The main types of geometric models. Electronic geometric model of an object in design Benefits of solid modeling

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Geometric modeling systems

Geometric modeling systems allow you to work with shapes in three-dimensional space. They were created in order to overcome the problems associated with the use of physical models in the design process, such as the difficulty of obtaining complex shapes with accurate dimensions, as well as the difficulty of extracting the necessary information from real models to accurately reproduce them.

These systems create an environment similar to that in which physical models are created. In other words, in a geometric modeling system, the developer changes the shape of the model, adds and removes parts of it, detailing the shape of the visual model. The visual model may look the same as the physical one, but it is intangible. However, a three-dimensional visual model is stored in a computer along with its mathematical description, which eliminates the main drawback of a physical model - the need to perform measurements for subsequent prototyping or mass production. Geometric modeling systems are divided into wireframe, surface, solid and non-shaped.

Wireframing systems

In wireframe modeling systems, a shape is represented as a set of lines and endpoints characterizing it. Lines and points are used to represent three-dimensional objects on the screen, and reshaping is done by changing the position and size of lines and points. In other words, the visual model is a wireframe drawing of the shape, and the corresponding mathematical description is a set of curve equations, point coordinates, and curve-to-point connectivity information. Connectivity information describes the belonging of points to specific curves, as well as the intersection of curves with each other. Wireframe systems were popular back when GM was in its infancy. Their popularity was due to the fact that in wireframing systems, the creation of forms was carried out through a series of simple steps, so that it was quite easy for users to create forms on their own. However, a visual model consisting of lines alone can be ambiguous. Moreover, the corresponding mathematical description does not contain information about the internal and external surfaces of the modeled object. Without this information, it is not possible to calculate the object's mass, determine motion paths, or generate a mesh for finite element analysis, even though the object appears to be three-dimensional. Since these operations are an integral part of the design process, wireframe modeling systems have been gradually superseded by surface and solid modeling systems.

Surface modeling systems

In surface modeling systems, the mathematical description of the visual model includes not only information about characteristic lines and their end points, but also data about surfaces. When working with the model displayed on the screen, surface equations, curve equations, and point coordinates change. The mathematical description may include information about the connectivity of surfaces - how the surfaces connect to each other and along what curves. In some applications, this information can be very useful.

There are three standard methods for creating surfaces in surface modeling systems:

1) Interpolation of input points.

2) Interpolation of curved points.

3) Translation or rotation of a given curve.

Surface modeling systems are used to create models with complex surfaces, because the visual model allows you to evaluate the aesthetics of the project, and the mathematical description allows you to build programs with accurate calculations of motion paths.

Solid modeling systems

Are intended for work with the objects consisting of the closed volume, or a monolith. In solid modeling systems, unlike wireframe and surface modeling systems, it is not allowed to create a set of surfaces or characteristic lines if they do not form a closed volume. The mathematical description of the object created in the solid modeling system contains information by which the system can determine where the line or point is located: inside the volume, outside it or on its border. In this case, any information about the volume of the body can be obtained, which means that applications that work with the object at the volume level, and not on surfaces, can be used.

However, solid modeling systems require more input data compared to the amount of data that gives a mathematical description. If the system required the user to enter all the data for a complete mathematical description, it would become too complicated for users and they would abandon it. Therefore, the developers of such systems try to present simple and natural functions so that users can work with three-dimensional forms without going into the details of the mathematical description.

The modeling functions supported by most solid modeling systems can be divided into five main groups:

1) Functions for creating primitives, as well as functions for adding, subtracting volume - Boolean operators. These features allow the designer to quickly create a shape close to the final shape of the part.

2) Functions for creating three-dimensional bodies by moving the surface. The sweep function allows you to create a three-dimensional body by translation or rotation of an area specified on a plane.

3) Functions designed primarily to modify an existing form. Typical examples are the fillet or blend and lift functions.

4) Functions that allow you to directly manipulate the components of volumetric bodies, that is, along vertices, edges and faces.

5) Functions, using which the designer can model a rigid body using freeforms.

