Handbook of Research on Geoinformatics - Hassan A. Karimi Part 3 pps

68 Map Overlay Problem lie strictly either on the left or right side of the line are stored in the corresponding subtree. The segments that intersect the line are stored in both. The process continues until there are only a few segments of one color, and the brute force algorithm can be run for them. • R-tree allows ( ) ( )( ) knkn +∗+Ο log run- ning time and ( ) kn +Ο space in the worst case (van Oosterom, 1994). It is based on the spatial data structure R-Tree (Guttman, 1984). It could also be used to deal with distributed data in a non-uniform way. The insertion algorithm creates more nodes in the areas with a larger number of line seg- ments. Thus, it will automatically adapt to the distribution of the data. The R-Tree has been designed to work efciently with large amounts of data that do not t in primary memory. Therefore, it can handle the expen- sive operations of swapping memory pages between primary and secondary memory. Spatial Ordering Algorithms: Because al- gorithms based on plane partitions divide the space without considering the topology of the input, they run in quadratic time. However, we can consider spatial ordering algorithms that use an aboveness ordering to reduce the number of comparisons between segments. A segment A is above B if some vertical line intersects A at a greater y-coordinate than B. Some examples are (Andrews et al. 1994): • Bentley-Ottman sweep has a running time in the worst and average case of ( ) nknn loglog ∗+∗Ο , and a ( ) nΟ size space (Bentley & Ottman, 1979). This algorithm is based on the plane sweep tech- nique. A vertical line is moved across the plane, stopping at each node (event) of the subdivisions. It uses two data structures: a priority queue for storing the nodes of both subdivisions, ordered by the x-coordinate, in aboveness order; and a binary search tree for the segments that intersect the line at the events if corresponding. The initial subdivisions are stored in a Doubly-Con- nected Edge List (DCEL) (frequently used for representing planar subdivisions of the space, keeping a record for each node, edge and face. It also considers the topological relationships among elements, which allow operations such as traversing the edges along a face boundary, reporting the incident edges of a node, or determining which edges are shared by two faces. • Trapezoid Sweep provides ( ) knn +∗Ο log average and worst case running time, and ( ) nΟ space (Chan, 1994). This algorithm is an extension of a classical algorithm based on plane-sweep (Bentley-Ottman sweep), which also works with other types of curves. In the algorithm by Bentley and Ottman the intersections are sorted according to the X coordinate. Then, each intersection takes logarithmic time, which leads to nk log* complexity. On the other hand, the Trap- ezoid Sweep avoids the logarithmic factor by computing the segment intersections in a way that does not require sorting. Blue trapezoidation is dened as the decompo- sition of the plane into closed trapezoids formed by the blue segments and some vertical segments which are vertical extensions of the endpoints of the segments spanning from the next blue seg- ment above the point to the next blue segment below it (Figure 1). The algorithm is based on a sweep from left to right stepping through each blue trapezoid. Once the whole plane is swept, all intersections are reported. • Hereditary Segment Tree allows ( ) nn log∗Ο running time in the average and worst cases. It also takes ( ) nΟ size space (Chazelle et al. 1990; Palazzy & Snoeyink, 1994). The Hereditary Segment Tree is the basic data 69 Map Overlay Problem structure for computing the red-blue inter- sections. Firstly, all the segments and end- points are sorted using a horizontal sweep. The result is a sweep tree which as traversed in-order and returns the sorted list of points and segments. As input the algorithm takes two lists of segments and points in topologi- cal order. Then, it assigns each segment and point to its respective slab and computes the number of intersections per slab. r aster Algorithms Overlay algorithms for raster maps are much simpler than vector ones. The former make use of map algebra, which is the application of boolean operators (AND, OR, NOT and XOR) to the corresponding layers (Chang, 2006). To apply these operators, both layers must have