Geometric shapes as models of real objects. Geometric model. Surface modeling systems

Geometric model A model is a data representation 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 different nature using geometric data types.

Classification by the method of formation By the method of formation Rigid-dimensional modeling or with explicit specification of geometry (analytical models) Parametric model Kinematic model (lofting, sweep, Extrude, revolve, stretched, sweeping) Model of constructive geometry (using basic form elements and Boolean operations on them - intersection, subtraction, union) Hybrid model

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. Parameterization types and Hierarchical parameterization Variational (dimensional) parameterization Geometric parameterization Tabular parameterization

Geometry based on structural and technological elements (features) FEECHERS - single or composite structural geometric objects containing information about their composition and easily changed during the design process (chamfers, edges, etc.) into a geometric model of change. FEATURES - parameterized objects that are attached to other elements of the geometric model.

Hierarchical parameterization Parameterization based on construction history. During the construction of the model, the entire construction sequence, for example, the order of the geometric transformations performed, is displayed in the form of a construction tree. Making changes at one of the stages of modeling leads to a change in the entire model and the construction tree. The introduction of cyclic dependencies in the model will lead to the failure of the system to create such a model. The possibilities of editing 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)

Hierarchical parameterization can be referred to as rigid parameterization. With rigid parameterization, all links are fully specified in the model. When creating a model using rigid parameterization, it is very important to determine the order and the nature of the superimposed links that will control the change in the geometric model. Such links are most fully reflected in the construction tree. For rigid parametrization, the presence of cases is characteristic when, when changing the parameters of the geometric model, the solution cannot be solved at all. found because some of the parameters and the established links conflict with each other. The same can occur when changing individual stages of the construction tree.

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

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

Geometric parametrization 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 parameterization Parallelism Perpendicularity Tangency Concentricity of circles Etc. Geometric parametrization uses the principles of associative geometry

Geometric and variational parametrization can be attributed to soft parametrization. Why? soft parameterization is a method for constructing geometric models based on the principle of solving nonlinear equations describing the relationship between the geometric characteristics of the object. The constraints, in turn, are specified by formulas, as in the case of variational parametric models, or geometric relationships of parameters, as in the case of models created on the basis of geometric parameterization.

Methods for creating geometric models in modern CAD Methods for creating models based on three-dimensional or two-dimensional workpieces (basic form elements) - creation of primitives, Boolean operations Creation of a volumetric body or surface model according to the kinematic principle - sweeping, lofting, sweep, etc. The principle of parameterization is often used. Modification of bodies or surfaces by smooth fillet, rounding, extrusion. Editing methods of boundaries - manipulation of the components of solid bodies (vertices, edges, faces, etc.). Used to add, remove, modify elements of a solid or planar shape. Methods for modeling the body using free forms. Object Oriented Modeling. The use of structural elements of the form - features (chamfers, holes, fillets, grooves, notches, etc.) (for example, to make such and such a hole in such and such a place)

Classification of modern CAD systems Classification parameters degree of parameterization Functional richness Application areas (aircraft, automobile, instrumentation) Modern CAD systems 1. Low level (small, light): AutoCAD, Compass, etc. 2. Mid-level (mid-range): 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

Tasks solved by CAD systems of various levels 1. Solution of tasks of the basic level of design, parameterization is either absent, or implemented at the lowest simplest level 2. Have a fairly strong parameterization, 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. The systems are built on a modular basis. The entire cycle of work is performed without loss of data and parametric connections. The basic principle is end-to-end parameterization. In such systems, it is allowed to change the product model and the product itself at any stage of work. Support at any level of the product lifecycle. 4. The tasks of creating models of a narrow area of ​​use are being solved. All possible ways of creating models can be implemented

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

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. This leads to another advantage - when the image is resized, the file size does not change. Raster graphics is a rectangular matrix consisting of many very small indivisible points (pixels).