Few Simulation Systems

Solid modeling systems allow the user to create bodies with a closed volume, that is, in mathematical terms, bodies that are manifolds. In other words, such systems prohibit the creation of structures that are not manifold. Violations of the diversity condition are, for example, the tangency of two surfaces at one point, the tangency of two surfaces along an open or closed curve, two closed volumes with a common face, edge or vertex, as well as surfaces that form structures like honeycombs.

The prohibition on the creation of small-sized models was considered one of the advantages of solid modeling systems, since thanks to this, any model created in such a system could be manufactured. If the user wants to work with the geometric modeling system throughout the development process, this advantage turns into another side.

An abstract model with a mixture of dimensions is convenient because it does not constrain the designer's creative thought. A model with mixed dimensions can contain free edges, layered surfaces, and volumes. An abstract model is also useful in that it can serve as a basis for analysis. At each stage of the design process, different analytical tools can be applied. For example, by the finite element method, directly on the initial representation of the model, which allows you to automate the feedback between the stages of design and analysis, which is currently implemented by the designer independently. Small-sized models are indispensable as a stage in the development of a project from an incomplete description at low levels to a finished three-dimensional body. Diversified modeling systems allow you to use wireframe, surface, solid and honeycomb models simultaneously in the same modeling environment, expanding the range of available models.

Description of surfaces

An important component of geometric models is the description of surfaces. If the surfaces of the part are flat faces, then the model can be expressed quite simply with certain information about the faces, edges, and vertices of the part. In this case, the method of constructive geometry is usually used. Representation using flat faces also takes place in the case of more complex surfaces, if these surfaces are approximated by sets of flat sections - polygonal meshes. Then the surface model can be specified in one of the following forms:

1) the model is a list of faces, each face is represented by an ordered list of vertices (vertex cycle); this form is characterized by significant redundancy, since each vertex is repeated in several lists;

2) the model is a list of edges, each edge has incident vertices and faces. However, approximation by polygonal meshes at large mesh cell sizes gives noticeable shape distortions, and at small cell sizes it turns out to be inefficient in terms of computational costs. Therefore, descriptions of non-planar surfaces by cubic equations in the form of Bezier or 5-splines are more popular.

It is convenient to get acquainted with these forms by showing their application for describing geometric objects of the first level - spatial curves.

Note. Geometric objects of zero, first and second levels are respectively called points, curves, surfaces.

The MGIGM subsystems use parametrically defined cubic curves

geometric structural modeling surface

x(t) = axt3 + bxt2 + cxt + dx ;

y(t) = ay t3 + X by t2 + cy t + dy ;

z(t) = a.t3 + b_t2 + cj + d_,

where 1 > t > 0. Such curves describe the segments of the curve being approximated, i.e., the curve being approximated is divided into segments and each segment is approximated by equations (3.48).

The use of cubic curves provides (by an appropriate choice of four coefficients in each of the three equations) the fulfillment of four conditions for conjugation of segments. In the case of Bezier curves, these conditions are the passage of the segment curve through two given end points and the equality at these points of the tangent vectors of neighboring segments. In the case of 5-splines, the conditions for the continuity of the tangent vector and curvature (i.e., the first and second derivatives) at two end points are satisfied, which ensures a high degree of smoothness of the curve, although the passage of the approximating curve through the given points is not ensured here. The use of polynomials higher than the third degree is not recommended, since the likelihood of waviness is high.

In the case of the Bezier form, the coefficients in (3.48) are determined, firstly, by substituting into (3.48) the values ​​(=0k(=1) and the coordinates of the given end points Р, and Р4, respectively, and secondly, by substituting the derivatives into the expressions

dx / dt \u003d For t2 + 2b + c, X X x "

dy/dt = Za, G2 + 2byt + s,

dz/dt = 3a.t2 + 2b.t + c.

the same values ​​/ \u003d 0 and / \u003d 1 and the coordinates of the points P2 and P3, which specify the directions of the tangent vectors (Fig. 3.27). As a result, for the Bezier form we get

Bezier curve. (3.27)

for which the matrix M has a different form and is presented in Table. 3.12, and the vectors Gx, Gy, G contain the corresponding coordinates of the points P, 1; P, P, + 1, P, + 2.

