Geometric models for solids under the surface of the Earth are discussed in this paper.
We have developed a two and a half dimensional model and program for geologic applications.
We introduce a conceptual scheme for the full 3D geometric model as well

Siki, Z.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

MATHEMATICAL MODEL FOR SOLIDS UNDER THE SURFACE OF THE EARTH* Z. SIKI Department of Surveying, Institute of Geodesy, Surveying and Photogrammetry, Technical University, H-1521 Budapest Received July 3, 1989 Presented by Prof. Dr. F. Sarkozy Abstract Geometric models for solids under the surface of the Earth are discussed in this paper. V;'e have developed a tv;o and a half dimensional model and program for geologic applications. V;' e introduce a conceptual scheme for the full 3D geometric model as well. Preface :Many systems were developed to handle and manipulate three dimen-sional objects. The developers of these systems tried to implement general purpose systems for mechanical engineering or architecture. There is no standard to model mineral resources for geological or mining aspects. We have been trying to develop a t·wo and a half and a three dimen-sional geometric model for these purposes. We examined the method to model three dimensional objects, then developed a mixed model to descrihe natural and artifical ohjects under the surface of the earth. We considered some of the necessary algorithms to manipulate our model and the restrictions in-voh'ed in it. An overview of solid modeling methods Well known geometric models [5, 8, 9, 16, 17, 19] are discussed shortly in this paragraph. We have considered only the methods usahle for our aims: sweeping Constructive Solid Geometry (CSG) cell decomposition Boundary Representation (BR) Sweeping is useful to model prism-like or solid of rotation-like objects. An object is created by mo'wing a volume or polygon along a trajectory. There are t·wo types of sweeping, translational and rotational. * The ~ational Scientific Research Foundation gives financial assistance to the research discussed in this paper. ~ 144 z. SIKI Constructive Solid Geometry (CSG) uses set operations to represent a solid. Generally three operations are used: section (n), union (U) and set difference (-). Sometimes a fourth operation is introduced called glueing (G). A solid is represented as a binary tree where the nodes are set operations and the leaves are primitives (half spaces or previously defined solids). A half space divides the space into two parts inside and outside. CSG is a very good model for regular objects and the set operations are useful not only for storing hut creating and modifying them. Cell decomposition is a special case of CSG. There is only one type of primitives (cells) topologically and the cells are glued (no holes or common interiors among the adjacent primitives). Two types of primitives are usually used: regular (cub", or block) or irregular (tetrahedron). Cube cells are usually encoded into octrce. In easE' of tetrahedrons the points of the solid to model are used to make the decomposition. We get a linear approxmation of the solid. With regular cells the approximation is worse but the regular elements can be handled more easily. Cell decomposition is used to model sculptured objects and it is suitable for finitE' element analysis. Boundary representation is the nearest to human sense. The solids are described by their houndaries. Boundary is represented hy faces, edges and vertices with the topology. BR indicates a hierarchic data structure with one to manv relations. The developed models At the heginning of our research we divided our task into two part3: not fully 3D model (so called 21/2 D model) full 3D model A two and a half dimensional model for geologic applications The 21/2 D model supports geologic applications for larger areas with lower resolution. The model is based on layers. Each layer has an upper and a lower boundary which must be single valued, with only one z value for each x, y pair. The houndaries which do not fulfil this condition have to be divided into single valued parts. The boundaries must he given by scattered points or contour lines. From these data a triangle mesh is generated [14] for each boundary. The model is limited to a block called model space. A layer can he defined for the whole model space or for one or more parts of it. The bound-aries of partly defined layers must be closed (e.g. the upper and lower border must meet exactly) otherwise the calculated volumes and surfaces will be false. .,UTHEMAnCAL JWDEL FOR SOLIDS 145 Besides layers faults aTe allowed. This part of the model is being worked out. Faults can he stored as planes with a validity polygon. The elevations at the two sides of the face of a fault can be calculated in two ways. It can he interpolated from points of border on the same side of the face of fault or from given values at the intersection of the fault and the boundary. A program was developed for IBM PC XT/AT or compatibles to manipu-late the 21/2 D model. This program is really a graphic inquiry system which can he used with some IBM PC standard graphic cards CGA, Hercules, EGA, VEG A. The program makes it possihle to display the whole model or a part 30r:jer S'Jr70CeS 'ncreosing Z In-- I I . I 81 82 83 Bi, (L7 mu~t;;Qyer. :"'3 \er's sno;Je [oyer) linnl'! 