A raster image can be compared to a children's mosaic, when a picture is composed of colored squares. The computer memorizes the colors of all squares in a row in a specific order. Therefore, bitmaps require more storage space. They are difficult to scale and even more difficult to edit. To enlarge the image, you have to increase the size of the squares, and then the picture is "stepped". To reduce the bitmap image, it is necessary to transform several neighboring points into one or to discard unnecessary points. As a result, the image is distorted, its small details become illegible. Vector graphics are devoid of these shortcomings. In vector editors, a drawing is remembered as a collection geometric shapes- contours presented in the form of mathematical formulas. To proportionally enlarge an object, simply change one number: the scaling factor. No distortion occurs either when enlarging or reducing the picture. Therefore, when creating a drawing, you can not 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 required to fully describe the circle is as follows:

radius r;

coordinates of the center of the circle;

color and thickness of the outline (possibly transparent);

fill color (possibly transparent).

The 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 remains smooth. On the other hand, if the curve is a broken line, magnification will show that it is not actually a curve.

When enlarging or reducing objects, the line thickness 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 state the sizes in device independent units ((English)), which lead to the best possible rasterization on raster devices.

Vector graphics have two fundamental flaws.

Not every object can be easily vectorized. In addition, the amount of memory and display time depends on the number of objects and their complexity.

Converting vector graphics to raster is quite simple. But there is usually no way back - raster tracing usually does not provide high quality vector graphics.

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

More sophisticated transformations include Boolean operations on closed shapes: union, complement, 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 created from Bézier 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 treat a bitmap as a primitive object that behaves like a rectangle.

Basic 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 subdivided into two-dimensional and three-dimensional. Sketches, diagrams, drawings, graphics, paintings are examples of two-dimensional geometric models, and models of buildings, cars, airplanes, etc. Are three-dimensional geometric models.

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

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

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

rotation matrix

shift matrix

scaling matrix

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

When solving most problems in the field of computer-aided design (C) and technological preparation of production (TPP), it is necessary to have a model of the design object.

Under object model understand its some abstract representation that satisfies the condition of adequacy to this object and allows its representation and processing using a computer.

That. model- a set of data displaying the properties of an object and a set of relationships between these data.

Depending on the nature of its execution, the model of the PR object may include a number of various characteristics and parameters. Most often, object models contain data on the shape of an object, its dimensions, tolerances, materials used, mechanical, electrical, thermodynamic and other characteristics, processing methods, cost, as well as microgeometry (roughness, shape deviations, dimensions).

For processing a model in graphic CAD systems, it is not the entire amount of information about the object that is essential, but the part that determines its geometry, i.e. shapes, sizes, spatial arrangement of objects.

The description of an object in terms of its geometry is called geometric model of the object.

But the geometric model can also include some technological and auxiliary information.

Information about the geometric characteristics of an object is used not only to obtain a graphic image, but also to calculate various characteristics of an object (for example, by FEM), to prepare programs for CNC machines.

In the traditional design process, information exchange is carried out on the basis of sketch and working drawings using reference and technical documentation. In CAD, this exchange is implemented on the basis of an in-machine representation of the object.

Under geometric modeling understand the entire multi-stage process - from the verbal (verbal) description of the object in accordance with the task at hand to obtaining an intramachine representation of the object.

In geometric modeling systems, 2-dimensional and 3-dimensional objects can be processed, which in turn can be analytically descriptive and non-descriptive. Analytically indescribable geometric elements such as curves and freeform surfaces are used primarily to describe objects in the automotive, aircraft and shipbuilding industries.

The main types of GM

2-dimensional models that allow you to form and modify drawings, were the first models that have found use. Such modeling is often used to this day, because it is much cheaper (in terms of algorithms, use) and is quite suitable for industrial organizations in solving a variety of problems.

In most 2-D geometric modeling systems, the object is described interactively in accordance with algorithms similar to those of the traditional design method. An extension of such systems is that contours or flat surfaces are assigned a constant or variable image depth. Systems operating on this principle are called 2.5-dimensional. They allow you to obtain axonometric projections of objects in drawings.

But 2D representation is often not convenient for fairly complex products. In traditional design methods (without CAD), drawings are used, where the product can be represented by several views. If the product is very complex, it can be presented as a model. The 3D model is used to create a virtual representation of the product in all 3 dimensions.

There are 3 types of 3-dimensional models:

Frame (wire)

Surface (polygonal)

· Volumetric (models of solid bodies).

Historically, the first wireframe models... They store only the coordinates of the vertices ( x, y, z) and the edges connecting them.

The figure shows how a cube can be perceived ambiguously.