the same level of resolution, otherwise the layer with the coarsest resolution is converted into the nest resolution. It is also necessary that both layers cover the same geographic area; otherwise the operation is performed only on the intersecting area. Other problems could be introduced by a wrong orientation of the pixels. In this case, rotations or resampling can be applied in order to adjust the layers. T he m ost widely used data structures to represent raster maps in GIS are as follows: • Qua dtree is one of the best known data structures for representing raster maps. The initial map must be a power of two sizes. The map is recursively divided along the axes coordinates into four quadrants of equal size, which are labeled as NW, NE, SW, and SE. The division stops when the whole quadrant is in a homogeneous region and therefore the value of the region is stored in it. The data structure is represented as a tree with the root as the whole map. Every time a quadrant is divided, the four children that represent the new smaller quadrants are inserted to the corresponding node into the tree. • Run -length codes stores each row’s rst and last columns as the index of the region along with its thematic value. • Block codes uses the same idea as run-length, and is extended into two dimensions using square blocks to ll in the map regions. The data structure consists of the origin (center or bottom left corner) and the size of each square. • Cha in codes encodes the contour of a region by means of an origin dened by the user and a clockwise sequence of vectors in the direction of the cardinal points: (east=0, north=1, west=2, south=3). The length of a vector indicates the number of pixels. For example: 010, 37, 23, 13, 25, 33, 24, 17 (Figure 5), starting from the pixel on row 4 and column 6, where the rst number of each pair represents the direction and the second number (after the symbol ) represents the length of the vector. conc Lus Ion One of the principal applications of GIS is to analyze geographic data. In order to implement complex operations that involve more than one layer it is necessary to overlay several maps or Figure 1. A blue trapezoidation 70 Map Overlay Problem Figure 2. QuadTree Figure 3. Run Length codes Figure 4. Block codes Figure 5. Chain codes partitions so that a single map that relates all the inputs is obtained. In this article we have covered the principal techniques and algorithms for the Map Overlay Problem, which is one of the widely used spatial analysis operations in GIS. There are several problems that can be solved through map overlay: ranging from complex queries involv- ing more than one layer, and Boolean operations between polygons, to windowing and combined buffering. 71 Map Overlay Problem references Andrews, D. S., Snoeyink, J., Boritz, J., Chan, T., Denham, G., Harrinson, J., & Zhu, C. Fur- ther Comparison of Algorithms for Geometric Intersection Problems. In Proceeding of the 10 th International Symposium of Spatial Data Handling. Balaban, I. J. (1995). An optimal algorithm for nding segment intersections, In Proceeding of the 11 th Annual ACM Symposium of Computational Geometry, 211-219. Bentley, J. L. & Ottmann, T. A. (1979). Algorithms for reporting and counting geometric intersec- tions, IEEE, Transactions on Computers, 28(9), 643-647. Chan, T. M. (1994). A simple trapezoid sweep algorithm for reporting red/blue segment intersec- tions, Proceeding of the 6 th Canadian Conference of Computational Geometry. Chang, Kang-tsung. (2006). Introduction to Geographic Information Systems. Third Edition. McGraw-Hill. New York, NY. Chazelle, B., & Edelsbrunner, H. (1992). An Optimal Algorithm for Intersecting Line Seg- ments in the Plane. Journal of the Association for Computing Machinery, 39(1), 1-54. Chazelle, B., Edelsbrunner, H., Guibas, L. & Shair, M. (1990). Algorithms for bichromatic line segment problems and polyhedral terrains. Technical Report UIUC DCS-R-90-1578, Dept. Comp. Sci., Univ. Ill. Urbana. de Berg, M., van Kreveld, M., Overmars, M. & Schwarzkopf, O. (2000). “Computational Geom- etry: Algorithms and Applications”, Springer, 2, 19-42. de Hoop, S., van Oosterom, P., & Molenaar, M. (1993). Topological Querying of Multiple Map Layers. In COSIT’93, Springer-Verlag, 139-157. Finke, U. & Hinrichs, K. (1995). “Overlaying sim- ply connected planar subdivision in linear time”, In Proceeding of the 11 th