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AUTHOR(S): Valerian Muftejev, Fairuza Ziganshina, Vadim Gumerov
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TITLE Functional Curves of High Quality – Innovation in Geometric Modeling |
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ABSTRACT Curves and surfaces that form the geometry of technical products often directly determine the functional characteristics of the designed products. It is logical to call such curves and surfaces functional. Often, the aesthetics of a product is one of the important consumer properties of the product. Therefore, aesthetic curves can also be classified as functional. The optimal curve is not always defined by an analytical curve, such as the profile of a tooth (involute of a circle), the trajectory of a load transportation as a line of the fastest descent (brachistochrone), or the profile of a dome (catenary). Free-form curves in the form of spline curves are more commonly used. Methods for constructing functional curves must satisfy the following requirements: - Isogeometric construction of a curve on the initial polyline with fixed end and intermediate parameters. - Construction of a fair curve. - Low value of potential energy of the curve. Regardless of the specific product, functional curves must be fair. Functional curves must meet the following fairness criteria: - High order of smoothness (not lower than the 4th order). - Minimum number of curve vertices (or minimum number of curvature extremes). - Low value of curvature variation and rate of curvature change. - Low value of potential energy of the curve. Spline curves that meet these criteria are called F-curves or curves of class F. The authors have developed the FairCurveModeler software and methodological complex (SMC) for modeling F-curves. Based on the functionality of the FairCurveModeler SMC, universal and specialized applications for CAD systems (KOMPAS 3D, nanoCAD / ZWCAD / AutoCAD), mathematical systems (MathCAD / Mathematica / Wolfram Cloud), an Excel application, and a Web application have been developed. The FairCurveModeler SMC has been adapted and implemented into the C3D geometric core as the C3D FairCurveModeler section. The philosophy of the FairCurveModeler SMC is based on the theory of calculating parameters of the Soviet school of applied geometry. The initial data for constructing or editing curves are presented in the form of geometric determinants (GD). The following innovations have been implemented based on the parametric approach: - A new paradigm for constructing spline curves based on the theory of parameter calculus has been proposed. A spline basis is formed as a sequence of 5-parametric conical curves of double contact, with 4 common parameters of adjacent conical curves. Then, on the spline basis, points of a virtual curve are generated in the lenses of contacting conical curves. It is shown that the generated points belong to the curve of class C5. - The method for isogeometric approximation of a virtual curve by means of a rational cubic Bezier spline curve has been developed. - The method for isogeometric approximation of a virtual curve by means of a B-spline curve has been developed. The FairCurveModeler SMC is characterized by the following system properties: 1) The methods for constructing F-curves ensure isogeometricity (shape preserving) of the constructing curves on the original polylines. The shape of the modeled curve is similar to the shape of the original polyline. The designer is provided with a wide range of tools: - Base polyline. The spline curve passes through the vertices of the base polyline. In the general case, the spline nodes do not coincide with the vertices of the polyline. - A set of tangent lines (in particular, in the form of a tangent polyline). The curve passes tangent to the lines (tangent to the links of the tangent polyline). - Hermite GD. The base polyline is equipped with tangent vectors and curvature vectors at its vertices. - GB-polygons of Bezier spline curves. - S-polygons of B-spline curves. 2) The methods provide flexibility in construction and editing. This is the ability to locally control the shape of a global spline with fixed parameters at intermediate points of the polyline. 3) The unique feature of the methods is the ability to geometrically accurately model circles and, in general, conical curves. 4) The methods are invariant with respect to affine transformations. The article substantiates the importance of the property of minimizing the potential energy of curves in F-curves. The works of Mehlum and Livien are analyzed in detail. An experiment with a physical spline is conducted. The advantages of the methods for constructing F-curves in FairCurveModeler over spline curves of class A and over a physical spline and its approximation methods are proven. Innovative methods for constructing surfaces are proposed: a frame-kinematic method for constructing a spline surface, a method for constructing a topologically complex surface. |
KEYWORDS isogeometric approximation, fair curves, spline curves, FairCurveModeler, F-curves, B-spline surface, complex surfaces |
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Cite this paper Valerian Muftejev, Fairuza Ziganshina, Vadim Gumerov. (2025) Functional Curves of High Quality – Innovation in Geometric Modeling. International Journal of Mathematical and Computational Methods, 10, 300-320 |
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