Lecture 08: Map Transformation Geography 128 Analytical and Computer Cartography Spring 2007 Department of Geography University of California, Santa Barbara

Review of the transformational view of Cartography Transformations – Map scale – Dimension – Symbolic content – Data structures Why Transform? – We may wish to compare maps collected at different scales. – We may wish to convert the geometry of the map base. – We may wish or need to change the map data structure.

Robinson's Classification

Robinson's Classification (cnt.) Robinson's Classification was based on dimension and level of measurement Dimension of measurement – Zero dimensional – One dimensional – Two dimensional – Three dimensional ? Level of measurement idea is from Stevens (1946) – Nominal data assume only existance and type. An example is a text label on a map. – Ordinal data assume only ranking. Relations are like "greater than". – Interval data have an arbitrary numerical value, with relative value. Example: Elevation. – Ratio data have an absolute zero and scale.

Transformations as Stages in Map Production Transformation of level can be shown in making a choropleth map. This transformation is not invertible, but can be error measured and minimized.

David Unwin’s Extended Classification Robinson’s idea was extended by David Unwin. Unwin separated issue of data from issues of mapping method, (map type and data type)

State Changes and Transformations Cartographers are interested in the full set of state transformation. Each map has an optimal path through the set. Design cartography primarily concentrates on the last, or symbolization transformation. Four types of transformations shape the mapping process: – Geocoding (transforming entities to objects: levels, dimension, data structure) – Map Scale – Locational Attributes or Map Base – Symbolization

Scale Transformations Some transformations "collapse" space: e.g. area to point. Map scales of interest to cartography are 1:1,000 to 1:400M. Transformations from larger to smaller scale by the process of generalization. At the minimum, generalization involves simplification, elimination, combination and displacement.

Some Generalization Problems Length Shape Topology

Map Generalization and Enhancement These steps are conducted under specified and consistent rules. An example is the set of algorithms for point elimination along a line. The inverse of this adds points along a line: enhancement

Transformations and Algorithms In mathematics, transformations are expressed as equations. Solutions, inversion as so forth are by algebra, calculus etc. In computer science, a set of transformations defining a process is called an algorithm. Any process that can be reduced to a set of steps can be automated by an algorithm data structures + transformational algorithms = maps +=

Transformations and Algorithms (cnt.)

Transformations of Object Dimension The four dimensions of dimension, data can be represented at any one in one state Transformations can move data between states Full set of state zero to state one transformations is then 16 possible transformations Dimensional transformation are only one type When dimension collapses to "none" result is a measurement

Map Transformation Algebra Transformations map closely onto Matrix algebra Almost all spatial data can be placed into an (n x m) or (n x p) matrix Transformations can then be by convolution (iteration of a matrix over an array OR By selecting a small matrix (2 x 2) or (3 x 3) for multiplication Complex transformations can be compounded

Transformations as Multiple Steps (Dimensional Transforms)

Map Transformation Algebra (cnt.) Matrices have inverses, which reverse effect of multiplication to yield the identity matrix Error creep in when inversion does not result in identity matrix

Map Projection Transformations Map projections represent many different types of transformation Perfectly invertible (one-to-one) One-to-many Many-to-one Undefined (non-invertible) Imperfectly invertible, e.g. on ellipsoid and geoid, computational error, rounding etc. Some transformations use iterative methods i.e. algorithms, not formulas

Geographic Coordinate Transformation

Equatorial Mercator Transformation

Planar Geometry vs. Spherical Geometry Rule of Sines – Distance between points

Planar Map Transformations on Points - Length of a line Repetitive application of point-to-point distance calculation For n points, algorithm/formula uses n-1 segments

Planar Map Transformations on Points - Centroids Multiple point or line or area to be transformed to single point Point can be "real" or representative Mean center simple to compute but may fall outside point cluster or polygon Can use point-in-polygon to test for inclusion

Planar Map Transformations on Points - Standard Distance Just as centroid is an indication of representative location, standard distance is mean dispersion Equivalent of standard deviation for an attribute, mean variation from mean Around centroid, makes a "radius" tracing a circle

Planar Map Transformations on Points - Nearest Neighbor Statistic NNS is a single dimensionless scalar that measures the pattern of a set of point (point-> scalar) Computes nearest point-to-point separation as a ratio of expected given the area Highly sensitive to the area chosen

Planar Map Transformations Based on Lines - Intersection of two lines Absolutely fundamental to many mapping operations, such as overlay and clipping. In raster mode it can be solved by layer overlay. In vector mode it must be solved geometrically. Lines (2) to point transformation

Planar Map Transformations Based on Lines - Intersection of two lines (cnt.) When using this algorithm, a problem exists when b2 - b1 = 0 (divide by zero) Special case solutions or tests must be used These can increase computation time greatly Computation time can be reduced by pre-testing, e.g. based on bounding box.

Planar Map Transformations Based on Lines - Distance from a Point to a Line

Planar Map Transformations Based on Areas Computing the area of a vector polygon (closed) Manually, many methods are used, e.g. cell counts, point grid. For a raster, simply count the interior pixels Vector Mode more complex

Planar Map Transformations Based on Areas

Planar Map Transformations Based on Areas - Point-in-Polygon Again, a basic and fundamental test, used in many algorithms. For raster mode, use overlay. For vector mode, many solutions. Most commonly used is the Jordan Arc Theorem Tests every segment for line intersection. Test point selected to be outside polygon.

Planar Map Transformations Based on Areas - Theissen Polygons Often called proximal regions or voronoi diagrams Often used for contouring terrain, climate, interpolation, etc cNeel/PointsetReconstruction.html

Affine Transformations These are transformation of the fundamental geometric attributes, i.e. location. Influence absolute location, not relative or topological Necessary for many operations, e.g. digitizing, scanning, geo-registration, and display Affine Transformations take place in three steps (TRS) in order – Translation – Rotation – Scaling

Affine Transformations - Translation Movement of the origin between coordinate systems

Affine Transformations - Rotation Rotation of axes by an angle theta

Affine Transformations - Scaling The numbers along the axes are scaled to represent the new space scale

Affine Transformations Possible to use matrix algebra to combine the whole transformation into one matrix multiplication. Step must then be applied to every point

Statistical Space Transformations - Rubber Sheeting Select points in two geometries that match Suitable points are targets, e.g. road intersections, runways etc Use least squares transformation to fit image to map Involves tolerance and error distribution [x y] = T [u v] then applied to all pixels May require resampling to higher or lower density

Statistical Space Transformations - Cartograms also known as value-by-area maps and varivalent projections (Tobler, 1986) Deliberate distortion of geometry to new "space" Type of non-invertible map projection

Symbolization Transformations Screen coordinates are often reduced to a "satndard" device – Normalization Transformation Standard Device display dimensions are (0,0) to (1,1) World Coordinates-> Normalized Device Coordinates > Device Coordinates

Drawing Objects Most use model of primitives and attributes The Graphical Kernel System (GKS) has six primives, each has multiple attributes.

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