﻿ 几何迭代方法计算空间两圆之间的最近距离

# 几何迭代方法计算空间两圆之间的最近距离The Geometric Iteration Method for Computing the Minimum Distance between Two Spatial Circles

Abstract: Computing the minimum distance between two spatial circles is the base of collision detection and intersection in the fields of computer graphics, computer-aided design and computer-aided geo-metric design. This paper has completely analyzed and discussed the minimum distance problem between two spatial circles for their spatial position relationships. If the two central axes of two spatial circles are not paralleled, we have presented the algorithm for computing the minimum distance between two spatial circles based on the geometric iterative method. Besides, if two cen-tral axes of two spatial circles have an intersection, we also have presented two pairs of corres-ponding points of the minimum distance for two spatial circles based on the geometric iterative method; if two central axes of two spatial circles are paralleled or coincident, we have directly provided the analytical expressions for computing the minimum distance between two spatial cir-cles. Numerical examples illustrate that the algorithms are efficient and robust.

[1] Ho, S., Sarma, S. and Adachi, Y. (2001) Real-Time Interference Analysis between a Tool and an Environment. Com-puter-Aided Design, 33, 935-947. http://dx.doi.org/10.1016/S0010-4485(00)00117-2

[2] Johnson, D. and Cohen, E. (1998) A framework for Efficient Minimum Distance Computations. Proceedings of the IEEE Conference on Robotics and Automation, 36, 78-84.
http://dx.doi.org/10.1109/robot.1998.681403

[3] Cameron, S. and Enhancing, G.J.K. (1997) Computing Minimum and Penetration Distances between Convex Polyhedra. Proceedings of the IEEE Conference on Robotics and Automation, 3112-3117.
http://dx.doi.org/10.1109/ROBOT.1997.606761

[4] Gilbert, E.G., Johnson, D.W. and Keerthi, S.S. (1988) A Fast Procedure for Computing the Distance between Complex Objects in Three-Dimensional Space. IEEE Journal of Robotics and Automation, 4, 193-203.
http://dx.doi.org/10.1109/56.2083

[5] Lin, M.C. (1993) Efficient Collision Detection for Animation and Robotics. Ph. D Thesis, University of California, Berkeley.

[6] Snyder, J. (1995) An Interactive Tool for Placing Curved Surfaces without Interpenetration. Computer Graphics. SIGGRAPH, Los Angeles, 6-11 August 1995, 209-218.

[7] 陈小雕, 雍俊海, 郑国勤, 等. 圆环面/球面求交算法[J]. 计算机辅助设计与图形学学报, 2005, 17(6): 1202-1206.

[8] Neff, C.A. (1990) Finding the Distance between Two Circles in Three-Dimensional Space. IBM Journal of Research and Development, 34, 770-775.
http://dx.doi.org/10.1147/rd.345.0770

[9] 刘晓明, 刘长远, 胡强, 雍俊海. 计算两圆环面之间的最近距离[J]. 计算机辅助设计与图形学学报, 2011, 23(2): 240-246.

[10] Almohamad, H.A. and Selim, S.Z. (2003) An Algorithm for Computing the Distance between Two Circular Disks. Applied Mathematical Modelling, 27, 115-124.
http://dx.doi.org/10.1016/S0307-904X(02)00080-X

[11] Kim, I.S. (2006) An Algorithm for Finding the Distance between Two Ellipses. Communications Korean Mathematical Society, 21, 559-567.
http://dx.doi.org/10.4134/CKMS.2006.21.3.559

[12] 王日昀, 宁涛, 席平, 等. 圆环与圆环求交算法中初始点的计算[J]. 工程图学学报, 2004, 25(1): 47-51.

[13] Chen, X.D., Chen, L.Q., Wang, Y.G., Xu, G., Yong, J.H. and Paul, J.C. (2009) Computing the Minimum Distance between Two Bézier Curves. Journal of Computational and Applied Mathematics, 229, 294-301.
http://dx.doi.org/10.1016/j.cam.2008.10.050

[14] Chen, X.D., Ma, W.Y., Xu, G. and Paul, J.C. (2010) Computing the Hausdorff Distance between Two B-Spline Curves. Computer-Aided Design, 42, 1197-1206.

[15] Chang, J.W., Choi, Y.K., Kim, M.S. and Wang, W.P. (2011) Computation of the Minimum Distance between Two Bézier Curves/Surfaces. Computers & Graphics, 35, 677-684.
http://dx.doi.org/10.1016/j.cag.2011.03.025

[16] Bai, Y.B., Yong, J.H., Liu, C.Y., Liu, X.M. and Meng, Y. (2011) Polyline Approach for Approximating Hausdorff Distance between Planar Free-Form Curves. Computer-Aided Design, 43, 687-698.