Coordinate Geometry and Geometric Calculations

(from overheads, no links)

Objectives of this lecture
1 Review linkage of analytical geometry (trigonometry) for GIS
2 Examine simple calculations
3 Establish techniques for treating lines
4 Consider concerns of speed, efficiency, numerical stability

Basic Coordinate Geometry: A Thought-Experiment

1) How do you obtain coordinates if:

2) How would you determine bearing and distance from coordinates?

3) Take some sample parcel descriptions and figure out their coordinate representations.
(examples: 2209 Mass Ave, Lexington MA and 8687 Oddfellow Road Bainbridge Island WA)

Fundamental Geometric Algorithms

Wise starts off introducing equation for a line as:

the point (x,y) such that y = a + b*x.

This is a dubious choice. See snippet from Programmer's Geometry about ax +by +c =0. Even more useful if "normalized" so that a-squared plus b-squared =1. Then a and b are "direction cosines" of the angles that the normal to the line makes with the axes, c is - distance to the origin. Neat?

Distance:

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

Bearing of a line

• sin (bearing) = x-difference / distance
• cosine (bearing) = y-difference / distance
• use inverse functions to solve for the angle

Distance from a point to a line:

raises issues of how to represent a line; useful for generalization
Handout from Programmer's Geometry shows how simple it can be...

Topological algorithms

(for treatment later) Point-in-polygon, segment intersection, polygon overlay

Roundoff and numerical stability:

the view of Prof. Kahan
Paradoxes in concepts of accuracy (numerical representations)
example of area of triangle calculations