Let us show that at the conjugation points for the first and second derivatives of the approximating expression, the continuity conditions are satisfied, which is required by the definition of a B-spline. Let us denote the segment of the approximating B-spline corresponding to the segment [Р, Р +1] of the original curve by . Then for this section and the coordinate x at the conjugation point Q / + , we have t = 1 and

For the segment at the same point Qi+| we have t = 0 and

i.e., the equality of derivatives at the conjugation point in neighboring sections confirms the continuity of the tangent vector and curvature. Naturally, the x value of the x coordinate of the point Qi+1 of the approximating curve on the segment .

is equal to the value of x calculated for the same point on the section , but the values ​​of the coordinates of the nodal points x and x+] of the approximating and approximated curves do not match.

Similarly, one can obtain expressions for Bezier forms and 5-splines as applied to surfaces, taking into account that instead of (3.48) cubic dependences on two variables are used.

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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 of 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. The list of information needed to fully describe the circle is as follows:

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, as a rule, there is no way back - raster tracing usually does not provide high quality vector drawing.

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 are different types of curves (Catmull-Rom splines, NURBS, etc.) that are used in different 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 done 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

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, sections 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).

wireframe model represents a three-dimensional image of an object in the form of lines of intersection of the object's faces. As an example, Figure 10.1 shows the wireframe and data structure of a computer model of the tetrahedron's internal calculations.

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 plane section of various types of three-dimensional models of a parallelepiped are shown: 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 take advantage of 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.

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 at intermediate stages it is useful to work simultaneously with 3D and 2D models 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 geometric 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 the same 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.

To solve the problems of complex automation of machine-building industries, it is necessary to build information models of products. A machine-building product as a material object should be described in two aspects:

Like a geometric object;

Like a real physical body.

The geometric model is necessary to set the ideal shape that the product should correspond to, and the model of the physical body must characterize the material from which the product is made, and the permissible deviations of real products from the ideal shape.

Geometric models are created using geometric modeling software, and physical body models using database creation and maintenance tools.

A geometric model, as a kind of mathematical model, covers a certain class of abstract geometric objects and relationships between them. A mathematical relation is a rule that links abstract objects. They are described using mathematical operations that associate one (unary operation), two (binary operation) or more objects, called operands, with another object or set of objects (the result of the operation).

Geometric models are created, as a rule, in the right rectangular coordinate system. The same coordinate systems are used as local ones when defining and parameterizing geometric objects.

Table 2.1 shows the classification of basic geometric objects. According to the dimension of parametric models required to represent geometric objects, they are divided into zero-dimensional, one-dimensional, two-dimensional and three-dimensional. Zero-dimensional and one-dimensional classes of geometric objects can be modeled both in two coordinates (2D) on the plane, and in three coordinates (3D) in space. 2D and 3D objects can only be modeled in space.

SPRUT language for geometric modeling of engineering products and design of graphic and text documentation

There are a significant number of computer geometric modeling systems, the most famous of which are AutoCAD, ANVILL, EUCLID, EMS, etc. Of the domestic systems of this class, the most powerful is the SPRUT system designed to automate the design and preparation of control programs for CNC machines.

Zero-dimensional geometric objects

On surface

Point on the plane

Point on the line

A point given by one of the coordinates and lying on a straight line

In space

point in space

Point defined by coordinates in the base system

P3D i = Xx,Yy,Zz

Point on the line

Point, specified as the nth point of the space curve

P3Di = PNT,CCj,Nn

Point on the surface

Point, specified as the intersection point of three planes;

