Parameterization of a Circle Question

[royblaze]

This isn't really a HW question, it's just something that's been confusing me in my Calc class.

We recently went over how to find curvatures of curves in 3D space. In lecture, the professor went over a simple example: a circle of radius 3 at any given point.

Maybe it's because I don't remember my Calc II class that well, but he parameterized the equation of a circle,

x² + y² = 9

to become r(t) = 3cos(t)i + 3sin(t)j .

So from:

x² + y² = r²

If x = cos(t) and y = sin(t), the identity cos²(t) + sin²(t) = 1 still holds true.

If the radius is 3,

x² + y² = 3²

3(x² + y²) = 9(1²)

9cos²(t) + 9sin²(t) = 9(1²)

9(cos²(t) + sin²(t)) = 9(1²)

is how I understand it. If there's an easier way to picture it, please let me know!


So then r(t) = 3cos(t)i + 3sin(t)j describes the circle when "traced."


My question: why does the "x" part and the "y" part of the parameterization have to be the respective cos, sin trig functions? In class, one student said that x was sin; but the prof said that no, it's cos. Why does x have to be cos, rather than sin? Does it have to do something with the direction the curve "goes in?" How can I determine the "direction" the curve goes in? Or did I just hear my professor wrong...?

 

[jncarter]

Try drawing a picture. Do you remember your unit circle? For a right-handed coordinate system, x = rcos$\theta$ and y the sine function. Positive theta corresponds to counter-clockwise motion from the positive x-axis.
Now when you parametrize the equation $\theta$ becomes $\dot{\theta}$t, where $\dot{\theta}$ is how quickly you are going around the circle. In your case this is simply one.
Strictly speaking, x doesn't HAVE to be the cosine. It's just convention to use a right handed coordinate system. If you measured $\theta$ from a different axis or you switched positive x being in the vertical, your sines and cosines may change.

 

[LCKurtz]

So then r(t) = 3cos(t)i + 3sin(t)j describes the circle when "traced."


My question: why does the "x" part and the "y" part of the parameterization have to be the respective cos, sin trig functions? In class, one student said that x was sin; but the prof said that no, it's cos. Why does x have to be cos, rather than sin? Does it have to do something with the direction the curve "goes in?" How can I determine the "direction" the curve goes in? Or did I just hear my professor wrong...?

Yes, it has to do with the orientation of the curve and where you want it to be when t = 0. Parameterizations are in general not unique. If the curve is specified to start at (1,0) and go counterclockwise then R(t) = < cos(t), sin(t) > is a natural choice. If it went clockwise you might use <cos(t), -sin(t)>. If nothing is given about the orientation of the curve then you could use <sin(t), cos(t)> or other choices; they are all equally correct.

 

[HallsofIvy]

This isn't really a HW question, it's just something that's been confusing me in my Calc class.

We recently went over how to find curvatures of curves in 3D space. In lecture, the professor went over a simple example: a circle of radius 3 at any given point.

Maybe it's because I don't remember my Calc II class that well, but he parameterized the equation of a circle,

x² + y² = 9

to become r(t) = 3cos(t)i + 3sin(t)j .

So from:

x² + y² = r²

If x = cos(t) and y = sin(t), the identity cos²(t) + sin²(t) = 1 still holds true.

If the radius is 3,

x² + y² = 3²

3(x² + y²) = 9(1²)

9cos²(t) + 9sin²(t) = 9(1²)

9(cos²(t) + sin²(t)) = 9(1²)

is how I understand it. If there's an easier way to picture it, please let me know!


So then r(t) = 3cos(t)i + 3sin(t)j describes the circle when "traced."


My question: why does the "x" part and the "y" part of the parameterization have to be the respective cos, sin trig functions? In class, one student said that x was sin; but the prof said that no, it's cos. Why does x have to be cos, rather than sin? Does it have to do something with the direction the curve "goes in?" How can I determine the "direction" the curve goes in? Or did I just hear my professor wrong...?

They don't have to be. There exist an infinite number of parametric equations describing any curve. $x= 3sin(t)$, $y= 3cos(t)$ are perfectly good parametric equations and, in fact, have the same "direction" as $x= 3cos(t)$ $y= 3sin(t)$: with the first, as t goes from 0 to $\pi/2$, (x, y) goes from (0, 3) to (3, 0), counter-clockwise around the circle and the second goes from (3, 0) to (0, 3), also counterclockwise.

But before you go telling your teacher that we say he/she is wrong, note that they do have different points for specific values of t. The parametric equations $x= cos(t)$, $y= sin(t)$ "start" (t= 0) at (1, 0) and then go around the circle. Perhaps for some reason, that is important.

 

[royblaze]

Wow, thank you all for the absolutely fantastic replies. It REALLY cleared it up for me, big time.

Thanks again!