Transcript No Slide Title

Robert M. Guzzo
Math 32a
Parametric Equations
We’re used to expressing curves in terms
of functions of the form, f(x)=y.
What happens if the curve is too
complicated to do this?
Let’s look at an example.
An ant is walking along... only to be crushed by a
rolling wheel.
Question: What is the path traced out by its
bloody splat?
Why would we ask such a question?
Mathematicians are sick bastards!!!
Problem Posed Again
(in a less gruesome manner)
A wheel with a radius of r feet is marked at
its base with a piece of tape. Then we allow the
wheel to roll across a flat surface.
a) What is the path traced out by the tape
as the wheel rolls?
b) Can the location of the tape be determined at
any particular time?
Questions:
•What is your prediction for the shape
of the curve?
•Is the curve bounded?
•Does the curve repeat a pattern?
Picture of the Problem
Finding an Equation
•f(x) = y may not be good enough to express the
curve.
•Instead, try to express the location of a point, (x,y),
in terms of a third parameter to get a pair of
parametric equations.
•Use the properties of the wheel to our advantage.
The wheel is a circle, and points on a circle can be
measured using angles.
WARNING: Trigonometry ahead!
Diagram of the Problem
2r
r
C
q
P
Q
rq
O
X
T
We would like to
find the lengths
of OX and PX,
since these are
the horizontal and
vertical distances
of P from the
origin.
The Parametric Equations
r
C
r
q
r cosq
P
|OX| = |OT| - |XT|
= |OT| - |PQ|
x(q) = rq - r sinq
Q
r sinq
|PX| = |CT| - |CQ|
y(q) = r - r cosq
rq
O
T
X
rq
Graph of the Function
If the radius r=1,
then the parametric equations become:
x(q)=q-sinq, y(q)=1-cosq
Real-World Example:
Gears
For Further Study
• Calculus, J. Stuart, Chapter 9, ex. 5, p. 592:
The basic problem. Stuart also looks at
more interesting examples:
• What happens if we move the point, P, inside the
wheel?
• What happens if we move P some distance outside
the wheel?
• What if we let the wheel roll around the edge of
another circle?
•History of the Cycloid