Because only edges and vertices are known, different interpretations of the same model are possible. The wireframe model is simple, but it can only represent in space a limited class of parts in which the approximating surfaces are planes. Based on the wireframe model, you can get projections. But it is impossible to automatically remove hidden lines and get different sections.

· Surface Models allow you to describe fairly complex surfaces. Therefore, they often meet the needs of the industry (aircraft, ship, automotive) when describing complex shapes and working with them.

When building a surface model, it is assumed that objects are bounded by surfaces that separate them from environment... The surface of the object also becomes bounded by contours, but these contours are the result of 2 touching or intersecting surfaces. The vertices of an object can be specified by the intersection of surfaces, a set of points that satisfy some geometric property, in accordance with which the contour is defined.

Various types of surface definition are possible (planes, surfaces of revolution, ruled surfaces). For complex surfaces, various mathematical models of surface approximation are used (methods of Koons, Bezier, Hermite, B-spline). They allow you to change the nature of the surface using parameters, the meaning of which is available to the user who does not have special mathematical background.

Surface approximation general view flat edges gives advantage: for processing such surfaces, simple mathematical methods. Flaw: maintaining the shape and size of an object depends on the number of faces used for approximations. The> the number of faces, the< отклонение от действительной формы объекта. Но с увеличением числа граней одновременно увеличивается и объем информации для внутримашинного представления. Вследствие этого увеличивается как время на работу с моделью объекта, так и объем памяти для хранения модели.

If the distinction between points into internal and external is essential for the model of an object, then they talk about volumetric models... To obtain such models, the surfaces surrounding the object are first determined, and then they are collected in volumes.

Currently, the following methods of constructing volumetric models are known:

· IN boundary models volume is defined as a set of surfaces bounding it.

The structure can be complicated by introducing the actions of transfer, rotation, scaling.

Advantages:

¾ guarantee of generating the correct model,

¾ great opportunities for modeling forms,

¾ fast and efficient access to geometric information (for example, for drawing).

Flaws:

¾ a larger amount of initial data than with the CSG method,

¾ model logically< устойчива, чем при CSG, т.е. возможны противоречивые конструкции,

¾ the complexity of constructing variations of forms.

· IN CSG models an object is defined by a combination of elementary volumes using geometric operations (union, intersection, difference).

Elementary volume is understood as a set of points in space.

A tree structure is the model for such a geometric structure. Nodes (nonterminal vertices) are operations, and leaves are elementary volumes.

Dignity :

¾ conceptual simplicity,

¾ small amount of memory,

¾ design consistency,

¾ the possibility of complicating the model,

¾ simplicity of presentation of parts and sections.

Flaws:

¾ limiting the scope of boolean operations,

¾ computationally intensive algorithms,

¾ impossibility to use parametrically described surfaces,

¾ complexity when working with functions> than 2nd order.

· Cellular method. A limited area of ​​space covering the entire modeled object is considered to be divided into a large number of discrete cubic cells (usually of a single size).

The modeling system should simply record information about the belonging of each cube to the object.

The data structure is represented by a 3-dimensional matrix, in which each element corresponds to a spatial cell.

Advantages:

¾ simplicity.

Flaws:

¾ large amount of memory.

To overcome this drawback, the principle of dividing cells into subcells in especially complex parts of the object and on the border is used.

The volumetric model of the object, obtained by any method, is correct, i.e. in this model there are no contradictions between geometric elements, for example, a line segment cannot consist of one point.

Wireframe m.b. used not in modeling, but when reflecting models (volume or surface) as one of the visualization methods.

geometric model- geometric model; branch. layout A model in relation to the geometric similarity to the modeled object ... Polytechnic Terminological Explanatory Dictionary

geometric model- НПК layout Model, which is in relation to geometric similarity to the modeled object. [A collection of recommended terms. Issue 88. Foundations of the theory of similarity and modeling. USSR Academy of Sciences. Scientific and Technical Terminology Committee. 1973] ... ...

