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Query: "author" (Emil Žagar) .

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1.
Arc length preserving $G^2$ Hermite interpolation of circular arcs
Emil Žagar, 2023, original scientific article

Abstract: In this paper, the problem of interpolation of two points, two corresponding tangent directions and curvatures, and the arc length sampled from a circular arc (circular arc data) is considered. Planar Pythagorean–hodograph (PH) curves of degree seven are used since they possess enough free parameters and are capable of interpolating the arc length in an easy way. A general approach using the complex representation of PH curves is presented first and the strong dependence of the solution on the general data is demonstrated. For circular arc data, a complicated system of nonlinear equations is reduced to a numerical solution of only one algebraic equation of degree 6 and a detailed analysis of the existence of admissible solutions is provided. In the case of several solutions, some criteria for selecting the most appropriate one are described and an asymptotic analysis is given. Numerical examples are included which confirm theoretical results.
Keywords: geometric interpolation, circular arc, arc length, Pythagorean-hodograph curve, solution selection
Published in DiRROS: 20.03.2024; Views: 102; Downloads: 52
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2.
Optimal approximation of spherical squares by tensor product quadratic Bézier patches
Aleš Vavpetič, Emil Žagar, 2023, original scientific article

Abstract: In cited article E. F. Eisele considered the problem of the optimal approximation of symmetric surfaces by biquadratic Bézier patches. Unfortunately, the results therein are incorrect, which is shown in this paper by considering the optimal approximation of spherical squares. A detailed analysis and a numerical algorithm are given, providing the best approximant according to the (simplified) radial error, which differs from the one obtained mentioned article. The sphere is then approximated by the continuous spline of two and six tensor product quadratic Bézier patches. It is further shown that the $G^1$ smooth spline of six patches approximating the sphere exists, but it is not a good approximation. The problem of an approximation of spherical rectangles is also addressed and numerical examples indicate that several optimal approximants might exist in some cases, making the problem extremely difficult to handle. Finally, numerical examples are provided that confirm the theoretical results.
Keywords: Bézier patches, spherical squares, optimal approximation, sphere approximation
Published in DiRROS: 14.03.2024; Views: 95; Downloads: 51
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