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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 Unwins Extended Classification

- Robinsons 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 11,000

to 1400M. - 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-gt

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

http//en.wiki.mcneel.com/default.aspx/McNeel/Poin

tsetReconstruction.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-gt Normalized Device Coordinates

gt Device Coordinates

Drawing Objects

- Most use model of primitives and attributes
- The Graphical Kernel System (GKS) has six

primives, each has multiple attributes.

Next Lecture

- Data Structure Transformation