P3D i = PLs i1,PLs i2,PLs i3

Table 2.1 Geometric objects in the octopus environment

Object dimension	Dimension of space	Object type	SPRUT operator
	Flat(2D)	Points on the plane	Pi = Xx, Yy; Pi = mm, aa
	[SGR subsystem]	Dots on the line	Pi = Xx, Li; Pi = Ci, Aa
In space(3D)	Points in space	P3D i = Xx,Yy, Zz
[GM3 subsystem]	Dots on the line	P3Di = PNT,CCj,Nn
Points on the surface	P3Di = PLSi1,PLSi2,PLSi3
	Flat(2D)
	[SGR subsystem]	circles
	Ki = Pj, -Lk, N2, R20, Cp, Pq
	Ki = Mm, Lt, Pj, Pk,..., Pn, Cq
2nd order curves	CONIC i = P i1, P i2, P i3, ds
In space(3D) [GM3 subsystem]		P3D i = NORMAL, CYL j, P3D k; P3D i = NORMAL, Cn j, P3D k; P3D i = NORMAL, HSP j, P3D k; P3D i = NORMAL, TOR j, P3D k
	L3D i = P3D j,P3D k
	CC i = SPLINE,P3D i1,...,P3D j,Mm
Parametric curve on a surface	CC n = PARALL, BASES=CCi, DRIVES=CCk, PROFILE=CCp, STEPs
Surface Intersection Lines	SLICE K i, SS j, Nk, PL l; INTERS SS i, SS j, (L,) LISTCURV k
Projection of a line onto a surface	PROJEC Ki, CC j, PLS m
Wire models	SHOW CYL i; SHOW HSP i; SHOW CNi; SHOW TOR i
two-dimensional	In space [GM3 subsystem]	planes	PL i = P3D j,L3D k
cylinders	CYL i = P3D j,P3D k,R
	CN i = P3D j,R1,P3D k,R2; CN i = P3D j,R1,P3D k,Angle
	HSP i = P3D j,P3D k,R
	TOR i = P3D j,R1,P3D k,R1,R2
Surfaces of revolution	SS i = RADIAL, BASES = CC j, DRIVES = CC k, STEP s
Ruled surfaces	SS i = CONNEC, BASES = CC j, BASES = CC k, STEP s
Shaped surfaces	SS i = PARALL, BASES = CC j, DRIVES = CC k, STEP s
Tensor product surfaces
three-dimensional	In space [SGM subsystem]	Body of rotation	SOLID(dsn) = ROT, P3D(1), P3D(2), SET, P10, m(Tlr)
Shear body	SOLID(dsn) = TRANS, P3D(1), P3D(2), SET, P10, M(Tlr)
The body is cylindrical	SOLID(dsn) = CYL(1), M(Tlr)
Body conical	SOLID(dsn) = CN(1), M(Tlr)
The body is spherical	SOLID(dsn) = SPHERE(1), M(Tlr)
Body toric	SOLID(dsn) = TOR(1), M(Tlr)

One-dimensional geometric objects

On surface

Vectors Transfer vector MATRi = TRANS x, y

Lines Simple analytical

Direct (total 10 ways to set)

A straight line passing through two given points Li = Pi, Pk

Circle (total 14 ways to set)

Circle given by center and radius Ci = Xx, Yy, Rr

Curve of the second order (total 15 ways to set)

Curve of the second order passing through three points with a given discriminant Conic i = P i1, P i2, P i3, ds

Compound Contours - a sequence of segments of flat geometric elements, starting and ending with points lying on the first and last element, respectively K23 = P1, -L2, N2, R20, C7, P2 Piecewise polynomial

Spline. The first parameter in the operator is the identifier "M", which indicates the amount of deviation when fitted by spline curve segments. This is followed by the initial condition (line or circle), then enumeration of the points in the sequence in which they must be connected. The statement ends with the definition of the condition at the end of the spline curve (straight line or circle) Ki = Mm, Lt, Pj, Pk,..., Pn, Cq

Approximation by arcs Ki = Lt, Pj, Pk,..., Pn

In space Vectors Direction vector

Vector of the unit normal at a point to the hemisphere P3D i = NORMAL,HSP j,P3D k Vector of the unit normal at a point to the cylinder P3D i = NORMAL,CYL j,P3D k Vector of the unit normal at a point to the cone P3D i = NORMAL, Cn j,P3D k Unit normal vector at a point to the torus P3D i = NORMAL,TOR j,P3D k Transfer vector MATRi = TRANS x, y, z Lines

Independent Direct (total 6 ways to set)

By two points L3D i = P3D j,P3D k Spline curve CC i = SPLINE,P3D i1,.....,P3D j,mM Surface Parametric CC n=PARALL,BASES=CCi,DRIVES=CCk,PROFILE= CCp,STEPs Intersection of 2 surfaces Contour of a surface section by a plane SLICE K i, SS j, Nk, PL l ,LISTCURV k ; where L is the level of accuracy; 3<= L <= 9;

Projections onto a surface Projection of a spatial curve onto a plane with the PROJEC coordinate system Ki,CC j,PLS m.

