Geometric Measures and their limits

Objectives of Lecture:

1. Distance (length of lines) is easy (or is it?)
2. Area (with nuances of implementation)
3. Centroid
4. Fractals and the limits to calculating lengths of natural curves

Distance:

equation is bone-crushingly simple (Pythagoraean Theorem), but still raises issues of calculation. For example, why take square root? (square your tolerance instead...)

Pythagorean distance is on a flat surface (the projection, see later lectures...). Scale error of distance can be an issue to consider (length error of UTM for Oregon, of the State Plane North extended to the whole state)

Why not calculate on a sphere or the ellipsoid ?

• Australian site with three methods and a calculator
• Distance Calculate with four methods (from Joint Mapping Tool Kit - NIMA)
• Another calculator online; what method is being used?
• (Who actually cares about such nuances? See the radio industry/hobby that developed into the line of sight communications industry - $$$$$) Programs for radio. Notie it links to a calculation of BEARING (angle).

Area of a polygon:

equations (based on vector products of coordinates) given without showing why... Actually the formula is a simplification of the trapezoid rule. Note B&W concerns about large intermediate results (local origins); Result has a SIGN (total consternation? how can area be negative? - but of course, more information hiding in the way you calculate it...)

Centroid of polygon

Warboys just says it is the mean of the vertex vectors (but for REGULAR polygons).

implemented by summing up cross products (example from Clarke on overheads; handout from Bowyer and Woodwark)

Actually requires a calculation of AREA as a part of the process.

Fractals and cartometry:

Cartometry: The study of map measurements. Basically focused on geometric measurements: length and area, though some attribute issues could be considered cartometric. Mostly an obsolete issue, as digital representation can employ analytical methods to derive the measurements.

The Cartometric Equipment:
Measurement of area: dot grid; polar planimeter
Measurement of length: wheel-based gadgets, dividers, string...

The theory of the length of a line

Sounds simple: line is shortest path between two points, hence length of line is an attribute of the line, a necessary property...
Works fine for STRAIGHT lines...

More complex lines are not so simple...
If sampled at higher and higher resolutions, the length of the line increases continuously (and essentially without bounds!)

• Discovered by Hugo Steinhaus: How long is the Vistula River 1948 (the history of mathematics remembers him for other things...)
• Popularized by Benoit Mandelbrot (IBM with a courtesy appointment at Yale) in 1960s/70s
  How long is the coast of Britain? in Science (the journal)
• Richardson: emprical study of international boundaries
  length of boundary seems to rise according to an exponential rule

"Fractal Dimension" D is the exponent where distance is predicted from the resolution.
For lines, D=1 is a straight line; D=2 fills all space (a Peano curve)
Intermediate values mean partially space filling...

Mandelbrot asserts that natural curves are fractals, with a constant D.
emprially, this is a first approximation. More likely there are thresholds in which the regression fit is good, then some radical change in behavior.

Some resources on Fractals:

• UCGIS core curriculum (Brian Klinkenberg)
• paper on fractal computations (abstract)
• generally cool stuff on fractals as an art form...
• the plain old Mandelbrot set, etc.

We will attempt to estimate D for Bainbridge Island in the Lab.

Version of 22 January 2003