Annual ACM Symposium of Computational Geometry, 119-126. Franklin, W. R. (1989). Efcient intersection calculations in large databases. In International Cartographic Association 14 th World Conference, Budapest, A62-A63. Franklin, W. R., Kankanhalli, M. & Naraya- naswami, C. (1989). Geometric computing and uniform grid data technique. Comput. Aided Design, 410-420. Franklin, W. R., Kankanhalli, M., Narayanas- wami, C., Sun, D., Zhou, C. & Wu, P. Y. F. (1989). Uniform grids: A technique for intersection detection on serial and parallel machines. In Proceedings of Auto Carto 9: Ninth International Symposium on Computer-Assisted Cartography, Baltimore, Maryland, 100-109. Guttman, A. (1984). R-Trees: A Dynamic Index Structure for Spatial Searching. ACM SIGMOD, 13, 47-57. Kriegel, Hans-Peter, Brinkhoff, Thomas & Sch- neider, Ralf. (1992). An Efcient Map Overlay Algorithm Based on Spatial Access Methods and Computational Geometry, Springer-Verlag, 194-211. Mairson, H. G. & Stol, J. (1988). Reporting and counting intersections between two sets of line segments. Theoretical Foundations of Computer Graphics and CAD, NATO ASI Series F, Volume 40, Springer-Verlag, 307-325. Palazzy, L. & Snoeyink, J. (1994). Counting and reporting red/blue segment intersections. CVGIP: Graph. Mod. Image Proc. Preparata, F. P., & Shamos, M. I. (1985). Compu- tational Geometry, Springer-Verlag, New York. 72 Map Overlay Problem Shaffer, C. A., Samet, H. & Nelson, R. C. (1990). QUILT: A Geographic Information System based on Quadtrees. In J. GIS, 103-131. van Oosterom, Peter. (1994). An R-tree based Map-Overlay Algorithm. EGIS/MARI´94, 1-10. Van Oosterom, Peter. (1990). A modied Binary Space Partition for Geographic Information Sys- tems. Int. J. GIS, 133-140. Wu, Peter Y. (2005). The Notion of Vicinity: in Space, Time, and WorkFlow Automation. URISA Journal. key ter Ms Combined Buffer: Buffers are generated around points or lines. To obtain the combined buffer of two or more points and/or lines, the unions of the generated buffers for each one is computed. Line Segments Intersection: Consists in computing the intersections between two sets of line segments. It is also known as the red-blue line segments intersection when one set contains red segments and the other contains blue segments. In this case, intersections between red and blue segments are reported. Map Overlay: Overlay two or more thematic maps obtaining a single output map. Plane Sweep: A computational geometry technique, mostly used in nding intersections between lines. It consists of moving a line across a plane while stopping at certain events stored in a priority queue. The events are then taken ac- cording to their priorities. A binary search tree is also used to represent the state of the line at each moment. The problem is partially solved for all the area that has already been swept by the line at each event. It is completely solved when the line nishes with the last event. Subdivision Overlay: It is the process of gen- erating an output plane subdivision from overlay- ing two or more input plane subdivisions. A plane subdivision is a set of points, lines, and the regions determined by such points and lines. The output subdivision must contain all the points from the input maps plus the points of intersection between the input subdivision segments. The lines in the output subdivisions will be those made up by the input lines divided at the intersection points and the input lines that do not intersect. The output regions will be the union and intersection of all the input regions. Windowing: Overlaying a rectangular win- dow to a map and discarding everything that it is not inside of the window. 73 Chapter X Dealing with 3D Surface Models: Raster and TIN Mahbubur R. Meenar Temple University, USA John A. Sorrentino Temple University, USA Copyright © 2009, IGI Global, distributing in print or electronic forms without written permission of IGI Global is prohibited. Abstr Act Three-dimensional surface modeling has become an important element in the processing and visualiza- tion of geographic information. Models are created from a nite sample of data points over the relevant area. The techniques used for these activities can be broadly divided into raster-based interpolation methods and vector-based triangulation methods. This chapter contains a discussion of the benets and costs of each set of methods. The functions