Composite

Wireframe Wireframe Cylinder Wireframe Screen Display SHOW CYL i Hemisphere Wireframe Screen Display SHOW HSP i

Wireframe Cone Display SHOW CN i

Displaying the torus on the screen as a wire model SHOW TOR

2D Geometric Objects (Surfaces)

Simple analytical Plane (total 9 ways to set)

By point and line PL i = P3D j,L3D k

Cylinder (by two points and radius) CYL i = P3D j,P3D k,R

Cone Defined by two points and two radii; or by two points, radius and angle at the vertex CN i = P3D j,R1,P3D k,R2; CN i = P3D j,R1,P3D k,Angle

Sphere (hemisphere) Set by two points and radius HSP i = P3D j,P3D k,R

Tor Defined by two points and two radii; the second point together with the first determines the axis of the torus TOR i = P3D j,R1,P3D k,R1,R2

Composite Kinematic Surfaces of Revolution SS i = RADIAL, BASES = CC j, DRIVES = CC k, STEP s

Ruled surfaces SS i = CONNEC, BASES = CC j, BASES = CC k, STEP s

Shaped surfaces SS i = PARALL, BASES = CC j, DRIVES = CC k, STEP s

Piecewise Polynomial Tensor Product Surfaces (spline surfaces by point system) CSS j = SS i

Table 2.2 Geometric operations in the octopus environment

			OPERATOR SPRUT
	Transformations	Scaling
		MATRI = TRANS x, y, z
	Rotation	MATRi = ROT, X Y Z, Aa
	Display	MATRI = SYMMETRY, Pli
projections	Parallel	VECTOR P3Di, INTO P3Dj
		L = SURFAREA
parameters		S=SURFAREA
	S=SURFAREA
			S=AREA
			VS = VOLUME
		Moment of inertia	SURFAREA
			SURFAREA
			INERC SOLID i,L3d i1,INLN
			INERC SOLID i, P3Dj
		Center of mass	CENTER SOLID i,P3D j
			SURFAREA
BINARY	Parameter calculations	Distance	S = DIST P3Di, P3Dj
		S = DIST P3Di, L3Dj
		S = DIST P3Di, Pl j
			S = DIST P3Di, SS j
S = DIST P3Di, P3Dj
			Ang = SURFAREA
	intersection	two lines	Pi = Li, Lj; Pi = Li, Cj;
		Pi = Ki, Lt, Nn; Pi = Ki, Ct, Nn;
		Pi = Ki, Kt, Nn; Pi = Ki, Lt, Nn
		P3D i = L3D j,PL k
	surface	P3D i = L3D j,HSP k,n
			P3D i = L3D j,CYL k,n
			P3D i =L3D j,CN k,n; P3D i =CC i ,PL j
	L3D i = PL j, PL k
surfaces	INTERS SS i,SS j,(L,)LISTCURV k
	CROS SOLID(Top+2), RGT, SOLID(Top+3), RGT;
Subtraction	Bodies from the body	CROS SOLID(Top+2), RGT, SOLID(Top+3); SOLID(Top+1) = SOLID(Top+2), SOLID(Top+3)
Addition		CROS SOLID(Top+2), SOLID(Top+3); SOLID(Top+1) = SOLID(Top+2), SOLID(Top+3)
clipping	Body plane	CROS SOLID(Top+1), PL(1), SET
	An association	two surfaces	SSi=ADDUP,SSk,SSj,STEPs,a Angl
	An association	Merging surfaces	SS i = ADDUP,SS k,....., SS j,STEP s ,a Angl

Methods for presenting and transmitting information about the geometric shape of the product

The initial data on the geometric shape of the product can enter the CAM system in the Boundary Representation (B-Rep) format. Let's study this format in more detail.

