High-quality Functional Curves - Innovation in Geometric Modeling
Prof. Valeriian Mufteev
C3D Labs Moscow, Russia
Abstract: The curved lines and surfaces that form the geometry of technical products often directly determine the functional characteristics of the designed products. Such curves and surfaces are logically called functional curves and surfaces. Often the aesthetics of a product is one of the important consumer properties of the product. Therefore, aesthetic curves can also be referred to as functional curves. The optimal curve is not always defined by an analytical curve, such as the profile of a gear tooth (involute of a circle), the trajectory of a load transportation as 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. Regardless of the specific product, functional curves must be fairing. Functional curves shall meet the following fairness criteria: - High order of smoothness (at least 4th order). - Minimum number of curve vertices (or minimum number of curvature extrema). - Small value of curvature variation and rate of curvature change. - A small value of the potential energy of the curve. Spline curves satisfying these criteria are called F-curves or class F curves. The authors have developed a software and methodological complex (SMC) FairCurveModeler for modeling F-curves. Based on the functionality of FairCurveModeler, universal and specialized applications of CAD-systems (KOMPAS 3D, nanoCAD / ZWCAD / AutoCAD), mathematical systems (MathCAD / Mathematica / Wolfram Cloud), Excel application, Web application were developed. FairCurveModeler is adapted and implemented in the geometric kernel of C3D in the form of C3D FairCurveModeler section. The philosophy of FairCurveModeler is based on the theory of parameter calculus of the Soviet school of applied geometry. The input data for constructing or editing curves are represented as geometric determinants (GDs). On the basis of the parametric approach the following innovations are realized: - The new paradigm of spline curves construction based on the theory of parameter calculus is proposed. The spline basis is formed as a sequence of 5 parametric double contacted conic curves with 4 common parameters of adjacent conic curves. Next, virtual curve points are generated on the spline basis in the lenses of the contiguous conic curves. It is shown that the generated points belong to a curve of class C5. - The method of isogeometric approximation of the virtual curve by means of rational cubic spline Bézier curve is developed. - The method of isogeometric approximation of virtual curve by means of B-spline curve is developed. The designer is provided with a wide range of tools: - Base polyline. The spline curve passes through the vertices of the base polyline. In general, the nodes of the spline do not coincide with the vertices of the base polyline. - A set of tangent lines (in particular in the form of a tangent polyline). A curve runs tangent to the straight lines (tangent to the links of a tangent polyline). - Hermite GD. A base polyline is equipped with tangent vectors and curvature vectors at its vertices. - GB-polygons of Bézier spline curves. - S-polygons of B-spline curves. 2) The methods provide flexibility of modeling and editing. This is the possibility of local control of the global spline shape with fixed parameters at intermediate points of the polyline; 3) A unique feature of the methods is the possibility of geometrically accurate modeling of circles and, in general, conic curves. 4) The methods are invariant with respect to affine transformations. The paper justifies the importance of the property of minimization of potential energy of F- curves. The works of Mehlum and Livien are analyzed in detail. An experiment with a physical spline is carried out. Advantages of methods of construction of F-curves in FairCurveModeler over spline curves of class A and over physical spline and methods of its approximation are proved. Innovative methods of surface construction are described: frame-kinematic method of B- spline surface construction, method of topologically complex surface construction. The frame-kinematic scheme of construction allows to reduce the procedure of surface construction to two stages: construction of the frame of forming F-NURBS-curves on a uniform grid; construction of the frame of guiding F-NURBS-curves on the frame of control spline S- polygons. Advantages of methods of isogeometric modeling of F-curves are generalized to methods of construction of surfaces. In modeling the integral surface of a product, we propose to use a “mosaic” composed of compartments of B-spline surfaces. In this case, adjacent compartments have common subarrays of S-frames to ensure overall smoothness. When creating a mosaic, there are inevitably irregular areas that cannot be covered with a “rectangular patch”. Based on Forrest's method, we propose a procedure for constructing a topologically complex patch in the form of a recursive scheme of successive “embedding” of compartments of B-spline surfaces into a topologically complex patch. The listed characteristics and advantages of the SMC FairCurveModeler allow us to say that today it is the best geometric kernel for modeling functional curves and surfaces of high quality. We are looking for an investor to create a group of strong programmers to - To bring existing applications to the level of commercial products; - To develop applications for top CAD-systems (Catia, NX, Alias Studio, etc.) that declare high quality A-class curves. The users of these CAD-systems will get the desired high quality of curves in FairCurveModeler applications. - It is especially important to realize based on FairCurveModeler Web-applications with all options of F-curves and surfaces creation. This will enable engineers all over the world to design products of high quality in terms of aesthetics and functionality.
Brief Biography of the Speaker: I, Valeriian Mufteev, was born on 20 January 1950 in the USSR, in the Republic of
Bashkortostan.
In 1967 I graduated from secondary school No. 3 in Ufa.
In 1967 I entered the Ufa Aviation Institute (UAI).
In 1972 I graduated from the Ufa Aviation Institute (UAI).
From 1972 to 1974 I served in the Soviet Army in the Air Force with the rank of Lieutenant as an
aircraft technician.
In 1974 I was employed at the Ufa Aviation Institute.
From 1974 to 1990 I worked in the scientific research department of the UAI and then in the
scientific research department of the Ufa Oil Institute (UNI). I took part in work under contracts
with leading machine-building enterprises of the USSR. The subject of the contracts was the
development of programs for geometric modeling and production of products with curved
surfaces.
On the basis of my innovative scientific achievements achieved in the process of software
development, I wrote a thesis at the Moscow Aviation Institute under the guidance of Prof.
Osipov Vadim, and in 1986 defended my thesis in Kiev for the degree of Candidate of Technical
Sciences.
From 1990 to 2005 a break in scientific activity.
From 2005 to 2019 I worked as an associate professor at the Bashkir State Agrarian University
(BSAU) and then at the Ufa State Aviation and Technical University (UGATU).
On my own initiative, I continued theoretical and software development of programs for
geometric modeling and manufacturing of products with curved surfaces. I created a software
product (geometric core FairCurveModeler) for modelling high quality curves and surfaces
according to fairness criteria. Based on the FairCurveModeler geometric core, I have developed
a number of CAD system applications.
In 2018, at the invitation of Keimyung University (South Korea, Daegu), I gave an overview
course of lectures on the functionality of the geometric core of FairCurveModeler.
From 2019 to 2024, I worked as a leading mathematician-programmer at 'C3D Labs LLC' and
adapted and developed FairCurveModeler as a part of the C3D geometric kernel. The C3D
FairCurveModeler program is patented in the Russian Federation (author V. Mufteev).
At present, I continue to develop the geometric kernel of FairCurveModeler on my own initiative.
There are innovative results. A geometric core functionality has been developed for modelling
topologically complex surfaces with high quality according to fairness criteria.
I am looking for an investor and a team of programmers to refine FairCurveModeler-based
applications to the level of commercial software products with patentable methods and
programs.