85 86 B~ 88 > Fig. 1. Logical structure of thE' 2'/ cD model of it and to modify the hOTders and faults locally. Each fUIlction of the pro-gram can be reached through pop-up menus which can be manipulated "with keyboard or mouse. Different display methods are implemented to get the most from stored data. The following types of projection are supported: perspective axonometry axonometry to the xy plane (map) axonometry to thc plane of section (profilc) A part of thc model can be displayed cut by Hrtical planes which may he given by a convex or concave polygon in the xy plane. The graphic screen can he diyided into several vie-wports to yiew the model from different points of view at the same time. The resolution of the graphic screen highly influ-ences the rational numhers of non-overlapping viewports on the screen. There are different methods to display the upper boundary of the highest lwrspectiyic or axollometrie projected layers. Triangle mesh, rectangle mesh. contours can he selected to mak(' the picture more spectacular. Only the selected layers are included in the visualisation. Selection can he made by attrihutes of layers or illdiyidnally. The surface of the horders and the volume of selectC'cl layers can he calculated. 10 146 z. SIKI Besides the screen only Epson printers are allowed as output device (alphanumeric or graphic hardcopy) but we plan to implement plotter drivers and logical output to other systems. 3D ~Iodel for mining and geological applications lVIany 3D solid modeling systems were developed for engineering applica-tions. However, they cannot be applied in our case because complete informa-tion is needed for the modeling solid. We have only incomplete information about the distribution of substances under the surface of the Earth. Infor-mation can be coHeeted from different sources: drill holes maps geophysical observations surveying of caves, mines, the surface of the Earth. i - -------------- -_._----- --'-- ------~.-----"-- -----Fig. 2. Output of the 21/2D system MATHEMATICAL JWDEL FOR SULIDS 147 Another significant difference to engineering applications is that the horder among different minerals cannot be drawn unambiguously. The border can be defined by a cut off value (a given percent of a mineral). The cut off value depends on the kind of mineral, economic mining etc. From this point of view scalar fields would be the most suitable to model them. The distribution of minerals can he estimated from the incomplete information like a scalar-vector function: U f(v) = f(x, y, ;;) The scalar-vector function and a cut off value determine the shape of the solid. The function and thc cut off value define a set (half space) as: f(x, y, ;;) cut off value These types of sets can he used ,,-ith CSG because only an inside-outsids function is needed for CSG to manipulate solids or can be trallsform(~d to cell decomposition. Fig. 3. Tetrahedron cells If we consider that not only natural hut artificial (man-made) ohjects are under the surface of the earth (drifts, pits etc.) it would be effective to use the same or similar geometric-model for hoth natural and artificial ohjects. In the case of artificial objects BR, CSG or sweeping would he suitahle, in contrast with natural objects where cell decomposition would he used as the most efficient method to store sculptured solids. BR would he good in both cases if the volume were not so important in mining applications. So it seems that different methods would he good to model the two types of objects. But if we consider that cell decomposition is a special case of CSG then we can realize that cell decomposition and CSG can be comhined easily without the permanently repeated transformations from one model to the other. Only linear geometric elements are allowed in our model except scalar fields. This linear approximation [23] does not deform significantly our model hecause: 10* 148 Z. SIKI a) any geometric plement can be approximated by linear elements with arbitrary precision b) this error is smaller than the error of the estimation from the in-eomplete information. \Ve have to distinguish cayes from other natural objects because they can be suryeyed. If n~presentatiye cross sections are measured in the CUYC then its boundary can he approximated 'with a spatial triangle mesh [11. 14]. It is true for (existing artificial objects as well. Geopnysic obSer'/::11 ons Fig. 4. Triangle mesh from cross sections k>l Seater fields rlcut O":;oiCle ~ ~,,; ____ -'1 p ~------~------~ Tr ong:,= mest--! fer the bounar:es of d:fferent IsurveYin<;J L L-11~_,~r_;~_,~e_e ,_01_' _...J11 . ,;-;cngte rnes!! r-I" "-fer the surface of the Earth I ~ ~ne EClrtn . Surveying of CQves ,cnd ex 'st ing man-made objects (crOS5 sections) Tr i angle mesh for the boundaries of caves I Data from 2 1/20 system Piannec objects created by set operation or sweepln9 J I ~-------~) I (SS I Fig. 5. Cunceptual SChell1e for :iD model .HATHEJfATICAL JIODEL FOR SOLIDS 14!:J We prefer tetrahedrons in cell decomposition to blocks because tetra-hedrons give a better approximation and the number of tetrahedrons is inde-pendent of the resolution. Trigonal prisms may be used as cells if only vertical drift holes are given. A trigonal prism can be decomposed into three tetra-hedrons. Fig. 6. Prisms and tetrahedrons Tetrahedrons can be stored simply by their four -wrtices. NI) topology IS needed because any three points of the tetrahedron make a face. Another possibility to store tetrahedrons is hy their four planes like CSG. There is no Deed to store the binary tree because it is the samp for vvery tetrahedron. )-Ianipulation of solids given by CSG hinary tree Binary trees are widely used in computing hecause many data structures can be modeled by them and their structure is fairly simple. Recursive data structures and algorithms are generally used in connection with hinary trees. The following Pascal declarations define a minimal hinary tree structure to store solids with plane faces. type node_type = (section, union, difference, plane); node_ptr = ~node; node = record ease typ: node_type of section, union, difference: (left_child: node_ptr; right_child: node_ptr); plane: (a, h, c, cl: real); end; Using this data structure a recursive inside outside function can be defined for a spatial point. 