The author considered the data structures of the ACIS geometric kernel from Spatial Technology, the Parasolid geometric kernel from Unigraphics Solutions, the Cascade geometric kernel from Matra Datavision, and the representation of the model in the IGES specification. In all four sources, the representation of the model is very similar, there are only slight differences in terminology, in the ACIS core there are non-principled data structures associated with the optimization of computational algorithms. The minimum list of objects required to represent the B-Rep model is shown in Fig. 1. It can be divided into two groups. The left column represents geometric objects, while the right column represents topological objects.

Rice. 1. Geometric and topological objects.

Geometric objects are the surface (Surface), curve (Curve) and point (Point). They are independent and do not refer to other components of the model, they determine the spatial arrangement and dimensions of the geometric model.

Topological objects describe how geometric objects are connected in space. Topology itself describes a structure or grid that is in no way fixed in space.

Curves and surfaces. As is known, there are two most general methods of representing curves and surfaces. These are implicit equations and parametric functions.

Implicit equation of a curve lying in a plane xy looks like:

This equation describes the implicit relationship between the x and y coordinates of the points on the curve. For a given curve, the equation is unique. For example, a circle with unit radius and center at the origin is described by the equation

In parametric form, each of the curve point coordinates is represented separately as an explicit function of the parameter:

Vector function of parameter u.

Although the interval is arbitrary, it usually normalizes to. The first quadrant of the circle is described by parametric functions:

Install, get a different view:

Thus, the representation of a curve in a parametric view is not unique.

The surface can also be represented by an implicit equation in the form:

The parametric representation (not unique) is given as:

Note that two parameters are needed to describe the surface. The rectangular region of existence of the entire set of points (u, v), limited by the conditions, will be called the region or the parameter plane. Each point in the parameter area will correspond to a point on the surface in the model space.

Rice. 2. Parametric specification of the surface.

Fixing u and changing v, we obtain transverse lines by fixing v and changing u, we get longitudinal lines. Such lines are called isoparametric.

To represent curves and surfaces inside a B-Rep model, the parametric form is most convenient.

Topological objects.Body is a bounded volume V in three-dimensional space. The body will be correct if this volume is closed and finite. The body may consist of several pieces (Lumps) that do not touch each other, access to which must be provided as a whole. The figure shows an example of a body consisting of more than one piece.

Rice. 3. Four pieces in one body

A Lump is a single area in 3D space bounded by one or more Shells. A Lump can have an unlimited number of voids. Thus, one shell of a piece is external, the rest are internal.

Rice. 4. The body, consisting of two pieces

Shell- this is a set of limited surfaces (Faces), interconnected by means of common vertices (Vertexes) and edges (Edges). The normals to the surfaces of the shell must be directed from the zone of existence of the body. Limited surface (Face)- this is a section of an ordinary geometric surface, limited by one or more closed sequences of curves - loops (Loops). In this case, the loop can be specified by curves, both in the model and in the parametric space of the surface. A bounded surface is essentially a two-dimensional analog of a body. It can also have one outer and many inner restricted zones.

Rice. 5. Limited surface

Loop - is a section of the Face restriction zone. It is a set of parametric edges united in a doubly connected chain. For a correct body, it must be closed.

A parametric edge (Coedge) is a record corresponding to a section of a loop. It corresponds to the edge of the geometric model. A parametric edge has a link to a 2D geometric curve corresponding to a section of the constraint zone in parametric space. The parametric edge is oriented in the loop in such a way that if you look along the edge in its direction, then the surface existence zone will be located to the left of it. Thus, the outer loop is always directed counterclockwise, and the inner loop is always clockwise.

Parametric edge (Coedge) may have a link to a partner, to the same Coedge lying in another loop, but corresponding to the same spatial edge. Since in a correct body, each edge touches exactly two surfaces, so it will have strictly two parametric edges.

Rice. 6. Edges, parametric edges and vertices

Edge- a topological element that has a reference to a three-dimensional geometric curve. The edge is bounded on both sides by vertices.

Vertex- a topological element that has a link to a geometric point (Point). The vertex is the boundary of the edge. All other edges that come to a particular vertex can be found through the parametric edge pointers.

Rice. 7. Object implementation of the geometric model

There are two more undescribed objects in this diagram.