available using 3D surface models include elevation, queries, contours, slope and aspect, hillshade, and viewshed. Applications include modeling elevation, pollution concentration and run-off and erosion potential. The chapter ends with a brief discussion of future trends, and concludes that the choice among the methods depends on the nature of the input data and the goals of the analyst. Introduct Ion A surface can be dened as a continuously vari- able and measurable eld that contains an innite number of points. It typically embodies a great deal of information, and may vary in elevation or proximity to a feature (Bratt, 2004, O’Sullivan et al., 2003). The points that create a 3D topographic surface store elevation or z-values on the z-axis in a three-dimensional x, y, z coordinate system. Other examples of using continuous 3D surfaces are climate, meteorology, pollution, land cover, 74 Dealing with 3D Surface Models and natural resources (McCloy, 2006, Lo et al., 2002, Schneider, 2001). In a Geographic Information Systems (GIS) environment, a surface cannot measure or record the z-values of all of the innite points stored inside it. A surface model, commonly known as a Digital Terrain Model (DTM), or more gener- ally, a Digital Elevation Model (DEM), is used to create 3D surfaces in various areas of science and technology. The surface model primarily follows a two step process in recording and storing data: sampling and interpolation (O’Sullivan et al., 2003, Heywood et al., 2002). The model selects sample observation points, either systematically or adaptively; takes values of this nite number of points; interpolates the values between these points; and nally stores the output information (McCloy, 2006; Bratt, 2004, Heywood et al., 2002). There are six alternative ways, as shown in Figure 1, to express the continuous variation of a surface in digital representation: (1) regularly spaced sample points (e.g., a 10 meter spacing DEM); (2) irregularly spaced sample points (e.g., temperature recorded at different weather stations); (3) rectangular cells (e.g., values of re- ected radiation in a remotely sensed scene); (4) irregularly shaped polygons (e.g., vegetation cover class of different parcels); (5) irregular network of triangles (e.g., elevation in a Triangulated Irregular Network or TIN); and (6) polylines representing contours (e.g., elevation in contour lines) (Longley et al., 2005b). Among all these surfaces, numbers 1 and 3 are rasters and the remaining four are vectors, which use points (number 2), lines (number 6), or polygons (numbers 4 and 5). The DEMs or DTMs may be derived from a number of data sources, including contour and spot height information, Figure 1. Six alternative digital surface representations (Adapted from Longley et al., 2005b) ( 1) (2) (3) (4) ( 5) (6) 75 Dealing with 3D Surface Models stereoscopic aerial photography or photogramme- try, satellite images, cartography (existing maps), and ground surveys (Heywood et al., 2002, Lo et al., 2002, Cohen-Or et al., 1995). Table 1 gives an idea about different surface data collection methods and the techniques they use. In a raster, a DTM is structured as a regular grid consisting of a rectangular array of uni- formly-spaced equally-sized cells with sampled or interpolated z-values. In vector GIS, a more advanced, more complex, and more common form of DTM is the TIN, which is constructed as a set of irregularly located nodes with z-values, con- nected by edges to form a network of contiguous, non-overlapping triangular facets (Bratt, 2004, Longley et al., 2005b, Heywood et al., 2002, Lo et al., 2002). Figure 2 displays examples of raster and TIN surfaces. Both surfaces can be created using two main methods: interpolation and triangulation. bAckground creating and Interpolating a Raster Surface Spatial interpolation is the prediction of exact values of attributes at unsampled locations – a process of intelligent guesswork which is invoked without the user’s direct involvement (Longley et al., 2005b, O’Sullivan et al., 2003, Siska et al., 2001, Schneider, 2001, McLean et al., 2000). An interpolation method assumes that spatially- distributed objects are spatially correlated. Using this method, cell values are predicted in a raster from a limited number of sample geographic point data, such as elevation, rainfall, or temperature. Such points can be either