OBJECT MODELS IN THE OPENCAST MINING COMPUTER-AIDED DESIGN 
A. A. Sholomitski, A. Y. Anoprienko, S. G. Dzhura 
Address: DonNTU, 58, Artyom Street, Donetsk 83000, Ukraine 
E-mail: sholomitskij@gis.dgtu.donetsk.ua    ovs@pop.dgtu.donetsk.ua
The problems of modeling of mining engineering objects being the algebraic systems built on the points discrete set are examined in the paper. Their geometrical content and interaction within the object classes and among the classes in the computer-aided design systems are described.  
Introduction  
The object big sizes ( for the big casts the mining lease territories makes up 30 and more square kilometers) is a feature of the opencast mining. The accuracy of the mining engineering tasks solving under sizes like that should be very high and correspond to the requirements put forward to the accuracy of the computer-aided design systems. One the one hand the accuracy depends on the accuracy of the instrumental surveys and technology of their doing. On the other one the accuracy of the mining engineering tasks solving and their quality depends on the model’s object composition and the set of the allowable operations at the object and among the objects. The regular monitoring of the mining openings is one of the main tasks for the survey boards at open cast mining . 
The Object Models Algebraic Basics 
The mine survey gives the discrete description of the opening geometry, geological bodies and mining engineering objects of the opencast mining. At present when the open cast mining is modeled all information (survey, that is metric, geological and technical) is presented in the discrete form which can be given as the finite set  

 where  - the opencast mining model point. 
To describe the opencast mining (OCM) it’s not enough to have the points only. To be more precise, if only the points are used for the model,  , which is not allowable for the presentation of the model in the computer’s memory. For the finite set P to describe the linear objects and changes of the model features the limits for the model feature changes should be fixed at this set. The highwall edges, the upraise dip change lines, the surface edges etc. are the limits for the OCM. 
A set of segments can be built at the P set. These will be allocations,   that is the power allocation 2 on n elements [1]. The number of the allocations of the K power from n elements equals to 
		(1)			
In the concrete for the OCM this number is much more less as not all points form segments. The segments formed from the P points make up a set (fig.1) 

Each element of the set consists of references to a couple of points from the P set which have the following features: 

  и  
indices at  show the segment direction. 
There is the Boolean (or the family of sets) for the L set which correspond to the OCM linear objects aggregate. The linear objects are the ordered sets  
      
where  - the relation of the set strict aggregate , that is the binary relation having features of irreflexivity, skew symmetry and transitivity. 

Fig. 1 Forming of the segment  on the set 

Minorant and majorant of each object out of the aggregate of the linear objects is correspondingly the object’s initial and final point.  The OCM linear objects which delimit the features changes are the results of the surfaces or/and volume bodies crossing presented by the faces aggregates made up on the P aggregate in accordance with the definite laws. 

The number of possible placements for the faces is defined from (1) for the power 3. The faces aggregate itself is entered as 

where  i-face is defined by three references to P  (fig.2) 
,
 which has the features of 
   ,      и   
The face consists of three edges formed from the edge top points.:

There is the Booleanfor the G aggregate which corresponds to the OCM surfaces (fig.4a). Each surface should have closed frontier. For example, .is frontier (fig.4a). In general the segment ordered aggregate is the surface frontier 

for which the condition 
						(2)
is met, that is the i-segment final point is the initial point of the following i+1 segment. Thus, any surface frontier should begin and finish at one and the same point : 

Fig.2 Forming of the  edge on the aggregate

The surface frontier (s) can be obtained if the surface faces aggregates are presented as the segments aggregates excluding all faces for which the condition of the reflexiveness of points forming them is met.:

Thus, if there is the  frontier, the face aggregate form the surface (fig. 5a). If the frontier doesn’t exist (), it is the volume body (fig. 5b). 
The surface building for opencast mining that is the forming of the faces aggregate on the subaggregate , should be done with the help of Delaunay triangulation for which there is an additional condition of non-crossing of the faces edges with the OCM linear objects: 
				(3)
All OCM geological bodies and many other openings are modeled as volume bodies. The minimal volume body can be described by 4 points of the P aggregate (fig.4). It’s the tetrahedron which consists of 4 faces. 

Fig. 4 Forming of the tetrahedron on the P aggregate 

However, the division of some volume into tetrahedrons is not a trivial task though there is a version of the three dimensional Delaunay triangulation which solves this task for the convex figures. As to the models entering of the volume body as the closed surface consisting of the non-crossing faces aggregate is rational:

  и  
Thus, for the G aggregate there is the Boolean which corresponds to the OCM volume bodies. 
The volume figure correctness is the condition of non-self-intersection of the faces consists of:

that is the ratio of two faces V consist of should generate the contiguity ratio that is that of identity of one of the edges without taking into account their directions (commutativity law).
The second condition is that of reflexivity for all faces edges V consists of:  

that is for each face edge belonging to V there obviously is the other face edge adjacent to it and having backward direction (fig. 5a ). 