Geometric terrain model- (phototopography) a set of intersection points of the corresponding projection rays, obtained from a stereopair of oriented topographic photographs ... Source: GOST R 52369 2005. Phototopography. Terms and definitions (approved by Order ... ... Official terminology

geometrical terrain model (phototopography)- A set of points of intersection of the corresponding projection rays, obtained from a stereopair of oriented topographic photographs. [GOST R 52369 2005] Topics phototopography General terms, types of topographic photographs and their ... ... Technical translator's guide

geometric terrain model- 37 geometrical model of the terrain (phototopography): A set of intersection points of the corresponding projection rays, obtained from a stereopair of oriented topographic photographs. Source: GOST R 52369 2005: Phototopography. Terms and ... ...

electronic geometric model (geometric model)- electronic geometric model (geometric model): An electronic model of a product that describes geometric shape, dimensions and other properties of the product, depending on its shape and size. [GOST 2.052 2006, article 3.1.2] Source ... Dictionary-reference book of terms of normative and technical documentation

Electronic geometric model of the product- Electronic geometric model (geometric model): an electronic model of the product, describing the geometric shape, dimensions and other properties of the product, depending on its shape and size ... Source: UNIFIED SYSTEM OF DESIGN DOCUMENTATION. ... ... Official terminology

An abstract or real mapping of objects or processes, adequate to the objects (processes) under study in relation to some given criteria. For example, mathematical model of layering (abstract process model), block diagram ... ... Geological encyclopedia

Wireframe model- Wireframe model: a three-dimensional electronic geometric model, represented by a spatial composition of points, segments and curves that determine the shape of the product in space ... Source: UNIFIED SYSTEM OF DESIGN DOCUMENTATION. ELECTRONIC ... ... Official terminology

Product surface model- Surface model: a three-dimensional electronic geometric model, represented by a set of limited surfaces that determine the shape of the product in space ... Source: UNIFIED SYSTEM OF DESIGN DOCUMENTATION. ELECTRONIC MODEL ... ... Official terminology

Product model solid- Solid model: a three-dimensional electronic geometric model that represents the shape of a product as a result of the composition of a given set of geometric elements using Boolean algebra operations on these geometric elements ... ... ... Official terminology

Books

• Adaptive human norm. Symmetry and wave order of electrophysiological processes, NV Dmitrieva. In this work, we give new approach to the definition of a person's adaptive norm on the basis of generalization of the experience of polyparametric cognitive models of various physiological processes ...
• The theory of real relativity, E. A. Gubarev. In the first part of the book, based on the space of events of four-dimensional orientable points, the relativity of non-inertial (accelerated and rotating) frames of reference associated with real ...

Among all the variety of models used in science and technology, mathematical models are the most widely used. Mathematical models are usually understood as various mathematical constructions built on the basis of modern computer technology, describing and reproducing the relationship between the parameters of the modeled object. To establish the relationship between number and shape, there are various ways of space-number coding. The simplicity and availability of solving practical problems depends on a well-chosen frame of reference. Geometric models are classified into subject (drawings, maps, photographs, models, television images, etc.), calculated and cognitive. Object models are closely related to visual observation. Information obtained from subject models includes information about the shape and size of an object, about its location relative to others. Drawings of machines, technical devices and their parts are made in compliance with a number of conventions, special rules and a certain scale. Drawings can be assembly, general, assembly, tabular, dimensional, external views, operational, etc. Depending on the design stage, drawings are distinguished into technical proposal drawings, draft and technical designs, working drawings. Drawings are also distinguished by industry: 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, projections with numerical marks, affine projections, stereographic projections, cine perspective, etc. Geometric models differ significantly in the way they are executed: drawings are originals, originals, copies, drawings, paintings, photographs, films, radiographs, cardiograms, models, models, sculptures, etc. Among the geometric models are flat and volumetric models. Graphical constructions can be used to obtain numerical solutions to various problems. When calculating algebraic expressions numbers are represented by directional segments. To find the difference or sum of numbers, the corresponding segments are plotted on a straight line. Multiplication and division is carried out by constructing proportional segments, which are cut off at the sides of the corner by straight parallel lines. The combination of multiplication and addition operations allows you to calculate the sums of the products and the weighted average. Graphical exponentiation consists in sequential repetition of multiplication. The graphical solution of the equations is the value of the abscissa of the point of intersection of the curves. You can graphically calculate definite integral, plot the derivative, i.e. differentiate and integrate; and solve equations. Geometric models for graphical calculations must be distinguished from nomograms and computational geometric models (RGM). Graphic 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 to calculate and study functional dependencies. Calculations on RGM and nomograms are replaced by reading answers using elementary operations indicated in the key of the nomogram. The main elements of nomograms are scales and binary fields. Nomograms are subdivided into elementary and compound nomograms. Nomograms are also distinguished by the operation in the key. The fundamental difference between RGM and nomograms is that geometric methods are used to construct RGM, and analytical methods are used to construct 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, it does not matter whether the points of a straight line or a curve are connected. In graphs, only vertices, edges, and arcs are distinguished. For the first time, graphs were used in solving puzzles. Currently, graphs are effectively used in planning and control theory, scheduling theory, sociology, biology, in solving probabilistic and combinatorial problems, etc. The graphical model of dependence is called a graph. Function graphs can be built on a given part of it or on the graph of another function using geometric transformations. A graphical representation 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 thermodynamically equilibrium system. A bar chart, which is a collection of adjacent rectangles built on one straight line and representing the distribution of any quantities by quantitative criteria, is called a histogram.

