Geometric Modeling

Vector and raster graphics.

There are two types of graphics - vector and raster. The main difference is in the principle of image storage. Vector graphics describes an image using mathematical formulas. The main advantage of vector graphics is that when you change the scale of the image, it does not lose its quality. Another advantage follows from this - when resizing the image, the file size does not change. Raster graphics is a rectangular matrix consisting of many very small indivisible dots (pixels).

A raster image can be compared to a children's mosaic, when the picture is made up of colored squares. The computer remembers the colors of all the squares in a row in a certain order. Therefore, bitmap images require more memory to store. They are hard to scale and even harder to edit. To enlarge the image, you have to increase the size of the squares, and then the picture turns out to be "stepped". To reduce a raster image, several neighboring points must be converted into one or extra points should be discarded. As a result, the image is distorted, its fine details become illegible. These shortcomings are devoid of vector graphics. In vector editors, a drawing is stored as a set geometric shapes- contours presented in the form of mathematical formulas. To enlarge an object proportionally, all you need to do is change one number: the scaling factor. There are no distortions either when increasing or decreasing the picture. Therefore, when creating a drawing, you do not have to think about its final dimensions - you can always change them.

Geometric transformations

Vector graphics is the use of geometric primitives such as points, lines, splines, and polygons to represent images in computer graphics. Consider, for example, a circle of radius r. List of information required for complete description circles is:

radius r;

circle center coordinates;

color and thickness of the outline (possibly transparent);

fill color (possibly transparent).

Advantages of this way of describing graphics over raster graphics:

The minimum amount of information is transferred to a much smaller file size (the size does not depend on the size of the object).

Accordingly, you can infinitely increase, for example, the arc of a circle, and it will remain smooth. On the other hand, if the curve is represented as a broken line, magnification will show that it is not really a curve.

When objects are enlarged or reduced, the thickness of the lines can be constant.

Object parameters are stored and can be changed. This means that moving, scaling, rotating, filling, etc. will not degrade the quality of the drawing. Moreover, it is common to specify sizes in device-independent units ((English)), which lead to the best possible rasterization on raster devices.

Vector graphics have two fundamental drawbacks.

Not every object can be easily drawn in vector form. In addition, the amount of memory and time to display depends on the number of objects and their complexity.

Converting vector graphics to raster is quite simple. But way back, usually not - raster tracing usually does not provide high quality vector art.

Vector graphics editors typically allow you to rotate, move, reflect, stretch, bevel, perform basic affine transformations on objects, change z-order, and combine primitives into more complex objects.

More sophisticated transformations include boolean operations on closed figures: union, addition, intersection, etc.

Vector graphics are ideal for simple or composite drawings that need to be device independent or don't need photorealism. For example, PostScript and PDF use the vector graphics model.

Lines and broken lines.

Polygons.

Circles and ellipses.

Bezier curves.

Bezigones.

Text (in computer fonts such as TrueType, each letter is made up of Bezier curves).

This list is incomplete. There is different types curves (Catmull-Rom splines, NURBS, etc.) that are used in various applications.

It is also possible to think of a bitmap as a primitive object behaving like a rectangle.

The main types of geometric models

Geometric models give an external idea of ​​the original object and are characterized by the same proportions of geometric dimensions. These models are divided into two-dimensional and three-dimensional. Sketches, diagrams, drawings, graphics, paintings are examples of two-dimensional geometric models, and models of buildings, cars, aircraft, etc. are three-dimensional geometric models.

3D graphics operates with objects in three-dimensional space. Usually the results are a flat picture, a projection. Three-dimensional computer graphics is widely used in movies and computer games.

In 3D computer graphics, all objects are usually represented as a collection of surfaces or particles. The smallest surface is called a polygon. Triangles are usually chosen as a polygon.

