Vector based GIS Source: http://www.sli.unimelb.edu.au/gisweb/GISModule/GIST_Vector.h tm#gd General definitions Vector is a data structure, used to store spatial data. Vector data is comprised of lines or arcs, defined by beginning and end points, which meet at nodes. The locations of these nodes and the topological structure are usually stored explicitly. Features are defined by their boundaries only and curved lines are represented as a series of connecting arcs. Vector storage involves the storage of explicit topology, which raises overheads, however it only stores those points which define a feature and all space outside these features is 'non-existent'. A vector based GIS is defined by the vectorial representation of its geographic data. According with the characteristics of this data model, geographic objects are explicitly represented and, within the spatial characteristics, the thematic aspects are associated. There are different ways of organizing this double data base (spatial and thematic). Usually, vectorial systems are composed of two components: the one that manages spatial data and the one that manages thematic data. This is the named hybrid organization system , as it links a relational data base for the attributes with a topological one for the spatial data. A key element in these kind of systems is the identifier of every object. This identifier is unique and different for each object and allows the system to connect both data bases. 1
Vector is a data structure, used to store spatial data. Vector data is comprised of lines or arcs, defined by beginning and end points, which meet at nodes. The locations of these nodes and the topological structure are usually stored explicitly. Features are defined by their boundaries only and curved lines are represented as a series of connecting arcs. Vector storage involves the storage of explicit topology, which raises overheads, however it only stores those points which define a feature and all space outside these features is 'non-existent'.
A vector based GIS is defined by the vectorial representation of its geographic data. According with the characteristics of this data model, geographic objects are explicitly represented and, within the spatial characteristics, the thematic aspects are associated.
There are different ways of organizing this double data base (spatial and thematic). Usually, vectorial systems are composed of two components: the one that manages spatial data and the one that manages thematic data. This is the named hybrid organization system, as it links a relational data base for the attributes with a topological one for the spatial data. A key element in these kind of systems is the identifier of every object. This identifier is unique and different for each object and allows the system to connect both data bases.
Figure 1.Vector representation
Vector representation of dataIn the vector based model (figure 1), geospatial data is represented in the form of co-ordinates. In vector data, the basic units of spatial information are points, lines
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(arcs) and polygons. Each of these units is composed simply as a series of one or more co-ordinate points, for example, a line is a collection of related points, and a polygon is a collection of related lines.
Vector modelsThere are different models to store and manage vector information. Each of them has different advantages and disadvantages.
List of coordinates "spaghetti"
Figure 2. List of coordinates "spaghetti"
This model is simple easy to manage no topology lots of duplication, hence need for large storage space very often used in CAC (computer assisted cartography)
Vertex dictionary
Figure 3. Vertex dictionary
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The model has no duplication, but does not use topology
Dual Independent Map Encoding (DIME) (figure 4)Sides of each polygon (segments) are given an identifier. Each polygon is recognized by its identifies as well as coordinates.
Figure 4. Dual Independent Map Encoding (DIME) format developed by US Bureau of the Census
nodes (intersections of lines) are identified with codes assigns a directional code in the form of a "from node" and a "to node" both street addresses and UTM coordinates are explicitly defined for each
link
Arc/Node Format
This particular method uses arcs to store a series of points (coordinate pairs) that start and end with nodes and possible include vertices (shape points). A node is the intersection of two or more arcs. This means that areas are composed of
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connected lines and lines are composed of connected points. Each component in turn, having its own separate file. Actually three data tables are used to record the topology namely Polygon Topology table (lists the arcs that comprise each polygon), the Node Topology table (defines the arcs that belong to each node, and the Arc Topology table (defines the relationships of the nodes and polygons for each arc). Arc-node data structure is used in ArcInfo and ArcView because spatial relationships are definitely present and it is very efficient for spatial analysis.
Figure 5. ARC / NODE structure or POLYVRT
Tables below give details of the topology of the structure.