а) Surface  	б) volume body 
				
Fig 5. Building of the surface and volume body on the P aggregate 

The operations of grouping, crossing, difference and complement should be applied to all these aggregates .
 grouping of two aggregates  and  is the aggregate , which consists of the elements  and the elements of the aggregate :

This task arises when the OCM model is supplemented by the new surveying points.  
The  crossing of two aggregates  and  is the aggregate which consists of the elements which belong both to the aggregate  and the aggregate :

For the OCM  this operation is worth while under the comparison of two models to identify the spheres in which there were no changes during some period of time..
The  difference of two aggregates  and  is the aggregate which consists of two elements belonging to   and  not belonging to the aggregate :

For the OCM model this operation takes place when the points are removed from the initial model before the new surveying points are added to the model. 
The complement  of the aggregate is the aggregate:

The OCM model complements at a certain stage are the new surveying points until they are not added to the model or the points of the territory which are not yet added to the model. 
The geometrical composition of the OCM model and the operations made with her can be described by these aggregates. But besides the geometrical aspect the model has the objective one [3]. The OCM model object can be presented as  

where the aggregate describes the object geometry and,   
that is the geometry should be entered at least by one of possible elements,  - is the object features aggregate. 
By its genesis the model objective composition is divided into two classes. The first object has natural background –all geological (volume) bodies and, with some allowance, surfaces (faults).  

The second class are artificial objects, such as openings: benches, ditches, mining blocks etc.  

The natural objects fill in the OCM area without crossing each other. The only relation possible among them is adjacency, when they have a common point, frontier or surface. 
   or      or   
The natural objects algebra could be presented as follows: 

as the natural objects aggregate with the operation smk which defines the objects adjacency, ,  - the adjacency index, , 1 – for common points, 2 – for lines, , 3 – for surfaces.. The task to identify the adjacent objects arises under the deposits supplementary exploration when based on new data two objects are combined into one or one object is divided into several ones. The operations of grouping, crossing and complement are allowed to be made on the natural objects aggregate. Though if the objects connected to the resources measuring are deleted from the natural objects and removed to the artificial objects which they, in fact, are, the only operation of the adjacency identification can be defined for the natural objects. But historically the objects connected to the resources measuring are distinguished on the natural objects aggregate.  
The artificial objects algebra can be presented as 

that is as the aggregate of artificial objects and operations of grouping, crossing and complement. Thus, all OCM artificial objects grouping gives the surface called the OCM day surface. Different data bench surface crossing generates the volume body which gives the mined mass volume. The crossing of this volume body with the natural (or geological) objects gives the volumes of the mineral deposits and dirt which were extracted under processing.  
The operations of crossing natural and artificial objects which have a specific physical sense are possible. Thus, crossing of the volume geological body and the OCM day surface generates two volume bodies which demonstrate how much minerals  were mined and remained. 
The opencast mining complete model can be defined as the algebraic system: 

which is defined by the aggregate of the corresponding objects and operations made with them and a single operation of crossing among classes of the natural and artificial objects.

Conclusion
Taking into account all mentioned above it is obvious that the reference and object data models are different technologies of creation, saving and control of data on the reality objects. The reference model can be used under modeling of the volume bodies of bank type the surfaces of which have the features of the topography surface that is the surface of certain thickness. In this case we can economize the memory for the objects presentation as one seam soil surface is the top of the underlying seam and so on. The objects topological accuracy is provide by the reference system. If the seams are broken and have complicated hypsometry, nicks and pinches, the reference model becomes more complicated and loses the advantages of storage compactness. To make it simpler, if to present the surface I byte of the information is required, then to model k seams I(k+1) byte is required within the reference model and I(2k-1) byte within the object model. Now there is software support for the reference models of linear objects and surfaces [2]. 
The object form of the information presentation is preferable for ore deposits which have complicated forms of minerals [3]. Under the object modeling the object contains all information about itself which leads to data doubling but eliminates the reference complicated system which is difficult to support when the model is edited. The objects topological correctness is provided by the set of methods for the objects inside topology and the set of operations among the object classes for outside topology. 
Opencast modeling was always done on the objective basis. But whether these objects corresponded to the  classical definitions – is different matter [4]. In most cases they did not. Their behaviour and relations with other objects and classes were dependant on the designer’s experience and wish. Besides, the OCM modeling subject differs greatly from the abstract objects [4]  or programme interface objects. From our point of view the OCM object can both encapsulate data about itself and refer to common data. The most important requirement is that its features and behaviour should reflect the features of real objects in the computer aided design systems. 
The authors hope they managed to systematize the object composition of the OCM models, their methods and operations both within the object classes and among the classes.  

References:
1.	V. A. Gorbatov. Fundamental Basis of Discrete Mathematics. Informational Mathematics. – M.: Nauka. Fismatlit, 2000, 544 p. 
2.	 Michael Zeiler Modeling Our World. The ESRI Guide to Geodatabase Design. ESRI Press, 1999, 199p.
3.	A. A. Sholomitski. Principles of Digital Modeling of Opencast Mining.//DonSTU Proceedings. Mining and Geological Series, Issue 11, - Donetsk, 2000, p. 77-85. 
4.    G. Booch. Object Oriented Analysis and Design with the Examples of C++ Supplement,                 2nd Edition.-M.: Izdatelstvo Binom, CPb.:Nevsky Dialect, 2000.-560p..