Thread: Intersection of cylinder and plane

"Let S denote the elliptical cylinder given by the equation 4y²+ z²=4, and let C be the curve obtained by intersecting S with the plane y=x.

Parameterize C


I am not sure how to go about this. I tried solving the equations for each other

4y²= 4-z
y²= 1- z²/4

y= sqrt(1-z²/4)

and then do i do some sort of parameterization of sqrt(1-z²/4) -x=0 or am i supposed to parameterize the cylinder and plane separately? I know the cylinder parameterizes to
y=cos(theta)
z=2sin(theta)
but i don't know what to do from here. If anyone could hit me with some tips i would greatly appreciate it



EDIT:

would it simply be <cos (t), cos (t), 2 sin (t)> as x=y and y is equal to cos(t)? is it really this simple?

Originally Posted by bchemmath

would it simply be <cos (t), cos (t), 2 sin (t)> as x=y and y is equal to cos(t)? is it really this simple?

Yes, that's correct.

- Hollywood

The next part says:
" Use the parametrization above to compute the unit tangent vector, the principal normal vector, and the binormal vector at each of the two points where C intersects the xy-plane"

I know how to solve for unit tangent vector/ principal normal vector/binormal vector, etc but it says at each of the two points where C intersects the xy plane. Doesn't C intersect it in its entire domain? Should i have parameterized this differently without using sin/cos? or is it just at 0 and 2pi since that is the edge of the domain? i am slightly confused if anyone could please explain to me

The xy plane is where z is zero. Since in your parameterization of the curve, , it crosses the xy plane where , i.e. at and .

- Hollywood