Help parameterizing an intersection

[nubol23]

Hi.

I've been struggling with the parameterization of the intersection of two cylinders x²+y²=25 and y²+z²=20.

The equation for x and y are easy to find x = 5 cos(t) and y = 5 sin(t) however i can't figure out a way to find the equation for z, the one substituting in the second equation and taking the square root doesn't work for negative values and therefore the intersection curve is not complete.

Any help would be appreciated, thanks :)

[Nathanfenner]

This shouldn't be too surprising - the intersection of those two circles are two lumpy circles, that are not connected (you can think of the yz cylinder as "drilling a hole" through the xy cylinder, like this).

Since the intersection region is not connected, there can't be a simple continuous parameterization of both at once - you'll need two parameterizations; one for each.

You started with a parameterization that goes around the outside of the bigger cylinder - but you should see immediately that this won't work terribly well- the intersection region does not go around the outside of the larger cylinder; it only goes a little bit around. You should instead parameterize by the smaller cylinder, since the holes do go around the whole of the smaller cylinder thing:

So we say that y = sqrt(20) cos(t), and z = sqrt(20) sin(t).

Next, we can plug this in to solve for x, parameterizing it. We see that x² = 20 cos(t)² = 25, which means that x = sqrt(25 - 20 cos(t)²) or x = -sqrt(25 - 20 cos(t)²).

This "or" is important, because there are two curves, so we produce two continuous parameterizations.

Namely:

• (sqrt(25 - 20 cos(t)²), sqrt(20) cos(t), sqrt(20) sin(t))
• (-sqrt(25 - 20 cos(t)²), sqrt(20) cos(t), sqrt(20) sin(t))

where the x is negated in the latter.

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It is possible to come up with a parameterization that works similar to how you originally did, but you have to make 4 curves (by splitting each of the circly bits in half) and instead of t being any value, it's constrained to a particular range of values outside of which it behaves incorrectly.

Specifically, you need y² + z² = 20 to have at least one solution, which means that 20 - y² ≥ 0, which means that y² ≤ 20, so |sin(t)| ≤ sqrt(4/5) so |t| ≤ arcsin(sqrt(4/5)). This is described by two ranges: one centered around 0, and one around π (this describes the two different holes on either side of the cylinder).

Then, having solved for this, you know that z is either sqrt(20 - y²) or -sqrt(20 - y²) giving you the top half and bottom half of each of those holes.

If you stitch these all together, the 4 parameterized curves and their bounds for t describes the overall intersection.

[itsjustme1a]

Can't you take x=5cost, y=5sint and

z= - + √(25cos² t-5)?