All visual transformations in 3D graphics are controlled by matrices (see also: affine transformation in linear algebra). Three types of matrices are used in computer graphics:

rotation matrix

shift matrix

scaling matrix

Any polygon can be represented as a set of coordinates of its vertices. So, the triangle will have 3 vertices. The coordinates of each vertex are a vector (x, y, z). By multiplying a vector by the corresponding matrix, we get a new vector. Having made such a transformation with all the vertices of the polygon, we get a new polygon, and after transforming all the polygons, we get a new object rotated/shifted/scaled relative to the original

Subsystems of graphical and geometric modeling (GGM) occupy a central place in SAPS. The design of products in them, as a rule, is carried out in an interactive mode when operating with geometric models, i.e. mathematical objects that display the shape of the product, the composition of assembly units and possibly some additional parameters (mass, surface colors, etc.).

In HGM subsystems, a typical data processing route includes obtaining a design solution in an application program, its representation in the form of a geometric model (geometric modeling), preparing a design solution for visualization, visualization itself using a PC, if necessary, correcting the solution in an interactive mode.

The last two operations are implemented on the basis of GGM computing facilities. When talking about the mathematical software of HGM, they mean, first of all, models, methods and algorithms for geometric modeling and preparation for visualization.

There are two-dimensional (2D) and three-dimensional (3D) HGM software.

The main applications of 2D HGM are preparation of drawing documentation in CAD, topological design of printed circuit boards and LSI crystals in CAD of the electronics industry.

In the process of 3D modeling, geometric models are created, i.e. models that reflect the geometric properties of products. There are geometric models: frame (wire), surface, volumetric (solid).

wireframe model represents the shape of the product in the form of a finite set of lines lying on the surfaces of the product. For each line, the coordinates of the end points are known and their incidence to edges or surfaces is indicated. It is inconvenient to operate with a wireframe model in further EPS operations, and therefore wireframe models are rarely used at present.

surface model displays the shape of the product by specifying its bounding surfaces, for example, in the form of a set of data on faces, edges and vertices.

A special place is occupied by models of products with surfaces of complex shape, the so-called sculptural surfaces. Such products include, for example, cases of microcircuits, computers, workstations), etc.

Volumetric models differ in that they explicitly contain information about whether the elements belong to the internal or external space in relation to the product.

The considered models display bodies with closed volumes, which are the so-called manifolds. Some geometric modeling systems allow the operation of a few models ( nonmanifold), examples of which can be models of bodies touching each other at one point or along a straight line. Small-sized models are convenient in the design process, when it is useful to work simultaneously with 3D and 2D models at intermediate stages, without specifying the thickness of the walls of the structure, etc.

Systematization of geometric models

Mathematicians and physicists, engineers and designers, scientists and workers, doctors and artists, astronauts and photographers have to deal with geometrical models. However, there is still no systematic guide to geometric models and their applications. This is explained primarily by the fact that the range of geometric models is too wide and varied.

Geometric models can be the embodiment of the designer's intent and serve to create a new object. There is also a reverse scheme, when a model is made for an object, for example, during restoration or repair.

Geometric models are classified into subject (drawings, maps, photographs, layouts, television images, etc.), computational and cognitive. Object models are closely related to visual observation. The information obtained from object models includes information about the shape and size of the object, about its location relative to others.

Drawings of machines, structures, technical devices and their parts are performed in compliance with a number of symbols, special rules and a certain scale. There are drawings of parts, assembly, general view, assembly, tabular, overall, external views, operational, etc. Depending on the design stage, the drawings are divided into technical proposal drawings, draft and technical designs, working drawings. Drawings are also distinguished by industries: machine-building, instrument-making, construction, mining and geological, topographic, etc. Drawings of the earth's surface are called maps. Drawings are distinguished by the method of images: orthogonal drawing, axonometry, perspective, numerical marks, affine projections, stereographic projections, film perspective, etc.

Geometric models differ significantly in the way they are made: original drawings, originals, copies, drawings, paintings, photographs, films, radiographs, cardiograms, layouts, models, sculptures, etc. Among the geometric models, flat and volumetric ones can be distinguished.