File 1. Coordinates of nodes and vertex for all the arcsARC F_node Vertex T_node1 3.2, 5.2 1, 5.2 1, 32 1, 3 1.8, 2.6, 2.8, 3, 3.3, 4 3.2, 5.23 1, 2 3.5, 2, 4.2, 2.7 5.2, 2.7
Advantages/disadvantages of raster and vector data models raster vector
precision in graphicstraditional cartographydata volumetopologycomputationupdatecontinuous spaceintegrationdiscontinuous
Data basesThe elements in a vector based GIS are then the DBMS (Data Base Management System) for the attributes and the system that manages the topological data. In some GIS packages, the DBMS is based in an existing software, i.e. dBASE
Entity-relation modelThree elements are considered in this approach: (a) Entities as the relevant objects for the data base. In a GIS, an entity is any fact that can be localised spatially. (b) Attributes or characteristics attached to the entities. Each attribute has a limited domain of possible values, i.e. the quality of a road can be bad, average, good, very good. (c) Relations or mechanisms that allow to relate entities. Some examples are: ‘located in’, ‘contained in’, ‘crossed with’, etc.
DBMSThe data bases used in GIS are most commonly relational. Nevertheless, Object Oriented data bases are progressively incorporated.
Relational data basesIn a relational data base, data is stored in tables where rows represent the
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objects or entities and columns the attributes or variables. A data base is usually composed of several tables and the relations between them is possible through a common identifier that is unique for each entity. Most of the relational data bases in GIS present two variables with identifiers; one of them is unique and correlative, it could be numeric or alphabetic, and the second one might be repeated and helps to organise the attribute table.
The advantages of using this kind of data base are: The design is based in a methodology with heavy theoretical basis, which offers
confidence in its capacity to evolve. It is very easy to implement it, specially in comparison with other models such as
hierarchical, network, and object oriented. It is very flexible. New tables can be appended easily. Finally, many powerful DBMS using this approach contains query languages (like SQL)
which makes easy to include this tool in a GIS. Thus, some commercialised GIS packages include a DBMS pre- existent.
Object oriented data bases
Based on objects, it can be defined as an entity with a localisation represented by values and by a group of operations. Thus, the advantage in comparison with relational data bases is based on the inclusion, in the definition of an objet, not only its attributes but also the methods or operations that act on this object. In addition, the objects belong to classes that can have their own variables and these classes can belong to super-classes.
DEM derivatives: slope gradient and slope aspect
1. Definitions
Slope is defined by a plane tangent to a topographic surface, as modelled by the DEM at a point (Burrough, 1986). Slope is classified as a vector ; as such it has a quantity (gradient) and a direction ( aspect ) . Slope gradient is defined as the maximum rate of change in altitude (tan Q ) . Aspect (y) as the compass direction of this maximum rate of change (Cf. Fig. 1). More analytically, slope gradient at a point is the first derivative of elevation (Z) with respect of the slope (S), where S is in the aspect direction (y). At the same time the first derivative of a function (i.e. S stands for slope) at a point can be defined as the slope (angular coefficient or trigonometric tangent) of the tangent to the function on that particular point, hence:
tan Q = rise/run = ¶Z/¶S
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Figure 1. Slope components, note that slope gradient can be express in percent or in degrees.
1. 2. Different techniques for calculating gradient and aspectThe technique/algorithm adopted to calculate slope gradient and aspect varies according to the type of DEM selected to model the topography. Models for structuring elevation database can be square-grid, TIN (triangular irregular network), and contour-based ones (Check here for more details on DEMs). After conducting a literature review (though not a comprehensive one), it appears that a greater number of algorithms have been developed for gridded DEMs (Cf. Table 1).
Table 1. Techniques for calculating gradient on DEMs.
Type of DEM
Technique for calculating gradient
TIN 1. 1. Average triangle value (Tajchman, 1981)Contour-based
A study conducted by Srinivasan and Engel (1991) compared methods 3, 4, 5 and 6 (Cf. Table 1) against field measurements. According to their results the neighbourhood and quadratic surface methods would follow more faithfully ground measurements, the maximum slope method being the least reliable.
