Adjusting the energy of ball curves by modifying movable control balls. (English) Zbl 1476.65014
Summary: Ball curve plays a crucial role in modeling tubular shapes with variable thickness. In this paper, energy functionals for curve design are generalized to Ball curves and the variational design of Ball curves is investigated. Given a Ball curve with some of its control balls movable and having variable radiuses, we propose a method to determine the positions and radiuses of the movable control balls which minimize the energy functional. Based on this, we provide two efficient design tools: (i) to achieve \(C^k\) continuity across linked Ball curves at their joint, while decreasing their energy as low as possible, (ii) for blending disjoint Ball curves subject to \(C^k\) continuity constraints and energy minimization. The feasibility of the method is verified by several examples. By adjusting the weighted coefficients, different energy functionals are defined and thus Ball curves with different shapes and thickness can be obtained.
MSC:
| 65D07 | Numerical computation using splines |
| 65D10 | Numerical smoothing, curve fitting |
| 65D17 | Computer-aided design (modeling of curves and surfaces) |
| 65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |
| 68U05 | Computer graphics; computational geometry (digital and algorithmic aspects) |
| 68U07 | Computer science aspects of computer-aided design |
References:
| [1] | Farin, G., Curves and surfaces for computer aided geometric design (1990), Salt Lake City: Academic Press, Salt Lake City · Zbl 0702.68004 |
| [2] | Hagen H (1992) Variational principles in curve and surface design. In: IMA conference on the mathematics of surfaces, Edinburgh, pp 169-190 · Zbl 0805.65023 |
| [3] | Hagen, H.; Bonneau, G., Variational design of smooth rational Bézier curves, Comput Aided Geom Des, 8, 393-400 (1991) · Zbl 0753.65014 · doi:10.1016/0167-8396(91)90012-Z |
| [4] | Jou, E.; Han, W., Minimal-energy splines: I. Plane curves with angle constraints, Math Methods Appl Sci, 13, 351-371 (1990) · Zbl 0722.49033 · doi:10.1002/mma.1670130407 |
| [5] | Jou E, Han W (1990) Minimal energy splines with various end constraints. In: Curve and surface modeling. SIAM Frontiers in Applied Mathematics Series, Philadelphia, USA · Zbl 0789.41006 |
| [6] | Juhász, I.; Róth, Á., Adjusting the energies of curves defined by control points, Comput Aided Des, 107, 77-88 (2019) · doi:10.1016/j.cad.2018.09.003 |
| [7] | Kim, K., Minimum distance between a canal surface and a simple surface, Comput Aided Des, 35, 10, 871-879 (2003) · Zbl 1206.68325 · doi:10.1016/S0010-4485(02)00123-9 |
| [8] | Leng C, Wu Z, Zhou M (2011) Reconstruction of tubular object with ball b-spline curve. In: Proceedings of computer graphics international |
| [9] | Liu, X.; Wang, X.; Wu, Z.; Zhang, D.; Liu, X., Extending Ball B-spline by B-spline, Comput Aided Geom Des, 82, 12 (2020) · Zbl 1450.65008 · doi:10.14733/cadaps.2021.S2.12-24 |
| [10] | Seah H, Wu Z (2005) Ball b-spline based geometric models in distributed virtual environments. In: Proceedings of workshop towards semantic virtual environments. Villars, Switzerland, pp 1-8 |
| [11] | Veltkamp R, Wesselink W (1995) Modeling 3D curves of minimal energy. In: Eurographics 95, Maastricht, The Netherlands, pp 97-110 |
| [12] | Wang, X.; Wu, Z.; Shen, J., Repairing the cerebral vascular through blending Ball B-Spline curves with \(G^2\) continuity, Neurocomputing, 173, 768-777 (2016) · doi:10.1016/j.neucom.2015.08.028 |
| [13] | Wesselink, W.; Veltkamp, R., Interactive design of constrained variational curves, Comput Aided Geom Des, 12, 533-546 (1995) · Zbl 0875.68848 · doi:10.1016/0167-8396(94)00033-O |
| [14] | Wu Z, Seah H, Zhou M (2007) Skeleton based parametric solid models: Ball B-spline surfaces. In: 2007 10th IEEE international conference on computer-aided design and computer graphics, pp 421-424 |
| [15] | Wu Z, Wang X, Fu Y et al (2018) Fitting scattered data points with ball B-Spline curves using particle swarm optimization. Comput Graph 1-11 |
| [16] | Wu Z, Zhou M, Wang X, et al (2007) An interactive system of modeling 3D trees with ball b-spline curves. In: 2007 10th IEEE international conference on computer-aided design and computer graphics, pp 259-265 |
| [17] | Xu, G.; Wang, G.; Chen, W., Geometric construction of energy-minimizing Bézier curves, Sci China Inf Sci, 54, 7, 1395-1406 (2011) · Zbl 1267.65021 · doi:10.1007/s11432-011-4294-8 |
| [18] | Xu X, Leng C, Wu Z (2011) Rapid 3d human modeling and animation based on sketch and motion database. In: 2011 Workshop on digital media and digital content management (DMDCM), pp 121-124 |
| [19] | Yong, J.; Cheng, F., Geometric hermite with minimum strain energy, Comput Aided Geom Des, 21, 281-301 (2004) · Zbl 1069.65541 · doi:10.1016/j.cagd.2003.08.003 |
| [20] | Zhu, T.; Tian, F.; Zhou, Y., Plant modeling based on 3D reconstruction and its application in digital museum, Int J Virtual Real, 7, 1, 81-8 (2008) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
