*Proceedings of Donetsk National University series "Informatics, cybernetics and computer technology" Issue 10 (153), 2009*
**Muftejev V., Mardanov A. Isogeometric modeling of fair curves and surfaces.**
Geometric robust modeling is considered as isogeometric modeling. Base criteria of an estimation of quality of functional surfaces are developed. The new scheme of modeling of a curve on set of geometrically connected conic curves is generalized. The method of modeling of virtual fair curve (v-curve) of class G5, based on the new scheme is improved. Isogeometric modeling of NURBS curves and surfaces, uv-loft surfaces is investigated.
At all stages of product life cycle essential component of electronic product model is a geometric model. The main stages of the life cycle of a product - the stage of design and development, manufacturing products provides an integrated CAD / CAM / CAE system. The most important part, "core”, of integrated CAD / CAM / CAE system is a geometric software. Geometric modeling follows the whole process of design, technology and manufacture of object. Geometrical parameters of curved surfaces is often crucial for the design characteristics of the object. These objects include products with dynamic surfaces (or functional surfaces) in the aircraft industry, automotive, shipbuilding, energy and chemical engineering, agricultural engineering. Stability shaping parametric curves and surfaces is the main criterion for the quality of modeling techniques in engineering geometry and computer graphics. The authors of the stability of the method of modeling is seen as a isogeometric definition of the curve / surface on complex drawing [1,2,3]. Curve / surface is geometrically defined on the projection, if the shape of the projection of Geometric Determinant (GD) uniquely determines the shape of the projection of the curve / of the parametric lines of the surface. Curve / surface is isogeometrically defined in the space, if the curve / surface is isogeometrically defined in the orthogonal projections of complex drawing. Developed formal criteria for visual assessment of shape preserving certainty of NURBS curve / surface on configuration of s-polygon / s-mesh on projections of complex drawing [3]. Method of constructing a surface on a network of curves (e.g., uv-loft surface in AutoCAD [4]) is one of the main ways. An actual task is to develop formal criteria for the visual evaluation of geometric definition of the surface, modeled on a network of curves on the projections of complex drawing. Special demands are made to quality of geometrical characteristics of functional surfaces. Methods of modeling should correspond to demanded quality of a functional surface of a product. Recently development of methods of modeling of curves of high quality by criteria of fairness are intensively carried out. The main criteria for the quality of the curve are the order of smoothness and the fairness of the curve. Synonym "quality curve” is "fair curve”. This term appeared in works devoted to improving the quality of parametric cubic splines [5, 6]. An actual task is to develop scientifically based criteria for assessing the fairness of curves for modeling the functional surfaces of technical objects. The general scheme of modeling curve consists of the following main stages. I). Sketching of curve. Initial information about the curve can be defined as a) French curve or its discrete representation, b) the set of points taken from full-scale model using the measurement system, c) a line drawn by designer on paper or screen of graphical display and fixed in the form of a discrete set of points, d) rare discrete set of points through which there should pass curve, q) the fixed analytical curve. 2) Constructing on the sketch of curve a Geometric Determinant (GD), which determines the geometrical structure of the curve. 3). Geometric approximation of the GD by analytical (or piecewise analytical) curve of a given class, or constructing an algorithm generating points of a curve on the given parameters of the GD. 4). Transition to other type of a determinant of a curve by equivalent transformation or by geometrical approximation of a determinant of a curve for its editing by means of parameters new GD. 5). Transition to another type of determinant of the curve by equivalent transformation or by shape preserving approximation of the determinant of the curve for the solution of metric and positional problems in the CAD system. In this case, a new determinant of the curve is called the template of the curve. The most natural types of GD are basic polylines, composed of points coincident with curve, and tangent polylines. It is assumed that the curve is constructed sequentially passes through the point of basic polyline and configuration of a basic polyline isogeometrically determines the shape of the curve. When specifying the structure of the curve by a tangent polyline, the curve passes consistently on lines of a tangent polyline touching each line in one point. Form of the tangent polyline isogeometrically determines the shape of the curve. GD may be the control polygon (s-polygon) of a NURBS curve. S-polygon is commonly used as an initial approximation in the method of heuristic fitting by b-spline curve. Also in iteration [7,8,2,9,10] and evolution [11,12,13,14] methods for constructing b-spline curves. In any case, the form of s-polygon should clearly define the structure (shape) of the modeled curve. Often as an initial approximation s-polygon is taken tangent polygon [7,8,2,9,10] or a basic polyline [11,12,13,14]. In the works of authors [1,2,15] proposed a new scheme for constructing a spline curve. In this scheme the spline curve is considered as a geometric model (determinant) of the curve in engineering geometry. The description of a determinant of a curve in the offered scheme is carried out by attraction so-called " theory of parametrization ” or "calculations of parameters ”, for the first time acad. N.F.Chetveruhin entered into applied geometry [16,17]. In the scheme the global GD of a curve in the form of a basic polyline or a tangent polyline is set. The spline curve (piecewise curve) is considered as the complex figure composed of non-derivative analytical curves on certain geometrical conditions of connection with elements of global GD and conditions of connection of adjacent analytical curves. Such connections is coincidence of points, tangents and curvature in points of GD of a curve. The paper seeks development and generalization of the scheme for constructing the determinant of spline curve. Different types of GD have their advantages and disadvantages. Basic polyline allows precise positioning of the curve, tangent polyline is uniquely and precisely defines the shape of the modeled curve, s-polygon of high degree NURBS curve can locally edit the shape of the curve and can provide high quality space curves according to the criteria of smoothness of curvature and torsion. Method of v-curve [18,1,2,15] allows the use of two types of GD – basic polyline and tangent polyline for modeling plane curves of high quality (5-th order smoothness with a smooth change of curvature). A combined method for constructing curves of high quality on flat and spatial basic polylines or flat and spatial tangent polylines of arbitrary configuration [10]. The method represents sequence of construction of a v-curve for structuring of an initial polyline and isogeometrical construction of a b-spline curve any degree on the structured polyline [10]. The disadvantage is low accuracy at creation of a b-spline curve on a spatial polyline with torsion. Actual problem is development of an interactive method of isogeometrical modeling of a curve of high quality by criteria of fairness with simultaneous use of three kinds of GD in one session of modeling of a curve and with maintenance of exact transition from one kind of GD to another in any direction. The purpose of this article: to provide a systematic review of scientific research and development, the authors in the scientific field "Isogeometric modeling of curves and surfaces with high quality on fairness criteria in engineering geometry and computer graphics”, and solve the following problems: - to investigate and develop the scheme of construction of a determinant spline curve as the complex figure composed of non-derivative analytical curves on certain geometrical conditions of connection with elements of global GD and conditions of connection of adjacent analytical basic curves; - to develop scientifically proved formal criteria of an estimation of quality of geometrical characteristics of functional curves and surfaces; - to develop an interactive method of isogeometrical modeling of a curve of high quality by criteria of fairness with simultaneous use of three kinds of GD in one session of modeling of a curve and with maintenance of accuracy of transition from one kind of object to another in any direction; - to investigate a method of modeling of a surface on a network of curves, to develop formal criteria of a visual estimation of isogeometrical definiteness of a surface. |