Question

what is the curve of intersection of the surfaces x^2+z^2 = 1; and y = x^2

Answer 1

How would you come up with a parametrization of this curve?

Answer 2

or let's try it this way.
the curve of intersection of surfaces x^2+z^2 = 1; and y = x^2 can be parametrized :
r = (cost, cos(t)^2, sint) ..........(vector).. how did this happen?

Answer 3

Ah, I figured it out in case anyone was interested, wasn't really that hard. but i hate parametrizations.
x^2=1-z^2
x^2 = y

let x = cost
then. y = cos^(2)t
and z = sqrt(1-(cost)^2))

the function on the inside of the square root is alternately sin(t)^2
so z = sint
so r(t) = (cost, cos(t)^2, sint)