, Ann. Numer. Math., 3 (1996), 91-104.
[14]	C. C. Ho, E. Cohen, Surface self-intersection, In: Mathematical methods for curves and surfaces, 2001,183-194.
[15]	M. Hosaka, Modeling of curves and surfaces in CAD/CAM, Springer-Verlag, 1992.
[16]	S. Krishnan, D. Manocha, An efficient surface intersection algorithm based on lower dimensional formulation, ACM T. Graphic, 16 (1997), 74-106. doi: 10.1145/237748.237751
[17]	T. W. Sederberg, R. J. Meyers, Loop detection in surface patch intersections, Comput.-Aided Geom. Design, 5 (1988), 161-171. doi: 10.1016/0167-8396(88)90029-5
[18]	M. N. Patrikalakis, T. Maekawa, H. K. Ko, H. Mukundan, Surface to surface intersections, Comput.-Aided Design Appl., 1 (2004), 449-457. doi: 10.1080/16864360.2004.10738287
[19]	G. Craciun, M. Feinberg, Multiple equilibria in complex chemical reaction networks: Ⅰ. The injectivity property, SIAM J. Appl. Math., 65 (2005), 1526-1546. doi: 10.1137/S0036139904440278
[20]	G. Craciun, L. D. Garcıa-Puente, F. Sottile, Some geometrical aspects of control points for toric patches, In: Mathematical methods for curves and surfaces, Springer, 2010,111-135.
[21]	F. Sottile, C. G. Zhu, Injectivity of 2D toric Bézier patches, In: International conference on computer-aided design and computer graphics, 2011,235-238.
[22]	C. G. Zhu, X. Y. Zhao, Self-intersections of rational Bézier curves, Graph. Models, 76 (2014), 312-320. doi: 10.1016/j.gmod.2014.04.001
[23]	X. Y. Zhao, C. G. Zhu, Injectivity conditions of rational Bézier surfaces, Comput. Graph., 51 (2015), 17-25. doi: 10.1016/j.cag.2015.05.017
[24]	Y. Y. Yu, Y. Ji, J. G. Li, C. G. Zhu, Conditions for injectivity of toric volumes with arbitrary positive weights, Comput. Graph., 97 (2021), 88-98. doi: 10.1016/j.cag.2021.04.026
[25]	D. Lasser, Rational tensor product Bézier volumes, Comput. Math. Appl., 29 (1994), 95-108.
[26]	D. Holliday, G. Farin, A geometric interpretation of the diagonal of a tensor-product Bézier volume, Comput.-Aided Geom. Design, 16 (1999), 837-840. doi: 10.1016/S0167-8396(99)00004-7
[27]	H. Tao, T. S. Huang, A piecewise Bézier volume deformation model and its applications in facial motion capture, Int. J. Radiat. Oncol. Biol. Phys., 39 (2009), 39-49.
[28]	D. C. Lay, Linear algebra and its applications, 4 Eds., Addison-Wesley Longman. Inc, 2012.
[29]	Y. Y. Yu, Y. Ji, C. G. Zhu, An improved algorithm for checking the injectivity of 2D toric surface patches, Comput. Math. Appl., 79 (2020), 2973-2986. doi: 10.1016/j.camwa.2020.01.001

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