1. CS 128/ES 228 - Lecture 11b1 Spatial Analysis (3D)

2. CS 128/ES 228 - Lecture 11b2 Some (More) GIS Queries How steep is the road? Which direction does the hill face? What does the horizon look like? What is that object over there? Where will the waste flow? What’s the fastest route home?

3. CS 128/ES 228 - Lecture 11b3 Types of queries Aspatial – make no reference to spatial data 2-D Spatial – make reference to spatial data in the plane 3-D Spatial – make reference to “elevational” data Network – involve analyzing a network in the GIS (yes, it’s spatial)

4. CS 128/ES 228 - Lecture 11b4 3-D Computational Complexity 1984 technology 1997 technology

5. CS 128/ES 228 - Lecture 11b5 Approximations In the vector model, each object represents exactly one feature; it is “linked” to its complete set of attribute data In the raster model, each cell represents exactly one piece of data; the data is specifically for that cell THE DATA IS DISCRETE!!!

6. CS 128/ES 228 - Lecture 11b6 Surface Approximations With a surface, only a few points have “true data” The “values” at other points are only an approximation The are determined (somehow) by the neighboring points The surface is CONTINUOUS Image from: http://www.ian-ko.com/resources/triangulated_irregular_network.htm

7. CS 128/ES 228 - Lecture 11b7 Types of approximation GLOBAL or LOCAL Does the approximation function use all points or just “nearby” ones? EXACT or APPROXIMATE At the points where we do have data, is the approximation equal to that data?

8. CS 128/ES 228 - Lecture 11b8 Types of approximation GRADUAL or ABRUPT Does the approximation function vary continuously or does it “step” at boundaries? DETERMINISTIC or STOCHASTIC Is there a randomness component to the approximation?

9. CS 128/ES 228 - Lecture 11b9 Display “by point” Image from: http://www.csc.noaa.gov/products/nchaz/htm/lidtut.htm

10. CS 128/ES 228 - Lecture 11b10 Display “by contour” Image from: http://www.csc.noaa.gov/products/nchaz/htm/lidtut.htm

11. CS 128/ES 228 - Lecture 11b11 Display “by surface” Image from: http://www.csc.noaa.gov/products/nchaz/htm/lidtut.htm

12. CS 128/ES 228 - Lecture 11b12 Voronoi (Theissen) polygons Points on the surface are approximated by giving them the value of the nearest data point Exact, abrupt, deterministic

13. CS 128/ES 228 - Lecture 11b13 Smooth Shading Standard (linear) interpolation leads to smooth shaded images Local, exact, gradual, deterministic Xyw 1- W = *y + (1-)*x

14. CS 128/ES 228 - Lecture 11b14 TINs – Triangulated Irregular Networks Connect “adjacent” data points via lines to form triangles, then interpolate Local, exact, gradual, possibly stochastic or Image from: http://www.ian-ko.com/resources/triangulated_irregular_network.htm

15. CS 128/ES 228 - Lecture 11b15 Simple Queries? The descriptions thus far represent “simple” queries, in the same sense that length, area, etc. did for 2-D. A more complex query would involve comparing the various data points in some way

16. CS 128/ES 228 - Lecture 11b16 Slope and aspect A natural question with elevational data is to ask how rapidly that data is changing, e.g. “What is the gradient?” Another natural question is to ask what direction the slope is facing, i.e. “What is the normal?” slope aspect

17. CS 128/ES 228 - Lecture 11b17 What is slope? The slope of a curve (or surface) is represented by a linear approximation to a data set. Can be solved for using algebra and/or calculus Image from: http://oregonstate.edu/dept/math/CalculusQuestStudyGuides/vcalc/tangent/tangent.html

18. CS 128/ES 228 - Lecture 11b18 Solving for slope In a raster world, we use the equation for a plane: z = a*x + b*y + c and we solve for a “best fit” In a vector world, it is usually computed as the TIN is formed (viz. the way area is pre-computed for polygons)

19. CS 128/ES 228 - Lecture 11b19 Our friend calculus Slope is essentially a first derivative Second derivatives are also useful for… convexity computations

20. CS 128/ES 228 - Lecture 11b20 What is aspect? Aspect is what mathematicians would call a “normal” Computed arithmetically from equation of plane Image from: http://www.friends-of-fpc.org/tutorials/graphics/dlx_ogl/teil12_6.gif Shows what direction the surface “faces”

21. CS 128/ES 228 - Lecture 11b21 Visibility What can I see from where? Tough to compute!

22. CS 128/ES 228 - Lecture 11b22 What is an elevation? It could be an ELEVATION, i.e. an altitude BUT, it could be rainfall, income, or any other scalar measurement Bottom Line: It’s one more dimension on top of the geographic data

23. CS 128/ES 228 - Lecture 11b23 Network Analysis Given a network What is the shortest path from s to t? What is the cheapest route from s to t? How much “flow” can we get through the network? What is the shortest route visiting all points? Image from: http://www.eli.sdsu.edu/courses/fall96/cs660/notes/NetworkFlow/NetworkFlow.html#RTFToC2

24. CS 128/ES 228 - Lecture 11b24 Network complexities Shortest pathEasy Cheapest pathEasy Network flowMedium Traveling salesperson Exact solution is IMPOSSIBLY HARD but can be approximated All answers learned in CS 232!

25. CS 128/ES 228 - Lecture 11b25 Conclusions A GIS without spatial analysis is like a car without a gas pedal. A GIS without 3-D spatial analysis is like a car without a radio. It may still be useful, but you wish you had the “luxury”.