Graphical constructions can serve to obtain numerical solutions of various problems. When calculating algebraic expressions, numbers are represented by directed segments. To find the difference or sum of numbers, the segments corresponding to them are plotted on a straight line. Multiplication and division is carried out by constructing proportional segments, which are cut off on the sides of the angle by parallel lines. The combination of multiplication and addition operations allows you to calculate sums of products and a weighted average. Graphical exponentiation consists in successive repetition of multiplication. The graphic solution of the equations is the value of the abscissa of the point of intersection of the curves. Graphically, you can calculate a definite integral, build a graph of the derivative, i.e. differentiate, and integrate differential equations. Geometric models for graphical calculations must be distinguished from nomograms and computational geometric models (RGMs). Graphical calculations require a sequence of constructions each time. Nomograms and RGMs are geometric images of functional dependencies and do not require new constructions to find the numerical values. Nomograms and RGMs are used for calculations and studies of functional dependencies. Calculations on RGM and nomograms are replaced by reading answers using elementary operations indicated in the nomogram key. The main elements of nomograms are scales and binary fields. Nomograms are divided into elementary and composite. Nomograms are also distinguished by the operation in the key. The fundamental difference between the RGM and the nomogram is that geometric methods are used to construct the RGM, and analytical methods are used to construct the nomograms.

Geometric models depicting relationships between elements of a set are called graphs.. Graphs are models of order and mode of action. On these models there are no distances, angles, the connection of points with a straight or curved line is indifferent. In graphs, only vertices, edges, and arcs are distinguished. For the first time, graphs were used in the course of solving puzzles. Currently, graphs are effectively used in planning and control theory, scheduling theory, sociology, biology, electronics, in solving probabilistic and combinatorial problems, etc.

A graphical model of functional dependence is called a graph. Function graphs can be built from a given part of it or from a graph of another function using geometric transformations.

A graphic image that clearly shows the ratio of any quantities is a diagram. For example, a state diagram (phase diagram) graphically depicts the relationship between the state parameters of a thermodynamic equilibrium system. A bar chart, which is a collection of adjacent rectangles built on one straight line and representing the distribution of any values ​​according to a quantitative attribute, is called a histogram.

Theoretical geometric models are of particular importance. In analytic geometry, geometric images are studied by means of algebra based on the method of coordinates. In projective geometry, projective transformations and immutable properties of figures independent of them are studied. In descriptive geometry, spatial figures and methods for solving spatial problems are studied by constructing their images on a plane. The properties of flat figures are considered in planimetry, the properties of spatial figures - in stereometry. In spherical trigonometry, relationships between angles and sides of spherical triangles are studied. The theory of photogrammetry and stereophotogrammetry makes it possible to determine the shapes, sizes, and positions of objects from their photographic images.

The geometric model of an object is understood as a set of information that uniquely determines its configuration and geometric parameters.

Currently, there are two approaches to the automated creation of geometric models using computer technology.

The first approach, representing the traditional technology for creating graphic images, is based on a two-dimensional geometric model and the actual use of a computer as an electronic drawing board, which makes it possible to speed up the process of drawing an object and improve the quality of design documentation. The central place in this case is occupied by a drawing, which serves as a means of representing the product on a plane in the form of orthogonal projections, views, cuts and sections and contains all the necessary information for developing the technological process for manufacturing the product. In a two-dimensional model, the product geometry is displayed in a computer as a flat object, each point of which is represented using two coordinates: X and Y.

The main disadvantages of using two-dimensional models in computer-aided design are obvious:

The created object design has to be mentally represented in the form of separate elements of the drawing (orthogonal projections, views, cuts and sections), which is a difficult process even for experienced developers and often leads to product design design errors;

All graphic images in the drawing (orthogonal projections, views, sections, sections) are created independently of each other and therefore are not associatively connected, that is, each change in the design object leads to the need to make changes (editing) in each corresponding graphic image of the drawing, which is time-consuming process and the cause of a significant number of errors in the modification of product designs;

The impossibility of using the obtained drawings to create computer models of control assemblies of objects from constituent components (aggregates, assemblies and parts);

The complexity and high labor intensity of creating axonometric images of assembly units of products, their catalogs and manuals for their operation;

It is inefficient to use two-dimensional models at subsequent stages of the production cycle (after the creation of the product design).

