Parametrizing the intersection of a cylinder and a sphere I need to parametrize the intersection between the cylinder $
x^2 + y^2= \frac{1}{4}$ and the sphere $(x+ \frac{1}{2})^2 + y^2 +z^2 = 1$. 
I tried parametrizing the first equation which gives $r(t) = (\frac{cos(t)}{2}, \frac{sin(t)}{2})$ since the radius is 1/2. 
Then, I plugged in the values in the second equation which yields $(\frac{cos(t) +1}{2})^2  + \frac{sin^2(t)}{4} + z^2 = 1$. We isolate $z$ and get that $z = \sqrt{1-\frac{cos(t)}{2}}$. I'm stuck here because the parametrization would be incomplete if we choose the positive or negative root, am I doing something wrong? If so, what would a correct parametrization be?
EDIT: I've just recalled that $\sqrt{\frac{1-cos(t)}{2}} = sin(\frac{x}{2})$ or $cos(\frac{x}{2})$. Is this helpful?
 A: $x^2+y^2=\frac{1}{4}$
$\left( x+\frac{1}{2} \right)^2 + y^2 + z^2 = 1$
Expand
$x^2 + x + \frac{1}{4} + y^2 + z^2 = 1$
Collect
$\left(x^2+y^2\right)+x+\frac{1}{4}+z^2=1$
Substitute first equation
$\frac{1}{4}+x+\frac{1}{4}+z^2=1$
The key equations now being$\ldots$
$x+z^2=\frac{1}{2}$
$x^2+y^2=\frac{1}{4}$
We pick $z=t$ hoping for a cleaner solution...
$x=\frac{1}{2}-t^2$
$x^2+y^2=\left( \frac{1}{2}-t^2 \right)^2+y^2=t^4-t^2+\frac{1}{4}+y^2=\frac{1}{4}$
$y^2=t^2-t^4$
$y^2=t^2\left(1-t^2\right)$
$y=\pm t\sqrt{1-t^2}$
And we write...
$\left(\frac{1}{2}-t^2,\pm t\sqrt{1-t^2},t\right)$
for $t \in \left[-1,1\right]$
and I expect someone else to point out the algebra mistake if there is one because there's alot of expansion in my approach, but I checked it and I'm pretty sure it's consistent.

Because the cylinder is of lesser radius, the intersection is a single point at the xy plane. The vertical (xy) projection of the curve is a circle. The lateral (xz) cross section of the curve is a parabola.



A: Your parametrization is OK.
Note that the intersection of the cylinder and the sphere in this case is not just one but two closed curve meeting at the origin, sort of like a figure $8$.
I do not see how we can fit both in one parametric equation.