Especially interesting is the use of geometry to assess the theoretical and practical significance of mathematical reasoning and to analyze the essence of mathematical formalism. Note that the generally accepted means of transferring acquired experience, knowledge and perception (speech, writing, painting, etc.) are obviously a homomorphic projection model of reality. The concepts of projection schematism and design operations refer to descriptive geometry and have their generalization in the theory of geometric modeling. From a geometric point of view, any object can have many projections that differ both in the position of the design center and the picture, and in their dimensions, i.e. real phenomena of nature and social relations allow for various descriptions, differing from each other in the degree of reliability and perfection. The basis of scientific research and the source of any scientific theory is observation and experiment, which always has the goal of identifying some pattern. When starting to study a specific phenomenon, a specialist, first of all, collects facts, i.e. notes such situations that are amenable to experimental observation and registration with the help of the senses or special devices. Experimental observation is always of a projection nature, since the same name (projection) is assigned by many facts that are indistinguishable in a given situation (belonging to the same projecting image). The space referred to the studied phenomenon is called operational, and the space referred to the observer is called pictorial. The dimension of the picture space is determined by the capabilities and means of observation, i.e. Voluntarily or involuntarily, consciously and completely spontaneously, it is established by the experimenter, but it is always less than the dimension of the original space to which the investigated objects belong, due to various connections, parameters, reasons. The dimension of the original space very often remains undetected, since there are undetected parameters that affect the object under study, but are not known to the researcher or cannot be taken into account. The projection nature of any experimental observation is explained, first of all, by the impossibility of repeating events in time; this is one of the regularly occurring and uncontrollable parameters that do not depend on the will of the experimenter. In some cases, this parameter turns out to be insignificant, and in other cases it plays a very important role. This shows how important and fundamental are geometric methods and analogies in the construction, evaluation or verification of scientific theories. Indeed, every scientific theory is based on experimental observations, and the results of these observations are - as said - a projection of the object under study. In this case, the real process can be described by several different models... From the point of view of geometry, this corresponds to the choice of a different design apparatus. He distinguishes objects according to some characteristics and does not distinguish them according to others. One of the most important and urgent tasks is to identify the conditions under which the determinism of a model obtained as a result of an experiment or research is preserved or, conversely, disintegrated, since it is almost always important to know how effective and suitable a given homomorphic model is. The solution of the problems posed by geometric means turned out to be appropriate and natural in connection with the use of the above projection views. All these circumstances served as the basis for the use of analogies between different kinds projection geometric models obtained by homomorphic modeling, and models resulting from the study. A perfect model corresponds to the regularities that establish an unambiguous or ambiguous, but, in any case, a well-defined correspondence between some of the initial and sought parameters describing the phenomenon under study. In this case, the effect of schematization acts, a deliberate reduction in the dimension of the picture space, i.e. refusal to take into account a number of essential parameters that allow you to save money and avoid mistakes. The researcher constantly deals with such cases when intuitively irregular phenomena differ from regular phenomena, where there is some connection between the parameters characterizing the process under study, but the mechanism of action of this regularity is not yet known, for which an experiment is subsequently set up. In geometry, this fact corresponds to the difference between a disintegrated model and a perfect model with an implicitly expressed algorithm. The task of the researcher in the latter case is to identify the algorithm in the projection, the input elements and the output elements. The regularity obtained as a result of processing and analysis of a certain sample of experimental data may turn out to be unreliable due to an incorrect sample of the acting factors subjected to the study, since it turns out to be only a degenerate version of a more general and more complex regularity. Hence the need arises for repeated or full-scale tests. In geometric modeling, this fact - obtaining an incorrect result - corresponds to the propagation of the algorithm for a certain subspace of the input elements, to all input elements (i.e., the instability of the algorithm).