The second approach to the development of graphic images of design objects is based on using three-dimensional geometric models of objects, which are created in automated 3D modeling systems. Such computer models are a visual way of representing design objects, which makes it possible to eliminate the listed disadvantages of two-dimensional modeling and significantly expand the efficiency and scope of three-dimensional models at various stages of the production cycle of manufacturing products.

Three-dimensional models are used for computer representation of product models in three dimensions, that is, the geometry of an object is represented in a computer using three coordinates: X, Y and Z. This allows you to rebuild axonometric projections of object models in different user coordinate systems, as well as obtain their axonometric views from any point of view or visualize them as a perspective. Therefore, three-dimensional geometric models have significant advantages over two-dimensional models and can significantly improve design efficiency.

The main advantages of three-dimensional models:

The image is clearly and simply perceived by the designer;

Detail drawings are created using automatically obtained projections, views, sections and sections of a three-dimensional object model, which significantly increases the productivity of drawing development;

Changes in the three-dimensional model automatically cause corresponding changes in the associative graphic images of the object's drawing, which allows you to quickly modify the drawings;

It is possible to create three-dimensional models of virtual control assemblies and product catalogs;

Three-dimensional models are used to create operational sketches of technological processes for manufacturing parts and forming elements of technological equipment: dies, molds, casting molds;

With the help of three-dimensional models, it is possible to simulate the operation of products in order to determine their performance before manufacturing;

Three-dimensional models are used in automated program preparation systems for automatic programming of the trajectories of movement of the working bodies of multi-coordinate machine tools with numerical control;

These advantages make it possible to effectively use 3D models in automated product lifecycle management systems.

There are three main types of 3D models:

- frame (wire), in which the images are represented by the coordinates of the vertices and the edges connecting them;

- superficial , represented by surfaces bounding the created object model;

- solid state , which is formed from models of solid bodies;

- hybrid .

Three-dimensional graphic models contain information about all graphic primitives of an object located in three-dimensional space, that is, a numerical model of a three-dimensional object is built, each point of which has three coordinates (X, Y, Z).

Rice. 10.1. Tetrahedron wireframe data structure

The main disadvantages of wireframe models:

It is not possible to automatically remove hidden lines;

Possibility of ambiguous representation of an object;

In the section of an object, only the intersection points of the object's edges will be planes;

However, wireframe models do not require a lot of calculations, that is, high speed and large computer memory. Therefore, they are economical in terms of their use in creating computer images.

In surface models a three-dimensional image of an object is represented as a set of individual surfaces.

When creating three-dimensional surface models, analytical and spline surfaces are used.

Analytical Surfaces(plane, cylinder, cone, sphere, etc.) are described by mathematical equations.

Spline surfaces are represented by arrays of points, between which the positions of the remaining points are determined using mathematical approximation. On fig. Figure 10.2b shows an example of a spline surface created by moving a flat sketch (Figure 10.2a) in the selected direction.

Rice. 10.2. Spline surface example

Disadvantages of surface models:

In the section of the object, the planes will be only the lines of intersection of the surfaces of the object with the cutting planes;

It is impossible to perform logical operations of addition, subtraction and intersection of objects.

Advantages of surface models:

Unambiguous representation of an object;

Possibility to create models of objects with complex surfaces.

Three-dimensional surface models have found wide application in creating models of complex objects consisting of surfaces whose relative thickness is much smaller than the dimensions of the created object models (ship hull, aircraft fuselage, car body, etc.).

In addition, surface models are used when creating hybrid solid models using surface-constrained models, when creating a solid model is very difficult or impossible due to the complex surfaces of the object.

solid model is a real representation of the object, since the computer data structure includes the coordinates of the points of the entire body of the object. This allows you to perform logical operations on objects: union, subtraction and intersection.

There are two types of solid models: surface-constrained and volumetric.

In a surface-constrained solid model object boundaries are formed using surfaces.