N.B. DEM derivatives are also highly dependent on the type of interpolation adopted during the process of DEM creation (e.g. kriging, spline, etc.).
2. 3. ApplicationsAmong a number of possible applications, I’d like to stress in a Pacific Northwest context, how important is to automatically derive slope gradient and aspect in the prospective to compile more sophisticated thematic maps such as: Terrain Stability Assessment Snow Avalanche Risk Mapping Debris Flow Risk Mapping
3. 4. Literature citedBeasley, D.B. and Huggins, L.F. 1982. ANSWERS: User’s manual. U.S. EPA-905/9-82-
001, Chicago, IL. 54pp.Burrough, P.A. 1986. Principles of Geographical Information Systems for Land
Resources Assesment. Oxford: Clarendon Press.CERL, 1988. GRASS reference manual, version 3.0. Champaign, IL: U.S. Army, Corps
of Engineers, Construction Engineering Research Laboratory.Evans, I.S. 1980. An integrated system of terrain analysis for slope mapping. Zeitschrift
fur Geomorphologie. 36: 274-295.Moore, I.D., R.B. Grayson and A.R. Ladson. 1991. Digital Terrain Modelling: a review
of hydrological, geomorphological, and biological applications. Hydrological Processes. 5: 3-30.
Srinivasan, R. and B.A. Engel. 1991. Effect of slope prediction methods on slope and erosion estimates. Applied Engineering in Agriculture. 7: 779-783.
Tajchman, 1981. On computing topographic characteristics of a mountain catchment. Canadian J. Forest Res., 11: 768-774.
Travis, M.R., Elsner, G.H., Iverson, W.D., and Johnson, C.G. 1975. VIEWIT: computation of seen areas, slope, and aspect for land-use planning. USDA F.S. Gen. Tech. Rep. PSW-11/1975, 70p. Berkeley, California, U.S.A.
Zevenbergen, L.W. and C.R. Thorne. 1987. Quantitative analysis of land surface topography. Earth Surface Processes and Landforms, 12: 47-56.
4. 5. Further readingsEvans, I.S. 1972. General geomorphometry, derivatives of altitude and descriptive
statistics. In: Chorley, R.J. (Ed.), Spatial Analysis in Geomorphology, pp. 17-90. Methuen.
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Jenson, S. K. 1985. Automated derivation of hydrologic basin characteristics from digital elevation data. Proc. Auto-Carto 7:Digital Representation of Spatial Knowledge, Washington D.C., 301-310.
Kwamme, K.L. 1990. GIS algorithms and their effects on regional archaeological analysis. In: K.M. Allen, S.W. Green and E.B. Zubrow (Eds.), Interpreting space: GIS and archaeology, pp. 112 – 124. Taylor & Francis.
Skidmore, A.K. 1989. A comparison of techniques for calculating gradient and aspect from a gridded digital elevation model. International Journal of Geographical Information Systems. 4: 323-334.
5. 6. URLsA case study:http://www.consrv.ca.gov/radar/geosar/year2rpt/chap2_2.html A gridded DEM perspective:http://www.jarno.demon.nl/gavh.htm#ref4Slope of 3-dimentional equations:http://forum.swarthmore.edu/dr.math/problems/keenan11.12.97.htmlPlanar and profile curvature in a TIN environment:http://www.jarno.demon.nl/gavh.htm#ref4
Vector Data Model
Vector graphics is the use of geometrical primitives such as points, lines, curves, and shapes or polygon(s), which are all based upon mathematical equations, to represent images in computer graphics.
Vector graphics formats are complementary to raster graphics, which is the representation of images as an array of pixels, as it is typically used for the representation of photographic images. There are instances when working with vector tools and formats is best practice, and instances when working with raster tools and formats is best practice. There are times when both formats come together. An understanding of the advantages and limitations of each technology and the relationship between them is most likely to result in efficient and effective use of tools.
