Ray-triangular Bézier patch intersection using hybrid clipping algorithm
Yan-hong LIU, Juan CAO, Zhong-gui CHEN, Xiao-ming ZENG
Ray-triangular Bézier patch intersection using hybrid clipping algorithm
In this paper, we present a novel geometric method for efficiently and robustly computing intersections between a ray and a triangular Bézier patch defined over a triangular domain, called the hybrid clipping (HC) algorithm. If the ray pierces the patch only once, we locate the parametric value of the intersection to a smaller triangular domain, which is determined by pairs of lines and quadratic curves, by using a multi-degree reduction method. The triangular domain is iteratively clipped into a smaller one by combining a subdivision method, until the domain size reaches a prespecified threshold. When the ray intersects the patch more than once, Descartes’ rule of signs and a split step are required to isolate the intersection points. The algorithm can be proven to clip the triangular domain with a cubic convergence rate after an appropriate preprocessing procedure. The proposed algorithm has many attractive properties, such as the absence of an initial guess and insensitivity to small changes in coefficients of the original problem. Experiments have been conducted to illustrate the efficacy of our method in solving ray-triangular Bézier patch intersection problems.
Ray tracing / Triangular Bézier surface / Ray-patch intersection / Root-finding / Hybrid clipping
[1] |
Barth, W., Stürzlinger, W., 1993. Efficient ray tracing for Bézier and B-spline surfaces.Comput. Graph., 17(4):423–430. http://dx.doi.org/10.1016/0097-8493(93)90031-4
|
[2] |
Bartoň, M., Jüttler, B., 2007a. Computing roots of polynomials by quadratic clipping.Comput. Aided Geom. Des., 24(3):125–141. http://dx.doi.org/10.1016/j.cagd.2007.01.003
|
[3] |
Bartoň, M., Jüttler, B., 2007b. Computing Roots of Systems of Polynomials by Linear Clipping. SFB F013 Technical Report.
|
[4] |
Garloff, J., Smith, A.P., 2001. Investigation of a subdivision based algorithm for solving systems of polynomial equations.Nonl. Anal. Theory Methods Appl., 47(1):167–178. http://dx.doi.org/10.1016/S0362-546X(01)00166-3
|
[5] |
Haines, E., Hanrahan, P., Cook, R.L.,
|
[6] |
Hanrahan, P., 1983. Ray tracing algebraic surfaces. ACM SIGGRAPH Comput. Graph., 17(3):83–90. http://dx.doi.org/10.1145/964967.801136
|
[7] |
Joy, K.I., Grant, C.W., Max, N.L., et al., 1989. Tutorial: Computer Graphics, Image Synthesis. IEEE Computer Society Press, Los Alamitos, CA, USA.
|
[8] |
Jüttler, B., Moore, B., 2011. A quadratic clipping step with superquadratic convergence for bivariate polynomial systems.Math. Comput. Sci., 5(2):223–235. http://dx.doi.org/10.1007/s11786-011-0091-4
|
[9] |
Liu, L., Zhang, L., Lin, B.,
|
[10] |
Lou, Q., Liu, L., 2012. Curve intersection using hybrid clipping.Comput. Graph., 36(5):309–320. http://dx.doi.org/10.1016/j.cag.2012.03.021
|
[11] |
Lu, L., Wang, G., 2006. Multi-degree reduction of triangular Bézier surfaces with boundary constraints.Comput.-Aided Des., 38(12):1215–1223. http://dx.doi.org/10.1016/j.cad.2006.07.004
|
[12] |
Markus, G., Oliver, A., 2005. Interactive ray tracing of trimmed bicubic Bézier surfaces without triangulation. Proc. 13th Int. Conf. in Central Europe on Computer Graphics, Visualization and Computer Vision, p.71–78.
|
[13] |
Martin, W., Cohen, E., Fish, R.,
|
[14] |
Moore, R.E., Jones, S.T., 1977. Safe starting regions for iterative methods.SIAM J. Numer. Anal., 14(6):1051–1065. http://dx.doi.org/10.1137/0714072
|
[15] |
Nishita, T., Sederberg, T.W., Kakimoto, M., 1990. Ray tracing trimmed rational surface patches.ACM SIGGRAPH Comput. Graph., 24(4):337–345. http://dx.doi.org/10.1145/97880.97916
|
[16] |
Roth, S.H.M., Diezi, P., Gross, M.H., 2000. Triangular Bézier clipping. Proc. 8th Pacific Conf. on Computer Graphics and Applications, p.413–414. http://dx.doi.org/10.1109/PCCGA.2000.883971
|
[17] |
Rouillier, F., Zimmermann, P., 2004. Efficient isolation of polynomial’s real roots.J. Comput. Appl. Math., 162(1):33–50. http://dx.doi.org/10.1016/j.cam.2003.08.015
|
[18] |
Schulz, C., 2009. Bézier clipping is quadratically convergent.Comput. Aided Geom. Des., 26(1):61–74. http://dx.doi.org/10.1016/j.cagd.2007.12.006
|
[19] |
Sederberg, T.W., Nishita, T., 1990. Curve intersection using Bézier clipping.Comput.-Aided Des., 22(9):538–549. http://dx.doi.org/10.1016/0010-4485(90)90039-F
|
[20] |
Stürzlinger, W., 1998. Ray-tracing triangular trimmed freeform surfaces.IEEE Trans. Vis. Comput. Graph., 4(3):202–214. http://dx.doi.org/10.1109/2945.722295
|
[21] |
Sweeney, M.A.J., Bartels, R.H., 1986. Ray tracing free-form B-spline surfaces.IEEE Comput. Graph. Appl., 6(2): 41–49. http://dx.doi.org/10.1109/MCG.1986.276691
|
[22] |
Toth, D.L., 1985. On ray tracing parametric surfaces.ACM SIGGRAPH Comput. Graph., 19(3):171–179. http://dx.doi.org/10.1145/325334.325233
|
[23] |
Woodward, C., 1989. Ray tracing parametric surfaces by subdivision in viewing plane.In: Straßer, W., Seidel, H.P. (Eds.), Theory and Practice of Geometric Modeling. Springer Berlin Heidelberg, Germany, p.273–287. http://dx.doi.org/10.1007/978-3-642-61542-9_18
|
[24] |
Yen, J., Spach, S., Smith, M.T.,
|
[25] |
Zhang, R.J., Wang, G., 2005. Constrained Bézier curves’ best multi-degree reduction in the L2-norm.Progr. Nat. Sci., 15(9):843–850. http://dx.doi.org/10.1080/10020070512331343010
|
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