For a 3D solid model the internal calculation model represents the coordinates of the points of the entire rigid body. It is obvious that solid models of objects require a large number of calculations compared to wireframe and surface models, since in the process of their transformation it is necessary to recalculate the coordinates of all points of the body of the object and, in connection with this, large computing power of computers (speed and RAM). However, these models have advantages that allow them to be effectively used in the process of computer-aided design:

Automatic removal of hidden lines is possible;

Visibility and impossibility of ambiguous representation of the object;

In the section of the object by planes, cuts will be obtained that are used when creating drawings;

It is possible to perform logical operations of addition, subtraction and intersection of objects.

In Fig.10.3, as an illustration, the results of a section by a plane various types three-dimensional models of a parallelepiped: wireframe, surface and solid.

Rice. 10.3. Plane Sections of Different Types of 3D Models

This illustration shows that with the help of three-dimensional models it is possible to obtain cuts and sections, which must be done when creating product drawings.

The principle of creating a complex object model is based on the sequential execution of three logical (boolean) operations with solid models (Fig. 10.4): hybrid model , which is a combination of a surface-constrained model and a volumetric solid model, which allows you to use the advantages of both models.

The advantages of solid and hybrid models are the main reason for their widespread use in creating three-dimensional models of objects, despite the need to perform a large number of calculations and, accordingly, the use of computers with large memory and high speed.

The result of geometric modeling of some object is a mathematical model of its geometry. The mathematical model allows you to graphically display the simulated object, obtain its geometric characteristics, study many of the physical properties of the object by setting up numerical experiments, prepare production and, finally, manufacture the object.

In order to see what an object looks like, you need to simulate the flow of light rays falling and returning from its surfaces. In this case, the faces of the model can be given the necessary color, transparency, texture and other physical properties. The model can be illuminated from different directions with light of different colors and intensities.

The geometric model allows you to determine the mass-centering and inertial characteristics of the designed object, to measure the lengths and angles of its elements. It makes it possible to calculate dimensional chains and determine the collectability of the designed object. If the object is a mechanism, then on the model you can check its performance and calculate the kinematic characteristics.

Using a geometric model, it is possible to set up a numerical experiment to determine the stress-strain state, frequencies and forms of natural oscillations, the stability of structural elements, thermal, optical and other properties of the object. To do this, you need to supplement the geometric model with physical properties, simulate the external conditions for its operation and, using physical laws, perform the appropriate calculation.

From the geometric model, you can calculate the trajectory of the cutting tool for machining the object. With the chosen manufacturing technology of the object, the geometric model allows you to design the tooling and prepare for production, as well as to check the very possibility of manufacturing the object in this way and the quality of this manufacturing. In addition, a graphical simulation of the manufacturing process is possible. But in order to make an object, in addition to geometric information, information about technological process, production equipment and much more related to production.

Many of the listed problems form independent branches of applied science and are not inferior in their complexity, and in most cases even surpass the problem of creating a geometric model. The geometric model is the starting point for further actions. When building a geometric model, we did not use physical laws, the radius vector of each point of the interface between the outer and inner parts of the modeled object is known, therefore, when building a geometric model, we have to compose and solve algebraic equations.

Problems that use physical laws lead to differential and integral equations, the solution of which is more difficult than the solution of algebraic equations.

In this chapter, we will focus on performing calculations that are not related to physical processes. We will consider the calculation of purely geometric characteristics of bodies and their flat sections: surface area, volume, center of mass, moments of inertia and orientation of the main axes of inertia. These calculations do not require additional information. In addition, we will consider the problems of numerical integration that have to be solved when determining the geometric characteristics.

Determining the area, center of mass, and moments of inertia of a plane section of a body leads to the calculation of integrals over the area of ​​the section. For plane sections, we have information about their boundaries. We reduce the integrals over the area of ​​a plane section to curvilinear integrals, which, in turn, are reduced to definite integrals. Determining the surface area, volume, center of mass, moments of inertia of the body leads to the calculation of surface and volume integrals. We will rely on the representation of the body with the help of boundaries, i.e., on the description of the body as a set of surfaces limiting it and topological information about the mutual neighborhood of these surfaces. We reduce the integrals over the volume of the body to surface integrals over the surfaces of the faces of the body, which in turn reduce to double integrals. In the general case, the region of integration is a connected two-dimensional region. The calculation of double integrals by numerical methods can be performed for regions of simple types - quadrangular or triangular in shape. In this regard, at the end of the chapter, methods for calculating definite integrals and double integrals over quadrangular and triangular regions are considered. Methods for splitting the areas of definition of surface parameters into a set of triangular subdomains are considered in the next chapter.

