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Math 126, Calculus II, Spring 2016
Computer Lab 1
An Elliptical Unicycle
Due Date: This assignment is due Friday, April 8 in class. It is worth 4% of your grade in
the class.
Suppose the wheel of a unicycle takes the shape of the ellipse
x2 y 2
+
= 1.
25 24
instead of being perfectly circular. What would it feel like to ride such a unicycle? This is
what we?ll explore in this computer lab.
The seat of the unicycle is attached to the center of the wheel, so to understand what it
feels like to ride the unicycle we need to understand the path the center of the wheel makes
as the wheel rotates along the ?at road. To that end, complete the following steps.
We?ll start with the unicycle wheel resting on the point that is located at (5, 0). Consider
a point (x, y) on the ellipse that lies in the ?rst quadrant.

(1) Find the value of y in terms of x.
(2) Find an expression for the length of the arc along the ellipse starting from (5, 0) and
ending at (x, y). Call this length s. Don?t attempt to evaluate this yet; we?ll get the
computer to estimate it for us!
1

(3) Find the slope of the tangent line at the point (x, y) in terms of x and y.
(4) Find the tangents, cosines, and sines of the angles ? and ? shown in the picture above
in terms of x and y.
(5) Determine the sine and cosine of the angle ? shown in the picture above in terms of
x and y.
(6) Determine the distance d from the center of the ellipse to the point (x, y) in terms of
x and y.
Notice when the unicycle wheel is resting on the point (x, y), it will have moved a distance
s along the road and the tangent line to the ellipse at that point will be ?at along the ground.
So if the unicycle starts at the origin and rolls along the x-axis, this point will be located at
the point (s, 0) and the coordinates of the center of the wheel will be
x = s + d cos ?
?
y = d sin ?
?

Now write a matlab script m-?le that does the following.
a) Creates a vector x consisting of the x-coordinates of lots and lots of points along
the ellipse and in the ?rst quadrant, starting from the point (5, 0) and ending at the
?
point (0, 24). (You should use on the order of 5000 points.)
b) Creates a vector y consisting of the corresponding y-coordinates of these points.
c) Creates a vector s that estimates the value of s for each point (x, y) you have selected.
(If you have lots of points you can estimate this value relatively simplistically and it
will be su?ciently accurate.)
d) Creates vectors xtilde and ytilde that contain the x and y coordinates of the center
of the wheel when the wheel is resting at each of your selected points.
e) Plots xtilde against ytilde showing the path of the center of the wheel as the wheel
rolls on the x-axis.
f) Determine and plot what the path will continue to look like as the wheel continues
to rotate completing a full revolution.
What to Turn In:
? Turn in the work you did for items 1 – 6 above.
? Print out and turn in a copy of the script m-?le you wrote. It should contain detailed
comments explaining what each part of the code does.
? Print out of the plot produced by your code showing the path of the center of the
wheel.
? Write a few sentences describing what it feels like to ride this unicycle. Do you ever
go backwards even though you are always pedaling forward?