Can Parameterization be Redefined Any Simpler?

[royblaze]

Hello all. This is just a question I've been having while learning about parameterization of curves in my Calc III class.

Now, I've never taken parameterization lessons (?) which apparently are supposed to be covered in Calc II (which includes heavy integrals, series, and other stuff).

But now that we are revisiting paramerization, I've got a question.

Why is it defined in the way that it is?

I mean, for some HW questions I had to draw the resultant curve of something. But I didn't paramterize or anything. I just fit in values of t into a function of t and I got points. I connected the points. I checked my answer, and the graph looked really good compared to what the book says the answer is.

My teacher, on the other hand, said something along the lines of

"try saying that (_₁,_₂) can be redefined by something easier, for example, let's set _₁ as 'x' and then redefine _₂ in terms of our new x."


What I'm saying, then, is, why do we parameterize? If given a function r(t), and I'm asked to graph it, can't I just plug in values of t and then plot those points and connect them?

If anyone also has any good reference websites about parameterization, that would be great too.

 

[pwsnafu]

But now that we are revisiting paramerization, I've got a question.

Why is it defined in the way that it is?

You haven't told us what your definition of parametrization is. You just sight examples of (I assume) polar curves. What does that have to do with parametrization?

 

[royblaze]

I guess my definition of parameterization is just simply taking a given equation r(t) equation and using different values of t to find see how the function looks on a graph. The "parameter" comes from t being what defines the r(t)...

That's what I think. :P

 

[royblaze]

Is that right? I'm still unsure.

 

[AlephZero]

I'm not sure exactly what you are asking here, but parameterizations of curves are not unique. The only reason for using a partiular parameterization is because it happens to be useful.

As a trivial example, the non-parametric equation of a circle $x^2 + y^2 = a^2$ could be parameterized as $x = a \cos t$, $y = a \sin t$, or $x = 2at / (1+t^2)$, $y = a(1-t^2) / (1 + t^2)$, or in polar coordinates as $r = a$, (where the parameter $t$ doesn't appear at all in the parameterized eqations!) etc.

Parameterizations are useful for much more than just plotting curves from points - for example finding the length of a curve by integration, finding tangents, normals, and curvature at any point in terms of the parameter, etc.