- Exam 3 is not graded, and probably won't be by tomorrow. Sorry
about that.
- Today we begin working on parametric curves. You should think about motion, through time. That's the context for our discussion.
- And now for something completely different....
A good introduction to parametric curves is given by ballistics. If we shoot a bullet into the air 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, 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 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.)
- Right now I'm working on the "Orange
Peeling Problem" that a friend put me onto. It involves calculating
the lengths of parametric curves, something which we will do.
- Let's see how much of these two summaries we can explore:
- Sect. 10.1: Parametric equations (pdf, pdf summary)
- Sect. 10.2: Calculus of parametric equations (pdf, pdf summary)
- Sect. 10.1: Parametric equations (pdf, pdf summary)