At the beginning of the chapter, we consider the reduction of area integrals to curvilinear integrals and the reduction of volume integrals to surface integrals. This will be the basis for calculating the geometric characteristics of the models.

Geometric model A model is a representation of data that most adequately reflects the properties of a real object that are essential for the design process. Geometric models describe objects that have geometric properties. Thus, geometric modeling is the modeling of objects of various nature using geometric data types.

The main milestones in the creation of the mathematical foundations of modern geometric models The invention of the CNC machine - the beginning of the 50s (MIT) - the need to create a digital model of the part The creation of "sculptural surfaces" (the needs of the aircraft and automotive industry) - mathematician Paul de Casteljo proposed for Citroen construct smooth curves and surfaces from a set of control points - future Bezier curves and surfaces - 1959. The results of the work were published in 1974.

A bilinear patch is a smooth surface constructed from 4 points. Bilinear Koons patch (Coons patch) - a smooth surface constructed from 4 boundary curves - author Stephen Koons - MIT professor - 1967 Koons proposed the use of a rational polynomial to describe conic sections Sutherland - a student of Koons developed data structures for future geometric models, proposed a number of algorithms that solve the problem of visualization

Creating a surface that controls the smoothness between boundary curves, Bezier surface - author Pierre Bezier - Renault engineer - 1962. The basis for the development of such surfaces were Hermite curves and surfaces, described by the French mathematician - Charles Hermite (mid-19th century)

The use of splines (curves, the degree of which is not determined by the number of reference points on which it is built) in geometric modeling. Isaac Schoenberg (1946) gave a theoretical description of them. Carl de Boer and Cox considered these curves in relation to geometric modeling - their name is B-splines - 1972.

Use of NURBS (Rational B-Splines on a Non-uniform Parameterization Mesh) in Geometric Modeling - Ken Verspril (Syracuse University), then at Computervision - 1975 NURBS first used Rosenfeld in the Alpha 1 and Geomod modeling system - 1983 Ability to describe all types of conic sections using rational B-splines - Eugene Lee - 1981 This solution was found during the development of the TIGER CAD system used in the Boeing aircraft manufacturer. This company proposed to include NURBS in the IGES format Development of parametrization principles in geometric modeling, introduction of the concept of features (future) - S. Heisberg. Pioneers - PTC (Parametric Technology Corporation), the first system to support parametric modeling - Pro/E -1989

Mathematical knowledge necessary to study geometric models Vector algebra Matrix operations Shapes mathematical representation curves and surfaces Differential geometry of curves and surfaces Approximation and interpolation of curves and surfaces Information from elementary geometry in the plane and in space

Classification of geometric models by information richness By information richness Wireframe (wire) Wireframe Surface solid model or solid model

Classification of geometric models according to internal representation By internal representation Boundary representation –B-rep -analytical description - shell Structural model - construction tree Structure + boundaries

Classification according to the method of formation According to the method of formation Rigid-dimensional modeling or with an explicit geometry specification - specifying the shell Parametric model Kinematic model (lofting, sweep, Extrude, revolve, stretched, sweeping) Constructive geometry model (using basic shape elements and Boolean operations on them - intersection, subtraction, union) Hybrid model

Methods for constructing curves in Geometric Modeling Curves are the basis for creating a three-dimensional surface model. Methods for constructing curves in geometric modeling: Interpolation - Hermite curves and cubic splines Approximation - Bezier curves, Spline curves, NURBS curves

Basic Surface Modeling Methods Analytical Surfaces Plane Polygonal Meshes Quadratic Surfaces – Conic Sections Surfaces Built by Points Polygonal Meshes Bilinear Surface Linear and Bicubic Koons Surface Bézier Surfaces B-Spline Surfaces NURBS Surfaces Triangular Surfaces Kinematic Surfaces Revolving Surface Connection Surface Sweeping Surface Complex sweep and lofting surfaces

Solid model When modeling solid bodies, topological objects are used that carry topological and geometric information: Face; Edge; Vertex; Cycle; Shell The basis of a solid body is its shell, which is built on the basis of surfaces.