The simplest real object, which is convenient to describe and model using geometric representations, is the set of all observable physical bodies, things and objects. This totality fills the physical space, which can be considered as the initial object to be studied, geometric space - as its mathematical model. Physical connections and relationships between real objects are replaced by positional and metric relationships of geometric images. Description of the conditions of a real problem in geometric terms is a very responsible and most difficult stage in solving a problem, requiring a complex chain of inferences and a high level of abstraction, as a result of which real event clothed in a simple geometric design. Theoretical geometric models are of particular importance. In analytical geometry, geometric images are investigated by means of algebra based on the method of coordinates. In projective geometry, projective transformations and invariable properties of figures that do not depend on them are studied. In descriptive geometry, spatial figures and methods for solving spatial problems by constructing their images on a plane are studied. The properties of planar figures are considered in planimetry, and the properties of spatial figures are considered in solid geometry. In spherical trigonometry, the relationship between the angles and sides of spherical triangles is studied. The theory of photogrammetry and stereo photogrammetry makes it possible to determine the shapes, sizes and positions of objects from their photographic images in military affairs, space research, geodesy and cartography. Modern topology studies the continuous properties of figures and their relative position. Fractal geometry (introduced into science in 1975 by B. Mandelbrot), which studies general patterns processes and structures in nature, thanks to modern computer technology, has become one of the most fruitful and wonderful discoveries in mathematics. Fractals would be even more popular if they were based on the achievements of the modern theory of descriptive geometry.

When solving many problems of descriptive geometry, it becomes necessary to transform images obtained on projection planes. Collinear transformations on the plane: homology and affine correspondence are essential in the theory of descriptive geometry. Since any point on the projection plane is an element of the model of a point in space, it is appropriate to assume that any transformation on the plane is generated by a transformation in space and, conversely, a transformation in space causes a transformation on the plane. All transformations performed in space and on the model are carried out in order to simplify the solution of problems. As a rule, such simplifications are associated with geometric images of a particular position and, therefore, the essence of transformations, in most cases, is reduced to transforming images general position private.

The flat model of three-dimensional space constructed by the method of two images is quite unambiguous or, as they say, isomorphically compares the elements of three-dimensional space with their model. This makes it possible to solve on planes almost any problem that may arise in space. But sometimes, for some practical reasons, it is advisable to supplement such a model with a third image of the modeling object. Theoretical basis for obtaining an additional projection, a geometric algorithm proposed by the German scientist Gauck serves.

The tasks of classical descriptive geometry can be conditionally divided into positional, metric and constructive tasks. Tasks associated with identifying the relative position of geometric images relative to each other are called positional. In space, straight lines and planes may or may not intersect. Open positional problems in the original space, when no constructions are required in addition to specifying intersecting images, become closed on a flat model, since the algorithms for their solution disintegrate due to the impossibility of separating geometric images. In space, a straight line and a plane always have an intersection at a proper or improper point (a straight line is parallel to the plane). On the model, the plane is given by homology. On the Monge plot, the plane is specified by a related correspondence, and to solve the problem, it is necessary to implement an algorithm for constructing the corresponding elements in a given transformation. The solution of the problem on the intersection of two planes is reduced to the definition of a line, which is equally transformed in two given related correspondences. Positional problems on the intersection of geometric images occupying a projection position are greatly simplified due to the degeneracy of their projections and therefore play a special role. As you know, one projection of the projection image has a collective property, all points of a straight line degenerate into one point, and all points and lines of the plane degenerate into one straight line, therefore the positional problem of intersection is reduced to determining the missing projection of the desired point or line. Considering the simplicity of solving positional problems for the intersection of geometric images, when at least one of them occupies a projection position, it is possible to solve general positional problems using drawing transformation methods to transform one of the images into a projection position. There is a fact: different spatial algorithms on the plane are modeled by the same algorithm. This can be explained by the fact that there are an order of magnitude more algorithms in space than on a plane. To solve positional problems, various methods are used: the method of spheres, the method of cutting planes, and drawing transformations. The projection operation can be thought of as a way of forming and defining surfaces.

