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

Lecture 12: Map Transformations (Ctd)

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

• So Far have covered Point transformations

Transformations Based on Lines

Intersection of two lines

• Absolutely fundamental to many mapping operations, such as overlay and clipping.
• In raster mode can be by layer overlay.
• I vector mode must be solved geometrically.
• Lines (2) to point transformation

Basic Layout

• 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.

Alternative Forms (See Saalfeld)

1. Point-slope form. y - y1 = b(x - y1)

2. Slope-intercept y = a + bx

3. Two point form (y - y1)/(x - x1) = (y2 - y1)/(x2 - x1)

4. Two Point form (no DBZ) (y - y1)(x2 - x1) = (x - x1)(y2 - y1)

5. Linear equation ax + by + c = 0

6. Point-vector form [x y] = [x1 y1] + [rv1 rv2]

Distance from a Point to a Line

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
  
  
• Worked example
  
  

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.

Theissen Polygons

• Often called proximal regions or voronoi diagrams.
• Possible within Arc/Info
• Often used for contouring terrain, climate, interpolation, etc.

Affine Transformations

• These are transformation of the fundamental geotraic attributes, i.e. location.
• Influence absolute location, not relative or topological
• Necessary for many operations.
• Most critical are digitizing, scanning, georegistration, and display.
• Affine Transformations take place in three steps (TRS) in order

TRANSLATION

Movement of the origin between geocoding systems

ROTATION

• Alignment of coordinate systems
• Rotation of axes by an angle theta, given by georegistration step.

SCALING

• Scale change to bring images into alignment

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

Statistical Space Transformations

1. Rubber Sheeting

• Select points in two geometries that match.
• Suitable points are targets, 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

2. Cartograms

• Deliberate distortion of geometry to new "space"
• Type of non-invertible map projection

Sybolization Transformations

The Normalization Transformation

• Screen coordinates are often reduced to a "satndard" device
• Device display dimensions are (0,0) to (1,1)
• Some vary, e.g. MicroCAM
• Computer graphics Model is
• World Coordinates->NDC-> Device Coordinates
• Also need to track Windows for multiple displays etc.

Drawing Objects

• Accomplished using graphics standards
• Many e.g. PHIGS, Xwindows, GKS etc.
• Obvious advantage of standards
• Most use model of primitives and attributes
• GKS has six primives, each has multiple attributes.
• Most software uses some version of the approach

Keith Clarke Last Change 5/12/97 Copyright Prentice Hall, 1995