Methods of solid modeling: explicit (direct) modeling, parametric modeling. Explicit modeling 1. Constructive geometry model - using BEF and Boolean operations. 2. Kinematic principle of construction. 3. Explicit shell modeling. 4. Object-oriented modeling - the use of features.

Geometry based on structural and technological elements (features) (object-oriented modeling) FEATURES are single or composite structural geometric objects that contain information about their composition and are easily changed during the design process (chamfers, ribs, etc.) depending on the input in geometric model of change. FEATURES are parameterized objects linked to other elements of the geometric model.

Surface and solid models based on the kinematic principle Rotation Simple movement - extrusion Blending two profiles Simple movement of a profile along a curve Movement of a profile along a curve with its change in the section plane

Examples of rigid bodies built according to the kinematic principle 1. Mixing profiles according to a certain law (quadratic, cubic, etc.)

Parametric Models A parametric model is a model represented by a set of parameters that establish the relationship between the geometric and dimensional characteristics of the modeled object. Types of parametrization Hierarchical parametrization variational Parametrization Geometric or dimensional parametrization Tabular parametrization

Hierarchical parametrization Parametrization based on the history of constructions is the first parametric model. History turns into a parametric model if certain parameters are associated with each operation. During model building, the entire building sequence, such as the order in which geometric transformations were performed, is displayed as a construction tree. Making changes at one of the stages of modeling leads to a change in the entire model and the construction tree.

Disadvantages of hierarchical parameterization ü The introduction of cyclic dependencies in models will lead to the system refusing to create such a model. ü Editing capabilities of such a model are limited due to the lack of a sufficient degree of freedom (the ability to edit the parameters of each element in turn) ü Complexity and opacity for the user ü The construction tree can be very complex, recalculating the model will take a lot of time ü Deciding which parameters to change occurs only during construction ü Inability to apply this approach when working with heterogeneous and legacy data

Hierarchical parametrization can be attributed to rigid parametrization. With rigid parametrization, all links are completely specified in the model. When creating a model using rigid parameterization, it is very important to determine the order and nature of the superimposed relationships that will control the change in the geometric model. Such connections are most fully reflected in the construction tree. Rigid parametrization is characterized by the presence of cases when, with a change in the parameters of the geometric model, the solution cannot be obtained at all. found, because some of the parameters and established relationships are in conflict with each other. The same can occur when changing individual stages of the construction tree Using the construction tree when creating a model leads to the creation of a model based on history, this approach to modeling is called procedural

Parent/Child relationship. The basic principle of hierarchical parameterization is the fixation of all stages of model building in the construction tree. This is the definition of the Parent/Child relationship. When a new feature is created, all other features referenced by the feature being created become its Parents. Changing a parent feature changes all of its children.

Variational parametrization Creating a geometric model using constraints in the form of a system of algebraic equations that determines the relationship between the geometric parameters of the model. An example of a geometric model built on the basis of variational parametrization

An example of creating a parametric sketch model by means of variational parametrization in Pro / E The presence of a symbolic designation for each dimension allows you to set the ratio of dimensions using mathematical formulas.

