MHB How do I find the Euclidean Coordinate Functions of a parametrized curve?

[banananaz]

I've been given a curve α parametrized by t :

α (t) = (cos(t), t^2, 0)

How would I go about finding the euclidean coordinate functions for this curve? I know how to find euclidean coord. fns. for a vector field, but I am a bit confused here.

(Sorry about the formatting)

 

[HOI]

I'm not certain what you mean by "Euclidean Coordinates". Perhaps you mean what I would call "Cartesian Coordinates"- a direct relation of x and y. The equations you have are x= cos(t), y= t^2, z= 0. The second equation can be written $t= \pm\sqrt{y}$ so we get two equations, $t= \sqrt{y}$ and $t= -\sqrt{y}$ and then $x= cos(\sqrt{y})$ and $x= cos(-\sqrt{y})$.

But since cosine is an "even function", $cos(-\theta)= cos(\theta)$, those both give $x= cos(\sqrt{y})$.