How to find a parmetrization of..

Question

..of this curve $x^2+y^2=1$ and $z=x^2$ anticlockwise from (1,0,1) to (-1,0,1). I sort of understand how to find a parmetrization when the curve is simpler but here I'm lost.

Answer 1

We can take

• $x=\cos t$

• $y=\sin t$

• $z=\cos^2 t$

now we need to find a correct range for $t$.

Answer 2

Parameterized in polar angle $\theta,$ the space curve with periodic repetition for all integers $n:$

$$ (x,y,z) = (\cos \theta, \sin \theta, \cos^2 \theta),(0<\theta <2 n \pi) $$