Overview
Computer displays are made up from small dots called pixels. The picture is built up from these dots. The smaller and closer the dots are together, the better the quality of the image but the bigger the file needed to store the data. If the image is magnified it becomes grainy as the resolution of the eye enables it to pick out individual pixels.
Vector graphics files store the lines, shapes and colors that make up an image as mathematical formulae. A vector graphics program uses the mathematical formulae to construct the screen image by building the best quality image possible, given the screen resolution, from the mathematical data. The mathematical formulae determine where the dots that make up the image should be placed for the best results when displaying the image. Since these formulae can produce an image scalable to any size and detail the quality of the image is only determined by the resolution of the display and the file size of vector data generating the image stays the same. Printing the image to paper will usually give a sharper, higher resolution output than printing it to the screen but can use exactly the same vector data file.
A vector-graphics drawing software is used for creating and editing vector graphics. The image can be changed by editing screen objects which are then saved as modifications to the mathematical formulae. Mathematical operators in the software can be used to stretch, twist, and colour component object in the picture or the whole picture and these tools are presented to the user intuitively through the graphical user interface of the computer. It is possible to save the screen image produced as a bitmap/raster file or generate a bitmap of any resolution from the vector file for use on any device.
The size of the file generated will depend on the resolution required but the size of the vector file generating the bitmap/raster file will always remain the same. Thus it is easy to convert from a vector file to a range of bitmap/raster file formats but it is very much more difficult to go in the opposite direction, especially if subsequent editing of the vector picture is required. It might be an advantage to save an image created from a vector source file as a bitmap/raster format because different systems have different and incompatible vector formats and some might not support vector graphics at all. However, once the file is converted from the vector format it is likely to be bigger and it loses the advantages of scalability without losing resolution. Editing will also lose the convenience of being able to work on individual parts of the picture as discrete objects. Vector formats are not always appropriate in graphics work. For example, digital devices such as cameras and scanners produce raster graphics that are impractical to convert into vectors and so for this type of work the editor will operate on the pixels rather than drawing objects defined by mathematical formulae. Comprehensive graphics tools will combine images from vector and raster sources and might provide editing tools for both since some parts of the overall work could be sourced from a camera and others drawn using vector tools in the software.
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The W3C (World Wide Web Consortium )standard for vector graphics is svg (Scalable Vector Graphics. is an XML (Extensible Markup Language) specification and file format for describing two-dimensional vector graphics, both static and animated).
XML: Its primary purpose is to facilitate the sharing of structured data across different information systems, particularly via the Internet, and it is used both to encode documents and to serialize data.
The standard is complex and has been relatively slow to be established at least in part owing to commercial interests. Many web browsers have now some support for rendering svg data but full implementations of the standard are still comparatively rare.
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Vectorising is good for removing unnecessary detail from a photograph. This is especially useful for information graphics or line art.
Detail can be added or removed from vector art. Vector illustrations can have their own colours, allowing artists to achieve desired results.
One of the first uses of vector graphic displays was the US SAGE air defense system. Vector graphics systems were only retired from U.S. en route air traffic control in 1999, and are likely still in use in military and specialized systems. Vector graphics were also used on the TX-2 at the MIT Lincoln Laboratory by computer graphics pioneer Ivan Sutherland to run his program Sketchpad in 1963.
Subsequent vector graphics systems include Digital's GT40 [1]. There was a home gaming system that used vector graphics called Vectrex as well as various arcade games like Asteroids and Space Wars. Storage scope displays, such as the Tektronix 4014, could also create dynamic vector images by driving the display at a lower intensity.
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Modern vector graphics displays can sometimes be found at laser light shows, using two fast-moving X-Y mirrors to rapidly draw shapes and text on a large screen.
The term vector graphics is mainly used today in the context of two-dimensional computer graphics. It is one of several modes an artist can use to create an image on a raster display. Other modes include text, multimedia and 3D rendering. Virtually all modern 3D rendering is done using extensions of 2D vector graphics techniques. Plotters used in technical drawing still draw vectors directly to paper.