There is a wide range of tasks associated with measuring the lengths of segments, angles, areas of figures, etc. As a rule, these characteristics are expressed by a number (two points determine the number that characterizes the distance between them; two straight lines determine the number that characterizes the value of the angle formed by them and etc.), for the determination of which various standards or scales are used. An example of such standards is a regular ruler and protractor. In order to determine the length of a segment, it is necessary to compare it with a standard, for example, a ruler. And how to attach a ruler to a straight line of general position in a drawing? The scale of the ruler in projections will be distorted, and for each position of the line there will be its own scale of distortion. To solve metric problems in the drawing, it is necessary to set support elements (improper plane, absolute polarity, scale segment), using which you can build any scale. To solve metric problems on the Monge plot, transformations of the drawing are used so that the desired images are not distorted in at least one projection. Thus, by metric problems we mean the transformation of segments, angles and plane figures into positions when they are depicted in full size. In this case, you can use various methods. There is a general scheme for solving basic metric problems for measuring distance and angles. Of greatest interest are constructive problems, the solution of which is based on the theory of solving positional and metric problems. Structural problems are understood as problems associated with the construction of geometric images that correspond to certain theorems of descriptive geometry.

In technical disciplines, static geometric models are used to help form ideas about certain objects, their design features, about their constituent elements, and dynamic or functional geometric models that allow you to demonstrate kinematics, functional relationships, or technical and technological processes. Geometric models very often make it possible to trace the course of such phenomena that do not lend themselves to ordinary observation and can be represented on the basis of existing knowledge. Images allow not only to represent the device of certain machines, devices and equipment, but at the same time to characterize their technological features and functional parameters.

Drawings provide not only geometric information about the shape of the parts of the assembly. It understands the principle of operation of the unit, the movement of parts relative to each other, the transformation of movements, the occurrence of forces, stresses, the transformation of energy into mechanical work, etc. At a technical university, drawings and diagrams take place in all general technical and special disciplines studied (theoretical mechanics, strength of materials, structural materials, electromechanics, hydraulics, mechanical engineering, machine tools and tools, theory of machines and mechanisms, machine parts, machinery and equipment, etc. ). To convey various information, the drawings are supplemented with various signs and symbols, and for their verbal description, new concepts are used, the formation of which is based on the fundamental concepts of physics, chemistry and mathematics. In the process of studying theoretical mechanics and resistance of materials, qualitatively new types of visualization appear: a schematic view of a structure, a design diagram, a diagram. A diagram is a type of graph that shows the magnitude and sign of various internal force factors acting at any point in the structure (longitudinal and transverse forces, torsion and bending moments, stresses, etc.). In the course of resistance of materials, in the process of solving any computational problem, repeated re-coding of data is required by using images that are different in their functions and levels of abstraction. A schematic view, as the first abstraction from a real construction, allows you to formulate a task, highlight its conditions and requirements. The design scheme conventionally conveys the design features, its geometric characteristics and metric ratios, the spatial position and direction of the acting force factors and support reactions, points of characteristic sections. On its basis, a model for solving the problem is created, and it serves as a visual support in the process of implementing the strategy at different stages of the solution (when constructing a diagram of moments, stresses, torsion angles and other factors). In the future, when studying technical disciplines, the structure of the geometric images used is becoming more complex with the widespread use of conditional graphic images, sign models and their various combinations. Thus, geometric models become an integrating link in natural and technical academic disciplines, as well as methods professional activity future specialists. The development of the professional culture of an engineer is based on a graphic culture that allows different types activities to unite within one professional community. The level of training of a specialist is determined by how developed and mobile his spatial thinking is, since the invariant function of the intellectual activity of an engineer is to operate with figurative graphic, schematic and symbolic models of objects.

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