Lecture Notes for Clarke, K. C. Analytical and Computer Cartography

Lecture 11: Map Transformations

Transformations and Algorithms (Review)

• 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 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
• Lab exercises fall into several of these.
• Dimensional transformation are only one type
• When dimension collapses to "none" result is a measurement
  
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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
• Matrices have inverses, which reverse effect of multiplication to yield the identity matrix
• Error creep in when inversion does not result in identity matrix
  
  

Transformations as Multiple Steps (Dimensional Transforms)

• Also serve to generate measurements, scalars = cartometry

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

PLANAR MAP TRANSFORMATIONS

Transformations Based on Points

Distance Between Two Points

• Simple transformation requiring two points
• Point (line) to scalar (non-invertible).
• Precise and accurate for vector. Less so for raster.
• Accuracy depends on map projection!
• Can be done on sphere, hard on ellipsoid
  
  
  
• Law of sines translates to lengths as angles
  
  
• Mercator Projection
  
  
  

Length of a Line

• Repetitive application of point-to-point distance calculation
• For n points, algorithm/formula uses n-1 segments
  

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
  
  

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

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
  

Keith Clarke Last Change 5/8/97 Copyright Prentice Hall, 1995