It's easy to find parametric equations for some curves. Graphs of functions, straight lines, and circles are the most familiar examples. In this project we'll find parametric equations for some more complicated curves.
The
most general parametric equations for a point moving counterclockwise around
a circle are
x
= r*cos(2t/k + f)
+ a
y
= r*sin(2t/k + f)
+ b.
r is the radius of the circle; k is the increase in t required for the moving point to complete one revolution; f (phi) is the distance (in radians, measured counterclockwise along the circle) from the "standard" starting point (directly to the right of the center) to the actual starting point (the position at t=0); (a, b) is the center of the circle
Exercise
1: Describe the motion of a point whose position is given parametrically
by
x = 2cos(2t/5) + 1
y = 2sin(2t/5) + 2
Exercise 2: Now modify the above equations so that the center of the circle is moving. (Moving any way you like.)
Exercise
3: Now suppose the circle rolls along the line y=1 at the rate
of one rotation every 5 seconds.
(We want
the center above the line, moving from left to right.)
How fast
is the center moving?
What are
parametric equations for the position of the center?
Exercise
4: Modify the parametric equations from exercise #1 so that the center
of the circle moves as in exercise #3.
Graph
the resulting curve. (This curve is called a cycloid.)
Exercise
5: (For people willing to explore a little.)
Now suppose
the circle "rolls" along the outside of a circle of radius 5 centered
at the origin, starting with its center at (7,0) and rotating once
every 5 seconds. What does the center of the small circle do?
Find parametric
equations for the center of the small circle.
Then find
parametric equations for the path of the point on the small circle that
starts at (9,0) at time t=0.
Graph
the resulting curve. (This curve is called an epicycloid.)
Exercise
6: (For people who thought exercise #5 was interesting.)
Roll the
small circle around the inside of the large circle.
(The resulting
curve is called a hypocycloid.)
Try varying
the large and small radii to get more epicycloids and hypocycloids.
There's
no requirement that the fixed circle have a bigger radius than the moving
circle.