randomly, strategically, or regularly-spaced features. The several interpolation methods used to cre- ate raster surfaces are Inverse Distance Weighted (IDW), Kriging, Spline, Natural Neighbors, and Trend. The IDW is the simplest and most com- monly used method that weights the points closer to the processing cell more heavily than those located further away. Such weights are dened following the inverse square of distances: w i = 1/d i 2 . This method is useful when a variable being mapped decreases in inuence with distance from the sampled location (Kennedy, 2006, Bratt, 2004, Longley et al., 2005b, Clarke et al., 2002). Kriging, a popular geo-statistical method, assumes that the distance or direction between sample points reects a spatial correlation that can be used to explain variation in the surface. In this method, values between points are interpolated by considering the nearest eight points covaried Table 1. DTM data collection methods • Ground Survey ° Suitable for large-scale surface modeling for engineering projects ° Data can be acquired using GPS or Electronic Tacheometer • Photogrammetry ° Suitable for smaller-scale surface covering larger geographic areas ° Data can be acquired using Analog Stereoplotter, Analytical Plotter, and Digital Photogrammetry • Cartography ° Suitable for digitizing maps of smaller-scale areas having regional or na- tional coverage ° Data can be prepared by digitizing or scanning Source: Lo et. al, 2002 76 Dealing with 3D Surface Models within a circular kernel (Hupy et al., 2005). Krig- ing is mostly used in geology or soil science where the data might have spatially-correlated distance or directional bias (McCloy, 2006, Kennedy, 2006, Bratt, 2004, Longley et al., 2005b, Clarke et al., 2002). Studies have shown that the Kriging method provides reasonable results in regions with sufciently dense observations (Hoerka et al., 2002). The Spline method may be thought of as the mathematical equivalent of tting a thin sheet of rubber to a series of data points (Clarke et al., 2002). It is the best method for gradually- varying surfaces such as elevations or pollution concentrations, but not appropriate for surfaces with signicant changes within short horizontal distance or regions where there are no nearby control points (Kennedy, 2006, Bratt, 2004, O’Sullivan et al., 2003). The Trend method uses a least-squares re- gression t. In the Natural Neighbors method, the raster surface is interpolated using the input points that are natural neighbors of a cell. It cre- ates a Delaunay Triangulation of the input points, which is part of the TIN creation process and is discussed in the following section (McCloy, 2006, Kennedy, 2006, Bratt, 2004). creating a TIN Surface The TIN surfaces are usually created from a combination of point (mass point), line (breakline), and polygon data, at least some of which should have z-values. The vertices of the TIN triangles represent peaks, pits, depressions, and passes; the edges represent ridges, river channels, and valleys; and the surfaces of each triangle provide area, slope (gradient), and aspect (orientation) (Ken- nedy, 2006, Heywood et al., 2002, Lo et al., 2002, O’Sullivan et al., 2003). In vector GIS, TINs are stored as polygons, each with three sides. Mass points are the primary input features for creating a TIN, as they determine the overall shape of a surface. These mass points with their z- values become nodes in the triangulated network. Breaklines and polygons can provide more control over the shape of the TIN surface. Breaklines may or may not have height measurements, and they represent natural or built features such as a stream or a road. Two types of breaklines are available, hard and soft. Hard breaklines are used to show the abrupt changes in a surface, whereas the soft breaklines do not indicate any change in slope, but may display a study area boundary. Polygons are separately-interpolated areas, and they represent surface features that have areas, such as lakes or ponds (Bratt, 2004). According to Heywood et al. (2002), there are two methods available to achieve the linking of TIN vertices. In the distance-ordering method, the distance between all pairs of points is calculated, and then sorted, and nally the closest points are connected. This method can produce, however, too many short triangles. In the Delaunay Triangulation method, a convex hull is created for a dataset. Then