Вторник, 19.06.2018 FairCurveModeler
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## 1. Introduction in nanoGeometry

Appendix FairCurveModeler (further Application) is intended not just to model the beautiful curves and surfaces. First of all, the app is intended for the design of high-quality products. Namely, products with functional surfaces. And there are a lot of products. What is the functional surface? This surface which quality directly determines the quality of the product as a whole. This external surfaces of aircraft, ships, automobiles; the working surfaces of blades of pumps, compressors and turbines of aircraft engines, propellers; the working surfaces of tillage machinery; cam surface in the cam mechanism; road surface; canal surface.
The characteristic curves on which are based functional surfaces - a functional curves: guide curve of plow, profile of wing or of blade of compressor, turbine, pump; flat profile of the cam; plane trace of road, etc.
The authors performed a deep theoretical research on the analysis of quality requirements for functional curves, independent of the specific conditions of work and type of product. These requirements are summarized in the following concept as a set of necessary requirements for the geometry quality of functional curves:
1) a high order of smoothness. The order of for the the smoothness of function curves is required at least the third order. This is the minimum order that ensures the continuity of the torsion of a space curve. The order of determines the the smoothness of local smoothness of the curve.
2) The minimum number of extrema of curvature (vertices of the curve). This parameter determines the smoothness of the integral curve. Obviously, in terms of energy, for movement of the flow of gas / liquid / soil with the pulsation of curvature of the trajectory is required more energy than with its absence.
3) Small value of variation of the curvature (the difference between the maximum and minimum curvature) at the site with an extremum of curvature. This requirement complements the the second requirement.
4) other conditions being equal small value of the potential energy. Of the two curves of the same order of smoothness and the same number of extrema of curvature the curve with a lower potential energy - the best curve. The requirement to minimize the potential energy is justified as follows:
- milieu, streamlined functional surface at high speeds behaves as an elastic body. The deformed elastic body takes the form of the minimum potential energy. Hence the energy for elastic deformation of the medium that moves along a path with a lower potential energy is required less.
- in case of the milieu flow of concave surfaces with friction energy costs for the movement of the milieu is less than less than the potential energy of curve of movement.

If the original geometric determinant as a base polyline or a tangent polyline permits the construction of curve with the minimum number of vertices, so methods of construction curves should provide the minimum number of vertices. At specifically, if the points lie on a conical curve, the method should provide an accurate approximation of the conical curve.

Application FairCurveModeler provides these stringent requirements of quality of the functional surfaces. The application has a lot of innovative methods of geometric modeling and geometric approximation, which give sinergetic effect in efficiency of Application. In the words of Prof. Osipov V.A. "There are Geometry of breadth and Geometry of depth."
Methods of Application - a "geometry of depth." Methods are unique not only in existing CAD-systems, but also in the geometric nuclei CAD-systems. In a sense, these methods - methods of "geometric nanotechnology."
The basis of modeling the curves of high quality is a method for modeling a virtual curve (v-curve). V-Curve has no analytic or piecewise analytic expression. Points of the v-curve are generated algorithmically. In the limit, generated points belong to the curve of the class C5.
The application uses innovative methods of geometrically stable (isogeometrically, shape preserving) approximation of v-curve by a cubic NURBzS curve (cubic rational Bezier curve) and b-spline curve of high even degree m, (m = 6/8/10). These methods keep the quality of v-curve of the second step of subdivision and allow to use the industry standard representation of curves in the form of NURBS curves with the same number of segments as of original polyline.
The innovative entering by the authors of the dual determinant of v-curve - enables simultaneous modeling of the curve as on base polyline and on a tangent polyline. This capability expands the range of solved geometric problems, for example, 1) allows you to model a high-quality road trace on a tangent polyline of theodolite moves; 2) to model a plane convex cam profile at positions of the flat soles of the pusher; 3) to form a curve on points, but by the constraints of the form by tangent lines. At the same time base polyline and tangent polyline - a classic forms of geometric determinants, which are familiar to the designer and do not require a knowledge of sophisticated features of modeling of NURBS curves on s-frames.

Creating a surface, regardless of the method requires building a set or a net of curves. Methods of construction of v-curve and geometrically stable approximation by NURBzS curve and by b-spline curve allow to form sets and nets of high-quality curves.
The authors have developed innovative methods of geometrically stable (Isogeometric) constructing spline surfaces of high degrees on different kinds of geometric determinants. Dual determinant of v-curve is generalized to the determinant of surface. Spline surfaces can be modeled on base 3D Mesh, on 3D Mesh with the tangent lines, on 3D Mesh wirh tangent lines and tangent columns.
Developed innovative methods of testing and controlling the shape of the surfaces. You can control the shape of the families isoparms surface. Until the control and management of form of any arbitrary isoparms of surface.
An important promising direction of development of application is to develop the method of modeling of topologically complex surfaces with the smoothness of of high order at any point up to the system with a developed structure of options. The system will be an analog of the so-called T-Splines, but with high quality of integral surface.

