Are there any other ways of parametizing S?

[Sneaksuit]

If S is part of the plane $$z = y - 2$$ that lies inside the elliptic cylinder $$x^2 + 4y^2 = 4$$ and I want to parametrize S I will let
$$x = 2sin(t)$$
$$y = cos(t)$$
$$z = cos(t) - 2$$
I assume this is right but let me know if not. My question is are there any other ways of parametizing S?

 

[whozum]

I think you ahve your cosins and sins switched. Which direction does your parameter go? I am assuming counter clockwise.

Come to think of it keeping them that way will also work, provided that you aren't going for a specific parametric starting and ending point.

 

[Sneaksuit]

There is no specific starting point. Do you know of another way parametrizing S though?

 

[SpaceTiger]

There is no specific starting point. Do you know of another way parametrizing S though?

Basically as he said, make every cosine a sine and the sine a cosine. I suppose you could also add an arbitrary phase shift to the trig functions as well (the sine-cosine reversal is a special case of that).

 

[Sneaksuit]

So just switch my sin and cosin and that is another way of parametrizing S?

 

[kleinwolf]

I don't really understand your solution, since apparently you're parametrizing a surface (so you need 2 parameters). I suppose the solution is simply :

$$x=Rcos(t)\quad y=\frac{R}{2}sin(t)\quad z=\frac{R}{2}sin(t)-2\quad R\in[0;2]\quad t\in[0;2\pi]$$

 

[whozum]

So just switch my sin and cosin and that is another way of parametrizing S?

Yes. It will draw the same curve but from a different starting point and direction.

 

[whozum]

Those parameters would draw the curve of intersection. Parametrizing the entire surface, you would use klein's equation