Example: consider circle of radius r. The main pieces of information a program needs in order to draw this circle are
1. data are describing a circle 2. the radius r and equation of a circle 3. the location of the center point of the circle 4. stroke line style and color (possibly transparent) 5. fill style and color (possibly transparent)
Advantages to vector style of drawing over raster graphics:
a much smaller file size compared to large raster images (the size of representation doesn't depend on the dimensions of the object), though a vector graphic with a small file size is often said to lack detail compared with a real world photo.
One can indefinitely zoom in e.g. a circle arc, and it remains smooth. On the other hand, a polygon representing a curve will reveal being not really curved.
On zooming in, lines and curves need not get wider proportionally. Often the width is either not increased or less than proportional. On the other hand, irregular curves represented by simple geometric shapes may be made proportionally wider when zooming in, to keep them looking smooth and not like these geometric shapes.
The parameters of objects are stored and can be later modified. This means that moving, scaling, rotating, filling etc. doesn't degrade the quality of a drawing. Moreover, it is usual to specify the dimensions in device-independent units, which results in the best possible rasterization on raster devices.
From a 3-D perspective, rendering shadows is also much more realistic with vector graphics, as shadows can be abstracted into the rays of light which form them. This allows for photo realistic images and renderings.
Typical primitive objects
Lines and Polylines
Also known as polygonal chain, polygonal curve, polygonal path, or piecewise linear curve is a connected series of line segments. Formally it is a curve specified by a
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sequence of points called its vertices so that the curve consists of the line segments connecting the consecutive vertices.
A simple polyline
Intersecting polyline
Closed polyline (polygon)
Polygon: It is a plane figure that is bounded by a closed path or circuit, composed of a finite sequence of straight line segments.
Circle: It is a simple shape of Euclidean geometry consisting of those points in a plane which are at a constant distance, called the radius, from a fixed point, called the center.
Equation of the circle is x2 + y2 = r2 . Its eccentricity is 0.
Ellipse: It is a locus of points in a plane such that the sum of the distances to two fixed points is a constant (This distance is the length of the major axis). The two fixed points are called foci (singular- focus).
The ellipse and some of its mathematical properties.
in the Cartesian plane defined by an equation
such that B2 < 4AC, where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the ellipse, exists.
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The line segment AB, that passes through the foci and terminates on the ellipse, is called the major axis. The major axis is the longest segment that can be obtained by joining two points on the ellipse. The line segment CD, which passes through the center (halfway between the foci), perpendicular to the major axis, and terminates on the ellipse, is called the minor axis. The semimajor axis (denoted by a in the figure) is one half the major axis: the line segment from the center, through a focus, and to the edge of the ellipse. Likewise, the semiminor axis (denoted by b in the figure) is one half the minor axis.
If the two foci coincide, then the ellipse is a circle; in other words a circle is a special case of an ellipse, one where the eccentricity is zero.
The simple equation of an ellipse is
with eccentricity .
An ellipse with a semimajor axis a and semiminor axis b, centered at the point (h,k) and having its major axis parallel to the x-axis may be specified by the equation
This ellipse can be expressed parametrically as
where t may be restricted to the interval .
Bézier curves: In the mathematical field of numerical analysis, a Bézier curve is a parametric curve important in computer graphics and related fields. In early 1960s they were used in designing automobile bodies. In vector graphics, Bézier curves are an important tool used to model smooth curves that can be scaled indefinitely. Bézier curves are widely used in computer graphics to model smooth curves.
Linear Bézier curves
Given points P0 and P1, a linear Bézier curve is simply a straight line between those two points. The curve is given by
and is equivalent to linear interpolation.
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t in the function for a linear Bézier curve can be thought of as describing how far B(t) is from P0 to P1. For example when t=0.25, B(t) is one quarter of the way from point P0 to P1. As t varies from 0 to 1, B(t) describes a curved line from P0 to P1.
Quadratic Bézier curves
A quadratic Bézier curve is the path traced by the function B(t), given points P0, P1, and P2,
A quadratic Bézier curve is also a parabolic segment.