non- overlapping straight lines are drawn from interior points to points on the boundary of the convex hull and to each other. Thus the hull is divided into a set of polygons and then divided again into triangles by drawing more lines between the vertices of the polygons (McCloy, 2006, Longley et al., 2005b, Clarke et al., 2002, Okabe et al., 1992). This method is better than the other one for the following reasons: (1) the triangles are as equi-angular as possible, which reduces potential numerical precision problems created by long, thin triangles; (2) the triangles ensure that any point on the surface is as close as possible to a node; and (3) the triangulation is independent of the order in which the points are processed (Geocities web site, 2006). Delaunay Triangulation breaks down when (a) the observation points get further apart than the feature(s) being mapped; (b) the surface becomes more anisotropic; and (c) the data do not match the shape of the features in the surface (McCloy, 2006). 77 Dealing with 3D Surface Models Another type of irregular network, known as Tetrahedral Irregular Network and not often used, creates a volumetric representation of a 3D set of points by using tetrahedron cells to ll the 3D convex hull of the data (Clarke et al., 2002). r Aster And t In: pros And cons Some advantages of raster surfaces are as follows: (1) they give a uniform density of point data that is easy to process; (2) they do not need to store spatial coordinates explicitly for every control point; (3) each z-value’s position can be easily found (because of the GRID structure), as well as its spatial relationship to all the other points; and (4) they can be processed using array data structures available in most computer program- ming languages (O’Sullivan et al., 2003). Since an interpolation method relies heavily on the spatial correlation of points, the accuracy of the raster DTM depends on the complexity of its surface and the grid spacing (McCloy, 2006, Kennedy, 2006, Heywood et al., 2002, Siska et al., 2001). Any feature that is more precise than the size of the grid cell cannot be located in a raster. Whether an input point falls on the center or other part of a cell, there is no guarantee that the whole cell will have the same value as that input point (Bratt, 2004). For this reason, the more input points in a raster surface and the more even their distribution, the more reliable or accurate will be the results (Bratt, 2004, Lo et al., 2002, Siska et al., 2001). On the other hand, because of the irregular shapes of the triangles the TIN DTM can have higher resolution in areas where the surface is more complex (such as mountainous areas) by including more mass points, and have lower resolution in areas that are relatively at (Longley et al., 2005b, Bratt, 2004, Heywood et al., 2002). Thus TIN surfaces can produce more accurate results in spatial analysis, even if they are built with a signicantly smaller number of sample points compared to raster surfaces (Lo et al., 2002, Siska et al., 2001). The other advantage of a TIN model is the efciency of data storage (Heywood et al., 2002, Lo et al., 2002). In addition to nodes and edges, other features such as roads and streams can be used as input features in the TIN creation process which makes it more precise than a raster GRID (Bratt, 2004). However, because of the complex data structure, TIN surface models are considered more expensive and time consuming, and thus not as widely used as raster surfaces. TINs are used mostly in smaller areas, for more detailed, large-scale applications; raster surfaces are used in more regional, small-scale applications (Bratt, 2004). Some other limitations of TINs are their inability to deal with discontinuity of slope across triangle boundaries, the difculty of calculating optimum routes, and the need to ensure that peaks, ridges, and channels are recorded (Longley et al., 2005b). Raster and TIN surfaces are interchangeable. A TIN can be created from a raster to simplify the surface model for visualization; a raster can Figure 2. Raster and TIN surfaces of the same area [...]... solution of the point-in-polygon problem (Preparata, 1985) on which we will focus our attention Some of the most efficient solutions for the point-in-polygon problem reduce the solution to a solution of other fundamental problems in computational geom- etry, such as computing the triangulation of a polygon or computing a trapezoidal partition of a polygon to solve then, in an efficient way, the point-location... 