Description
The purpose of the first three problems in this assignment is to explore how to deal with sparse, realworld data in interpolation methods. You will need your interpolation code from previous problem sets
in order to do this assignment, and you will need to do some creative thinking. Do not start this one
the night before it’s due!
1. [5 pts] Here are (x, y, z) coordinates for 22 points on the Worthington glacier near Valdez, Alaska
— twelve on the top surface and ten on the bottom:
top surface bottom surface
(33.44 87.93 105.88) (15.59 35.07 12.88)
(8.81 84.07 103.11) (38.57 37.17 13.33)
(15.62 34.83 105.98) (61.10 67.15 17.31)
(40.16 38.71 108.13) (58.97 92.05 19.09)
(61.45 67.07 108.12) (36.98 63.24 16.51)
(58.81 91.44 107.72) (64.45 42.66 20.01)
(36.97 63.29 107.14) (89.18 46.85 27.71)
(64.71 42.38 109.07) (66.87 18.48 14.24)
(89.11 46.49 109.93) (65.90 31.93 21.0)
(67.24 18.32 109.99) (76.55 44.51 22.0)
(65.90 31.93 109.51)
(76.55 44.51 109.91)
Plot the points on the top surface (just the points, not the whole surface!) using your favorite
3D plotting tool. Repeat for the bottom set. Notice how changing the perspective affects your
ability to make any sense of the surface from the points. If your language/package supports it,
try connecting the points in a “wireframe” plot. Turn in one copy of the best version of each of
the two surfaces, and please write down what language/command you used.
2. [20 pts] Your task in this problem is to perform a 3D surface interpolation of these data. Fitting
a smooth surface to these data is nontrivial; each triple of numbers in the table above required
laser/GPS surveying (and, for the bottom points, it also entailed drilling a hole completely through
the glacier!), so the points are unevenly spaced and very sparse — too sparse for most surface
interpolation functions to deal with. You will have to improvise, creatively, in order to solve this
problem well.
Using whatever smooth surface interpolation function is available in your favorite programming
environment (e.g., Matlab’s griddata or Mathematica’s Interpolation command), fit a surface
to the top points. The surface may pass through the points or not, as you wish. NOTE! because
the glacier data are sparse and irregularly spaced, you will not be able to blindly toss them into
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these garden-variety tools and expect to get a reasonable solution. Simply plugging these points
into griddata will garner very little credit, and plugging them into Interpolation won’t work
at all. Rather, you will have to figure out what the glacier surface looks like between and beyond
the existing data points — using some sensible interpolation and extrapolation methods based on
your code from previous problem sets in this course — and use that information to add some new
(made-up, but hopefully consistent) points to the data set. Depending on the method you use,
you may have to improvise at the edges to get surfaces that end smoothly — e.g., adding extra
copies of the end points.
Plot your interpolated surface in 3D, experimenting with shading, point size, and other plotting
parameters — contour versus perspective plot, various shading or coloring schemes, etc. — until it
looks as good as possible. Turn in a printout of this plot, together with a one-paragraph discussion
of your results and observations, including at least a few sentences on how and why you added
interpolated points between those in the data set. (This description is worth half the credit for
this problem.)
There is no single right answer here, and this is the hardest part of this assignment. Your creativity
in working around this problem and manufacturing “good” points to add will determine how nice
your surface looks. Hint: start by using some form of interpolation to add points between the
existing ones. You may also use any other method you wish (e.g., neural nets); if you use a method
that does not appear in the textbook, please include an explanation of how it works.
3. [optional for 10 pts extra credit] Repeat problem 2 for the bottom surface. Now try superimposing
the two surfaces, removing the hidden part of the bottom surface if your plotting tool lets you do
that easily. Note that the scales in the two data sets are quite different, so when you plot them
on the same axes, some surface detail may be lost.
4. [10 pts] Problem 8(c) on page 198 of the textbook.
5. [10 pts] Fit these data to a power-law model by using linearization. Find the RMSE of the fit:
t y
1 2
1.5 3.7
2.5 7.9
3 10.4
(This last problem may go away, depending on how far I get in lecture on 3/15. Please check the
moodle for announcements if you missed that class.)
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