Singular points of curves

A. D. Uvarov

Modelirovanie i analiz informacionnyh sistem, Tome 25 (2018) no. 6, pp. 692-710.

Résumé

In this paper, we consider the key problem of geometric modeling, connected with the construction of the intersection curves of surfaces. Methods for constructing the intersection curves in complex cases are found: by touching and passing through singular points of surfaces. In the first part of the paper, the problem of determining the tangent line of two surfaces given in parametric form is considered. Several approaches to the solution of the problem are analyzed. The advantages and disadvantages of these approaches are revealed. The iterative algorithms for finding a point on the line of tangency are described. The second part of the paper is devoted to methods for overcoming the difficulties encountered in solving a problem for singular points of intersection curves, in which a regular iterative process is violated. Depending on the type of problem, the author dwells on two methods. The first of them suggests finding singular points of curves without using iterative methods, which reduces the running time of the algorithm of plotting the intersection curve. The second method, considered in the final part of the article, is a numerical method. In this part, the author introduces a function that achieves a global minimum only at singular points of the intersection curves and solves the problem of minimizing this function. The application of this method is very effective in some particular cases, which impose restrictions on the surfaces and their arrangement. In conclusion, this method is considered in the case when the function has such a relief, that in the neighborhood of the minimum point the level surfaces are strongly elongated ellipsoids. All the images given in this article are the result of the work of algorithms on methods proposed by the author. Images are built in the author's software environment.

Détail

Keywords: singular point, intersection, touching
Mots-clés : matrix, gradient.

Bibliographie

[1] Golovanov N. N., Geometric modeling, Create Space Independent Publishing Platform, 2014, 348 pp. | MR

[2] Shenen P. et al, Matematika i SAPR, in 2 vol., Mir, M., 1988 (in Russian)

[3] Faux I. D., Pratt M. J., Computational geometry for design and manufacture, Ellis Horwood Ltd., 1979 | MR | Zbl

[4] Hopcroft J., Kannan R., Computer Science Theory for the Information Age, Springer, 2012 | MR

[5] Golub J., Reinsch C., “Singular Value Decomposition and Least Squares Solutions”, Numerische Mathematik, 14:5 (1970), 403–420 | DOI | MR | Zbl

[6] Butterfield K., Ph. D. Thesis, Brunel University Uxbridge, 1978

[7] Kalitkin N. N., Chislennyye metody, uchebnoye posobiye, BKHV-Peterburg, 2011, 592 pp. (in Russian)

[8] Vermel A. V., Vermel V. D., Kalitin E. I., “Intersection of airplane component surfaces for design of an aerodynamic layout”, Uchenyye zapiski TsAGI, XLII:5 (2011), 92–106 (in Russian)

[9] Bajaj C. L., Hoffmann C. M., Lynch R. E., Hoptcroft J. T., “Tracing surface intersections”, Computer Aided Geometric Design, 5 (1988), 285–307 | DOI | MR | Zbl