Recovery of surface from its normals in the system of points

H. V. Velichko

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 1 (2015), pp. 23-29.

Résumé

The author studies the problem of approximation of functions defined in a domain of $R^2$. In the system of points which lies in this domain, the normals to the surface are known. The normals are selected by such a way, so that they form an acute angle with the positive direction of $Z$-axis. The searched function is looked for in the form of a polynomial in two variables, the coefficients that minimize the discrepancy. The discrepancy is the sum of squares of the differences of the normals to the set and restoring surfaces. The normals are normalized so that their applicate is equal to 1. In this case normals, which have a greater angle with the vertical axis, will have a greater length, hence be better approximated. There is the system of linear algebraic equations for the unknown parameters, determined that, completely restores the searched surface. For the uniqueness of recovery is set in one of the points of the domain. The article is also considering a polynomial in the second degree. In this case, the system is aimed for finding the unknown parameters is greatly simplified. There are numerical examples for algebraic and transcendental functions. According to the obtained normals, the surface is restored, that approximates the original. In the second case the surface is taken, which comprises a sine equation. Just consider the case of the five normals. Element that does not contain the degrees have been chosen from the test match conditions and restored function at the origin. Drawings, which show the test surface and the surface, their approximate. Within the accuracy of the drawings, they are practically the same. In both of the above examples of numerical calculation formulas give a qualitative approach for a given function. Note that if the function is given by a polynomial of the second degree, and the amount is no less than five normals, the resulting estimated function will coincide with it.

Détail

Keywords: regular surface, normals to the surface, approximation, Least Square Method (LSM), misalignment function.

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