TrueType fonts use Bézier splines composed of quadratic Bézier curves.
For quadratic Bézier curves one can construct intermediate points Q0 and Q1 such that as t varies from 0 to 1:
Point Q0 varies from P0 to P1 and describes a linear Bézier curve. Point Q1 varies from P1 to P2 and describes a linear Bézier curve. Point B(t) varies from Q0 to Q1 and describes a quadratic Bézier curve.
Bezigons In geometry, a beziergon (or polybezier) is a closed path composed of Bézier curves. It is similar to a polygon in that it connects a set of vertices by lines, but whereas in polygons the vertices are connected by straight lines, in a beziergon the vertices are connected by Bézier curves.
Beziergon - The red beziergon passes through the blue vertices, the green points are control points that determine the shape of the connecting Bézier curves
Text In computer and machine-based telecommunications terminology, a character is a unit of information that roughly corresponds to a grapheme, grapheme-like unit, or symbol, such as in an alphabet or syllabary in the written form of a natural language.
An example of a character is a letter, numeral, or punctuation mark. The concept also includes control characters, which do not correspond to symbols in a particular natural language, but rather to other bits of information used to process text in one or more languages. Examples of control characters include carriage return or tab, as well as instructions to printers or other devices that display or otherwise process text.
This list is not complete. There are various types of curves (Catmull-Rom splines, NURBS etc.), which are useful in certain applications.
Often, a bitmap image is considered as a primitive object. From the conceptual view, it behaves as a rectangle.
Vector operations
Vector graphics editors typically allow rotation, movement, mirroring, stretching, skewing, affine transformations, changing of z-order and combination of primitives into more complex objects.
An affine transformation or affine map or an affinity between two vector spaces consists of a linear transformation followed by a translation (moving every point a constant distance in a specified direction):
In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, satisfying certain properties described below.
Physically, an affine transformation is one that preserves
1. Collinearity between points, i.e., three points which lie on a line continue to be collinear after the transformation
2. Ratios of distances along a line, i.e., for distinct collinear points p1, p2, p3, the ratio | p2 − p1 | / | p3 − p2 | is preserved
In general, an affine transform is composed of zero or more linear transformations (rotation, scaling or shear) and a translation (or "shift"). Several linear transformations
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can be combined into a single matrix, thus the general formula given above is still applicable.
More sophisticated transformations include set operations on closed shapes (union, difference, intersection, etc.).
Vector graphics are ideal for simple or composite drawings that need to be device-independent, or do not need to achieve photo-realism. For example, the PostScript and PDF page description languages use a vector graphics model.
Advanced vector artists are developing more photo-realistic vector art every day
Printing
Vector art is key for printing. Since the art is made from a series of mathematical curves it will print very crisp even when resized. For instance one can take the same vector logo and print it on a business card, and then enlarge it to billboard size and keep the same crisp quality. A low-resolution raster graphic would blur incredibly if it were enlarged from business card size to billboard size.
3D modelingIn 3D computer graphics, vectorized surface representations are most common (bitmaps can be used for special purposes such as surface texturing, height-field data and bump mapping). At the low-end, simple meshes of polygons are used to represent geometric detail in applications where interactive frame rates or simplicity are important. At the high-end, where one is willing to trade-off higher rendering times for increased image quality and precision, smooth surface representations such as Bézier patches, NURBS or Subdivision surfaces are used. One can however achieve a smooth surface rendering from a polygonal mesh through the use of shading algorithms such as Phong and Gouraud
Formats
One example of vector graphics format is SVG (Scalable Vector Graphics), an open standard created and developed by the World Wide Web Consortium to address the need (and attempts of several corporations) for a versatile, scriptable and all-purpose vector format for the web and otherwise. Another example is VML (Vector Markup Language), a proposed standard that was adopted by Microsoft.
The SWF (Shockwave Flash ) Adobe's (f. Macromedia's) file format, is also a vector-based container to store animation.
