In this section we will look at some very famous interesting curves, many of which may be new to you, because some of the most interesting curves are best described by parametric equations. To learn more than what is offered here, check out the Famous Curves Index at the History of Mathematics archive. Indeed, many of these curves have a long history.
A Lissajous curve or figure is any curve from the family of curves described parametrically by the equations x(t) = sin(at) and y(t) = sin(bt), where a and b are constants. They were first studied in 1815 by Nathaniel Bowditch and later, in 1857, by the French mathematician Jules-Antoine Lissajous. Lissajous curves have applications in physics and astronomy. They are interesting curves.
Here's a plot of the Lissajous curve corresponding to a = 2 and b = 3; we denote this curve by L(2, 3).
- Experiment with Lissajous curves of the form L(a, 3) where a varies from 1 to 7. In other words, keep b fixed at 3, and let a vary from 1 to 7. What do you observe?
- Next explore those Lissajous curves of the form x(t) = sin(at), y(t) = sin((a + 1)t). That is, explore the situation where the parameter b is just one unit larger than a. What happens to the Lissajous curves L(a, a+1) when a gets larger and larger? You'll want to make a fairly large to see the pattern. What do you discover? (Isn't it cool?)
A cycloid is the curve traced by a point P on the rim of a wheel (or circle) rolling along a straight line in a plane. The parametric equations of a cycloid have the form:
x(t) = at - a sin(t) and y(t) = a - a cos(t),
where a is the radius of the wheel.
Suppose now that a wheel of radius 1 rolls around the outside of a circle of radius 2. The curve traced out by a point on the rim of the smaller circle is called an epicycloid. Here's the plot of the curve.
More generally, the epicycloid traced by a fixed point on a circle of radius B as it rolls around the outside of a circle of radius A is described parametrically by the equations:
x(t) = (A+B) cos(t) - B cos( [(A+B)/B] t ),
y(t) = (A+B) sin(t) - B sin( [(A+B)/B] t ).
Play around with this! Vary the values of A and B in the Epicycloid MAPLET to see how the epicycloid changes.
The hypocycloid is the curve traced by a fixed point on the rim of a wheel as is it is rolled around the inside of a circle. If you have ever spent any time playing with Spirograph®, then you are already familiar with these curves. (Think about how you produced curves with Spirograph!).
The parameteric equations of the hypocycloid traced by a fixed point of a circle of radius B as it rolls around the inside of a circle of radius A are given by:
x(t) = (A - B) cos(t) + B cos( [(A - B)/B] t ),
y(t) = (A - B) sin(t) - B sin( [(A - B)/B] t ).
Here's an example.
Change the values of A and B in the preceding exercise to see what you can create.
Okay...maybe this curve isn't so famous. But it is very nice nonetheless. Check it out...
Maple Leaf Plotter
Change the parameters in the Maple Leaf Plotter to see what other sorts of leaves you can create.