- We're at the end of lucky week 13, and now for
something completely different!
Today we begin parametric and polar equations.
- Your colleague Yimin found a problem with one of the IMath problems, in 11.9, and so I withdrew it. But I saved the work we did, so that you can check it out. It was #11.
- And now for something completely different....
A good introduction to parametric curves is given by ballistics. If we shoot a bullet into the air (or a motorcyclist) with speed v horizontally, and we neglect all forces but gravity, then the bullet will trace out a parabola (some bullets are larger than others):
Now how might we characterize the path of the bullet? The answer is a parametric curve, of the form $C(t)=(x(t), y(t))$.
$x(t)=vt$ $y(t)=h-\frac{1}{2}gt^2$ or $c(t)=(vt,h-\frac{1}{2}gt^2)$ If we wish, we can solve for t in the equation for x and use that to eliminate the parameter t from the equation for y, hence getting an equation for the parabola traced out: $y=h-\frac{1}{2}g\left(\frac{x}{v}\right)^2$ According to this parameterization, where is the "bullet" at time t=0? In which direction is the motion occuring -- left to right, or right to left? I guess it depends on $v$, doesn't it?
(You probably have a parametric graphing mode on your calculator.)
- Orbits of planets in the heavens, movements of ants on a hill, a robotic arm in an assembly plant:
- all these can be described by parametric curves.
- A couple of famous examples involving the cycloid:
- The Brachistochrone Problem (1696, reputedly solved by Newton in a single day after he heard the challenge)
- The Tautochrone Problem (1673)
- One of the most important applications of parametric
curves is illustrated in the Bezier curve lab at the end of
10.2. There are lots of important instances where we need to
be able to design custom curves, and this is one of the most
common ways of creating them. For example, you can use them to
create your own "sigmac",
using control
points. (I had all my students create their own in
numerical analysis.)
- Materials:
- Parametric Equations (.nb)
- Parametric Equations (.pdf)
- Long talks you through Parametric Equations.
- Nothing to report.