150 Z. SIKI function inside (x, y, z: real; root: node_ptr): boolean; begin inside: = false: {suppose outside} if root < > NIL then {shift out empty solid} with root '" do case typ of section: inside: = inside (x, J, z, lefcchild) and inside (x, J, z, right_child); nnion: inside: difference: inside: plane: inside (x, y. z, left_child) or inside (x, J. z. right_child): inside (x, -", z, lefcchild) and not inside (x, J', z. righcchild); inside: (a*x -;- b*y c*= + d» = 0: end; {case} end; {inside} The set operations can easily be made between two solids. Only their binary trees have to be connected with the given operator (section, union. difference). No calculation is needed at all. Besides this benefit there are some disadvantages of CSG. We cannot display a solid directly stored in binary tree. Another problem with CSG is that we cannot find out easily if the solid is empty (e.g. empty set). Unfortunately this type of description of solids is not unique, the same solid can be described by several different binary trees. But this problem cau be solved by transforming the tree into a standard format [12,20,24). Let us go down to the displaying problem of solids stored in CSG struc-ture. Two solutions can be imagined: a) by the transformation of the CSG structure to another for example to BR b) the binary tree contains implicitly every information needed to dis-play it, let us try to pull it out. First let us deal with the transformation of the CSG structure to BR· The faces and the vertices of the solid can also be described by the same binary tree. We can define the co-planar one or more facps of the solid by the same binary tree only one half space has to be replaced by its border (e.g. instead of a '" x -;- b '" y -'- c '" z + cl> 0 use a * x b '" Y -'- c * z -+ d = 0). To get co-linear onp or more edges two half spaces haye to be replaced by their border. At last three half spaces haye to be replaced by their J[ ATHE-ifATICAL MODEL FOH SOLIDS 151 border to get a vertex. Replacing half spaces by their border (in our case it is a planc) the result may be an empty set. This way the faces, edges, vertices of a solid cau he defined as set operations. It is not enough to draw them yet. Further recursive algorithms have to he applied to solve the whole problem [21, 22]. It is not a very easy and quick way to display a solid. Let us try to solvc the displaying problem from the original CSG tree without transformations. The z buffer algorithm can help us [7, 15]. It is a hidden surface eliminating algorithm. By the help of z buffer algorithm a shaded picture of the solid can be displayed. This algorithm supposes a colour raster display device. The z buffer algorithm calculates depth infor-mation at each pixel for every geometrical object which hides it (e.g. the projecting ray to the pixel intersects the horder of solid) and displays the ncarest with a convenient colour. Since thc horder of solid i:;: not known in CSG wc have to extend the algorithm. We can only calculate the point of intersection hetween projection ray and the border of half spaces (plane). After it we have to decide whether the point of intersection belongs to the solid hy the help of inside function. The colour of the pixel can be calculated from the normal vector of the horder at the point of intersection and a light direction [7, 15]. This modified z buffer algorithm can Le used for Honlinear half spaces as welL e.g. sphere, cylinder, hyperholoid, torus. References 1. R~R"I'HILL, R. K: Surfaces in Computer Aided Geometric Design, l"orth Holland, 198:3. 2. BAYSE, J. W-GILCHRIST, J. E.: GMSolid: Interactive Modeling for Design and Anah-,.i-of Solids. IEEE Computer Graphics and Applications. VoL 2. March. 1982. -3. CARDEI'AS. A. F.: Data Base Management Systems, Allyn and Bacon, Boston, 1979. 4. E!'\cAR!'\AcAo. J.: File Structures and Data Bases for CAD. North Holland, Amsterdam. 1982. 5. E:-;cAR!'\AcAo. J.: Computer Aided Design, ~lodeling, Systems, Engineering. CAD Systems. Lecture :\"otes in Computer Science. Springer-Verlag Darmstadt. Berlin. 1980. 6. E!'\cARI'ACAo. J.-SCHLECHTE!,\DAHL. E. G.: Computer Aided Design. Springer-Verlag. Berlin, Heidelberg, New York, Tokyo. 1983. 7. FOLEY, J. D.-VA!,\ DA~L A.: Fundamentals of Interactive Computer Graphics. Adisoll-Weslev Pub!. Co. New York, 1982. 8. 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B.: Set Member Classification: A Unified Approach to Geometric Intersection Problems, IEEE Transactions on Computers, Vol. C·29, No 10. 1980. 22. TILOVE, R. B.-REQrrCH.A, A. A. G.: Closure of Eoolean Operations on Geometric Enti-ties, Computer Aided Design, Vol. 12, No 5, 1980. 23. TrR:NER, J. U.: Accurate Solid yrodeling Using Polyhedral Approximations, IEEE Com-puter Graphics and Applications, Vol. 8, :\lay. 1988. 24. ,VOODWARK. J. R.: Eliminating Redundant Primith'es from Set-Theoretic Solid ylodels by a Consideration of Constituents. IEEE Computer Graphics and Applications, Vo!. 8. }Iay. 1988. Zoltall SIKI H-1521, Budapest 

MATHEMATICAL MODEL FOR SOLIDS UNDER THE SURFACE OF THE EARTH

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MATHEMATICAL MODEL FOR SOLIDS UNDER THE SURFACE OF THE EARTH

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Periodica Polytechnica (Budapest University of Technology and Economics)