Geometric parametrization is based on the recalculation of the parametric model depending on the geometric parameters of the parent objects. Geometric parameters affecting the model built on the basis of geometric parametrization ü Parallelism ü Perpendicularity ü Tangency ü Concentricity of circles ü Etc. Geometric parametrization uses the principles of associative geometry

Geometric and variational parametrization can be attributed to soft parametrization. Why? soft parametrization is a method of constructing geometric models, which is based on the principle of solving nonlinear equations that describe the relationship between the geometric characteristics of an object. Relationships, in turn, are specified by formulas, as in the case of variational parametric models, or by geometric ratios of parameters, as in the case of models created on the basis of geometric parametrization. The method of constructing a geometric model using variational and geometric parametrization is called - declarative

Tabular parameterization Creation of a table of parameters of typical parts. The generation of a new type object is made by selecting from the table of standard sizes. Example of a type table created in Pro/E

The concept of indirect and direct editing Indirect editing involves the presence of a construction tree for a geometric model - editing occurs inside the tree. Direct editing involves working with the boundary of a solid body, i.e. with its shell. Model editing not based on the construction tree, but as a result of changing the components of the rigid body shell

Geometric Modeling Kernels The geometric modeling kernel is a set of software tools for constructing three-dimensional geometric models based on mathematical methods for their construction. ACIS - Dassault System - Parasolid Boundary Representation - Unigraphics Solution - Granite Boundary Representation - Used by Pro/E and Creo - Supports 3D parametric modeling

Basic components of geometric modeling kernels Data structure for modeling - constructive representation - constructive geometry model or boundary representation - B-rep model. Mathematical apparatus. Visualization tools. A set of interfaces - API (Application Programming Interface)

Methods for creating geometric models in modern CAD Methods for creating models based on three-dimensional or two-dimensional blanks (basic form elements) - creating primitives, Boolean operations Creating a volumetric body or surface model according to the kinematic principle - sweeping, lofting, sweeping, etc. Often used parametrization principle Modification of bodies or surfaces by blending, rounding, extrusion Boundary editing methods - manipulation of the components of volumetric bodies (vertices, edges, faces, etc.). Used to add, remove, modify elements of a solid or flat figure. Methods for modeling the body using free forms. Object-oriented modeling. Using structural elements of the form - features (chamfers, holes, fillets, grooves, notches, etc.) (example, make such and such a hole in such and such a place)

Tasks solved by CAD systems of various levels 1. Solution of problems of the basic level of design, parameterization is either absent or implemented at the lowest, simplest level 2. They have a fairly strong parameterization, are focused on individual work, it is impossible for different developers to work together on one project at the same time. 3. Allow to implement parallel work of designers. Systems are built on a modular basis. The entire cycle of work is carried out without loss of data and parametric connections. The main principle is end-to-end parametrization. In such systems, it is allowed to change the model of the product and the product itself at any stage of work. Support at any level life cycle products. 4. Problems of creating models of a narrow area of ​​use are solved. All can be implemented possible ways creating models

Classification of modern CAD systems Classification parameters degree of parameterization Functional richness Applications (aviation, automotive, instrumentation) Modern CAD 1. low level(small, light): Auto. CAD, Compass, etc. 2. Intermediate level(s): Pro Desktop, Solid Works, Power Shape, etc. 3. High level(large, heavy): Pro/E, Creo (PTC), Catia, Solid Works (Dassault Systemes), Siemens PLM Software (NX Unigraphics) 4. Specialized: SPRUT, Icem Surf, CAD used in specific industries - MCAD, ACAD , ECAD

CAD examples of various levels Low level - Auto. CAD, Compass Intermediate - Inventor (Autodesk), Solid Edge (Siemens), Solid Works (Dassault System), T-Flex - Top Systems company High level - Pro/E-Creo Parametric(PTC), CATIA(Dassault System) ), NX(Unigraphics –Siemens PLM Software) Specialized – SPRUT, Icem Surf(PTC)

The main concepts of modeling at present 1. Flexible engineering (flexible design): ü ü Parameterization Designing surfaces of any complexity (freestyle surfaces) Inheritance of other projects Goal-dependent modeling 2. Behavioral modeling ü ü ü Creation of intelligent models (smart models) - creation of models adapted to the development environment. In a geometric model, m. intellectual concepts are included, for example, features Inclusion in the geometric model of requirements for the manufacture of the product Creating an open model that allows it to be optimized 3. Using the ideology of conceptual modeling when creating large assemblies ü ü Using associative links (set of associative geometry parameters) Separation of model parameters at various stages of design assembly