The theoretical foundations in details are given in the authors articles in the section
Библиотека плагинов и статей

The concept of quality of functional curves practically tested on a general-purpose design of the plow. Only by following the proposed concept and the use of Application that implement a list of demands of the concept, and only by improving the geometry of the prototype was produced a striking result: at the same time was improved the quality of plowing and was obtained a fuel economy.

The concept and implementation of application FairCurveModeler - versatile and cheapest means of improving the quality of the designed product. That is, you can only by following the requirements of the concept and using the this application, without design tweaks, but by improving the geometry of your previous project or well-known project get more quality project and the product.

Moreover, the application FairCurveModeler is not requires highly skilled designer. Even in a non-uniform arrangement of points FairCurveModeler creates high-quality v-curve. Without exhausting fit of curves to the desired quality in a shorter time you design a better product.

Authors can help you to develop specialized applications on the basis of this concept and application FairCurveModeler.
There are certain scientific and program backlogs in the development of of specialized applications on the following topics:
- Profiling plane convex cam;
- Tracing the road in the plane;
- Modeling and improve airfoils.
More about specialized applications you can find in the Using Specialized Applications

The authors also have an interest in the feasibility of the concept and application FairCurveModeler in the aircraft industry, shipbuilding, automotive, architecture, industrial design.

Requests for development of specialized applications we receive on mail muftejev@mail.ru.

This application is a clone of nanoCAD application for AutoCAD. Moreover, this application can work in nanoCAD and in AutoCAD. The application implements interactive modeling of curves and surfaces with all facilities working in a graphical environment nanoCAD / AutoCAD and with the ability to control the quality of curves and surfaces by the Application and by own commands of CAD-systems.

The application is developed on COM-automation technology. The application consists of DLL-component and software interface programmed on AutoLISP. Communication between COM-server and nanoCAD / AutoCAD is performed via the so-called Geometric buffer. Geometric buffer consists of 3 folders: Exec, Temp, Result, which are located in the Tools folder of the application.

This application enhances the basic functionality of web FairCurveModeler by following features:
- Function modeling of sites of clothoid;
- Editing function of sites of s-frames of NURBS surfaces of arbitrary degrees and format by the deformation of area on the basis of sample deformation function F (u, v);
- Editing function of plots of s-frames of NURBS surfaces of arbitrary degrees and format by the formula Coons;

## 2.1. Command V_Model. Команда V_Model

Command V_Model - basic command of Application. The command has a complicated structure of the string menu.
Depending on the selected object will be available to those or other functions of the application.

## 2.1.1. Modeling of curves. Моделирование кривых

On page Fair Curves. the functions of modeling of curves of high quality are described.

To demonstrate the functionality of Application the library of scripts are prepared in the folder "Examples / Examples Curve" of Application. Scripts can be used for learning the Application. On page Scripts modeling of curves given a list of scripts. Set of scripts covers all the basic features and options of Application. On page Examples. Fair Curves. videos are given showing the execution of the some scripts.

## 2.1.2. Modeling surfaces

On page Geometric determinants of surfaces are described geometric determinants, which are used in the construction of surfaces.

On page Options of construction are described the functions of modeling of set of forming curves, set of guide curves and net of curves on surface carrier 3D Mesh. Are described the functions of construction of uv-loft surfaces, NURBzS surfaces, b-spline surfaces, NURBS surfaces.

At Technology of construction are described in detail methods for constructing spline surfaces of different formats on various types of geometric determinants.

On page surfaces are described the options of working with spline surfaces of different formats.

To demonstrate the functionality of Application the library of scripts are prepared in the folder "Examples / Examples Surface" of Application. Scripts can be used for learning the Application. On page Scripts modeling of surfaces is given a list of scripts. Set of scripts covers all the basic features and options of Application. On page Examples. Fair Surfaces videos are given showing the execution of the main scripts.

## 2.2. Basic set of commands

On page Base commands is described basic commands of Application.
For ease of operation a number of options of modeling curves from the command V_Model are issued in the form of individual commands: command of creating a v-curve and approximating by cubic NURBzS curve; command of approximatiing v-curve by b-spline curve of even high degree m (m = 6/8 / 10); the command of subdividing the specification of curve; command of increasing the degree of NURBS curves; testing curves with display of graphs of curvature and evolute and printing the macroparameters of curve (variation of the curvature, the value of the potential energy).
Are developed advanced options in Application: LISP-program for approximation of curves on the GD Hermite presented by table of the coordinates of points, tangent vectors and curvature values; command of creating and approximation a site of Clothoid spiral.
Examples of work with the basic commands in AutoCAD. In the examples you just need certain LISP-text fragments to insert into the AutoCAD command line and press ENTER.

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