Body coordinate system (Transform). As is known, the coordinate system can be specified by a transformation matrix. Dimension of the matrix. If the coordinates of a point are represented as a row vector, in the last column of which there is one, then multiplying this vector by the transformation matrix, we obtain the coordinates of the point in the new coordinate system.

The matrix can reflect in itself all spatial transformations, such as: rotation, translation, symmetry, scaling and their compositions. As a rule, the matrix has the following form.

Dimensions (Box)- data structure describing the parameters of a rectangular parallelepiped with sides parallel to the coordinate axes. In fact, these are the coordinates of two points located at the ends of the main diagonal of the parallelepiped.

NURBS curves and surfaces

Currently, the most common way to represent curves and surfaces in parametric form is rational splines or NURBS (non-uniform rational b-spline). In the form of NURBS, such canonical forms as a segment, a circular arc, an ellipse, a plane, a sphere, a cylinder, a torus, and others can be represented with absolute accuracy, which allows us to talk about the universality of this format, and eliminates the need to use other representation methods.

The curve in this form is described by the following formula:

W(i) - weight coefficients (positive real numbers),

P(i) - control points,

Bi - B-spline functions

B-spline functions of degree M are completely determined by the set of knots. Let N=K-M+1, then the set of nodes is a sequence of non-decreasing real numbers:

T(-M),…,T(0),…,T(N),…T(N+M).

Rice. 8. (a) cubic basis functions; (b) cubic curve using basis functions with (a)

A curve segment represented as NURBS can be converted to polynomial form without loss of precision, i.e. represented by the expressions:

where and are polynomials of the degree of the curve. Methods for converting curves from NURBS to polynomial form and vice versa are described in detail in /1/.

NURBS surfaces are represented in a similar way:

Rice. 9. B-spline surface: (a) grid of control points; (b) surface

As can be seen from the figures, the complexity of the geometric shape of a curve or surface can be estimated from control points.

A NURBS surface segment can also be represented in polynomial form:

where and are polynomials of two variables and can be represented as:

The properties of NURBS curves and surfaces are described in more detail in /1,2/.

For any two-dimensional parametric curve, where, and are polynomials, there is an equation, where is also a polynomial, that exactly defines the same curve. For any parametric surface given by expression (6) there is an equation, where is also a polynomial, which exactly defines the same surface. Methods for obtaining an implicit form of a parametrically defined curve or surface are described in /33/.

Geometric Model Transfer Standards

For end-to-end automation of the production preparation process, it is necessary to use CAD systems in design departments and CAM systems in technological ones. If the design is carried out at one enterprise, and the production is carried out at another, options for using different software are possible. In this case, the main problem is the incompatibility of the formats of the geometric model of systems from different companies. Most often, to solve this problem, the designer generates the entire set of technical documentation in paper form, and the manufacturer, using the received drawings, restores the electronic model of the product. This approach is very time-consuming and negates all the advantages of automating individual stages. The solution of such problems is carried out either by means of a converter program, or by bringing the data to a single standard.

One such standard is IGES (Initial Graphics Exchange Specification). This standard provides for the transfer of any geometric information, including analytical and NURBS surfaces and solid models in B-Rep representation. Currently, the IGES standard is generally recognized and provides the transfer of any geometric information. It is supported by all the most advanced computer-aided design and production systems. Nevertheless, to solve some production problems, the transmission of only geometric information is not enough. It is necessary to store all information about the product throughout its entire life cycle. The transfer of such information can be carried out using the completely new ISO 10303 STEP standard, which is a direct development of IGES. However, in Russia there is practically no demand for STEP-compatible systems. The geometric model can also be transferred in the STL format (format for stereolithography). In this representation, the model is represented as a set of flat triangular faces. However, the representation of the model in this form, despite its obvious simplicity, has a serious drawback associated with a large increase in the amount of memory required to store the model with a slight increase in accuracy.

In addition to these, there are corporate formats for storing and transmitting information about the geometric shape of the product. These include, for example, the XT format of the Parasolid core from Unigraphics Solitions or the SAT format of the ACIS core from Spatial Technology. The key disadvantage of these formats is their focus on the company promoting them, and, accordingly, dependence on it.

Thus, at present, the most acceptable format for transferring geometric information about the shape of a product from one system to another is IGES.