MACM 316 – Computing Assignment 7

Submission Instructions: You must upload one .pdf file in Crowdmark that consists of two pages:

page 1 is your report which should fit all discussions, data and figures into a single page; and page 2 is a

listing of your code. The deadline is 11:00pm on the due date. The actual due time is set to 11:05pm

and if Crowdmark indicates that you submitted late, you will be assigned a grade of 0 on this assignment.

Your TA has emailed you a Crowdmark link that you should save since it will allow you to upload your

completed assignments.

❼ Please review the Guidelines for Assignments carefully.

❼ Acknowledge any collaborations or assistance from colleagues/TAs/instructor.

❼ If you have any questions about Matlab or aspects of this assignment, then you are strongly en￾couraged to attend tutorials and drop-in workshops.

Parametric Splines for Multi-Valued Functions

In this assignment, your aim is to extend the definition of a cubic spline to interpolate data that cannot

be described by a single-valued function.

(a) As a first “test problem”, consider the data points listed in the table (below, left) which are sampled

from some underlying smooth curve shown in the plot (below, right):

index (t) x y

0 0.0 1.0

1 1.0 3.0

2 2.0 3.0

3 2.0 4.0

4 3.0 5.0

Clearly, the given data have multiple y-values at x = 2, meaning that the underlying curve is

multiply-defined – it’s not a function!! This means that the usual cubic spline approach based on

trying to interpolate a function of the form y = f(x) will not work. The trick to handling such

data is to use a parametric spline and instead treat the data as coming from a parametric curve

x = R(t) and y = S(t) for some parameter t. In this example, it is easiest to simply assign the

parameter values t = 0, 1, 2, 3, 4 corresponding to the index of the data points in the table. The

particular choice of t values is actually not important as long as t is monotonically increasing (or

decreasing – try that out if you’re curious!).

Based on the t-x-y data from the table, generate two cubic splines R(t) and S(t) using the Matlab

spline function with its default not-a-knot end-point conditions. Provide plots of R versus t, S

versus t, and S versus R (the last of which should reproduce the plot above).

(b) You will now apply the parametric spline idea to the more complicated curve shown below:

The 13 points that define the curve have coordinates listed below:

0 1 2 3 4 5 6 7 8 9 10 11 12

x 2.75 1.3 -0.25 0.0 0.25 -1.3 -2.5 -1.3 0.25 0.0 -0.25 1.3 2.75

y -1.0 -0.75 0.8 2.0 0.8 -0.25 0.0 0.25 -1.3 -2.5 -1.3 -0.25 -1.0

Using this set of ordered pairs (x, y), determine the parametric spline x = R(t), y = S(t) that

interpolates the points using Matlab’s spline function. Then plot your parametric spline curve

and verify (by zooming in on your plot) that the right “leaf” of your spline is different from the

other three leaves in that it is not smooth – the spline endpoints meet at a cusp.

(c) For a periodic curve like the one in part (b), it is more appropriate to use the periodic end-point

conditions discussed in class, R0

0

(t0) = R0

nn 1

(tn) and R00

0

(t0) = R00

nn 1

(tn) (similarly for S(t)) instead

of the not-a-knot conditions. Use the Matlab code perspline.m posted on Canvas to generate

the two periodic splines approximating R(t) and S(t), and then compare your parametric curve to

what you obtained in part (b). Verify that the cusp is eliminated.

(d) In this part, you can get creative! Start by drawing a periodic parametric curve, which can be

any curve of your own design as long as the endpoints meet at the same location. Your curve

should be defined by at least 20 spline points (but not more than 40) and it should have at least

one location where it crosses itself (like the “leaves” in part (b)).

To generate your list of points, you may find it helpful to use Matlab’s built-in function ginput

(graphical input from mouse) which allows you to “draw” your parametric curve on the screen

and outputs the x and y coordinates for you. If you type help ginput, you can see that it reads

mouse clicks in the plotting window until the “enter” key is hit, and then returns lists of the point

coordinates. I found it best to draw my curve on a sheet of paper (the most “see-through” paper

you have), place the paper over the computer screen (increase the screen brightness as much as

possible), execute ginput, and then trace the curve on the paper, clicking on a sequence of points

lying along it. You may find the following sequence of Matlab commands helpful:

figure(’position’, get(0,’screensize’)) % biggest window possible

axes(’position’, [0 0 1 1]) % domain [0,1] x [0,1]

axis square % x,y axes equal

[x,y] = ginput; % record mouse clicks

close % get rid of huge window

save mydatafile.mat x y % save the data points

The save command saves your data points to a file that can later on be read back in using the

load command. You may then use your x, y data as input to perspline.m in the same way you

did in part (c) to construct the periodic spline and plot your results.