2005) C onc lus ion We have presented a new algorithm for the solution of the well-known point location problem and for the more particular problem of point-in-polygon Figure 4 Creation of a quaternary partition with identification of intervals Vi (a) Each Vi interval identifies in the creation of the quad-tree is marked as Vi interval, in the figure shadowed intervals V6 V8 V5 V7 V9 V4 determination The... http://www.w3.org/Graphics/GIF/spec-gif89a txt W3C (2001) WebCGM 1.0 Second Release http://www.w3.org/TR/2001/REC-WebCGM20011217 W3C (20 03) JPEG JFIF http://www.w3.org/ Graphics/JPEG/ W3C (20 03) Scalable Vector Graphics (SVG) 1.1 Specification World Wide Web Consortium http://www.w3.org/TR /20 03/ REC-SVG1120 030 114/ W3C (2004) Extensible Markup Language (XML) 1.0 (Third Edition) World Wide Web Consortium... the first-at-hand (Hsu, 1981) A quantitative analysis of the deformation on a projection would help to retard the tendency towards the selection of a too conventional map projection (Robinson 1951) The map projection selection process is very challenging: the geospatial user has to choose among an abundant variety of projections, determined by the software in use For the sole purpose of world-maps, most... Jersey: John Wiley & Sons Ltd, (pp 7 6-7 7, 189190, 33 3- 3 37 ) McCloy, K (2006) Resource Management Information Systems: Remote Sensing, GIS and Modeling, Boca Raton, Florida: Taylor & Francis Group (pp 41 4-4 26) McLean, C., & Evans, I (2000) Apparent Fractal Dimensions from Continental Scale Digital Elevation Models Using Variogram Methods Transactions in GIS, 4(4), 36 1 -3 78 Okabe, A., Boots, B., & Sugihara,... of point-in-polygon determination These previous approaches to the problem are presented in the first sections In the remainder of the paper, we present a quick location algorithm based on a quaternary partition of the space, as well as its associated computational cost introduct ion In this chapter the authors present a new approach for the general solution of the point-location problem1 and the particular... Sugihara, K (1992) Spatial Tessellations: Concepts and Applications of Voronoi Diagrams New York: Wiley, (pp 94) O’Sullivan, D., & Unwin, D (20 03) Geographic Information Analysis Hoboken, New Jersey: John Wiley & Sons Ltd (pp 20 9-2 43) Schneider, B (2001) Phenomenon-based Specification of the Digital Representation of Terrain Surfaces, Transactions in GIS, 5(1), 3 9-5 2 Hupy, J., Aldrich, S., Schaetzl,... Prentice-Hall, Inc Imprint of Pearson Education, (pp 5 9-6 3, 12 8-1 30 ) Hofierka, J., Mitasova, H., Parajka, J., & Mitas, L (2002) Multivariate Interpolation of Precipitation Using Regularized Spline with Tension Transactions in GIS, 6(2), 13 5-1 50 Longley, P., Goodchild, M., Maguire, D., & Rhind, D (2005b) Geographic Information Systems and Science – Principles, Techniques, Management, and Applications Hoboken,... of vertexes very closed do not contribute notoriously to the average cost Despite the focus of this paper on polygons for simplicity reasons, the BQV method can be extended to any polygonal area of the space Regarding the previous processing time, it can Figure 5 Identification of Ei intervals of the space partition V5 V4 V2 1- 2 1- 2 E3 E2 V6 V3 V1 V2 V3 1- 2 E1 V1 1 03 ... answer questions such as the following: What is the elevation value of a location? What is the temperature or concentration of a pollutant at a certain location? How steep would be a proposed trail from location A to location B? What will be the run-off direction? Some common applications of Raster and TIN surfaces are described in the following paragraphs and some examples of different application fields . concentration of a pollutant at a certain location? How steep would be a proposed trail from location A to location B? What will be the run-off direction? Some com- mon applications of Raster. information at a certain point. Querying TIN data also gives information of slope and aspect, the two ways of quantifying the shape of a surface at a particular location (Bratt, 2004). Contour: Contour. geographic areas ° Data can be acquired using Analog Stereoplotter, Analytical Plotter, and Digital Photogrammetry • Cartography ° Suitable for digitizing maps of smaller-scale areas having regional