# Intersection of cylinder and plane

1. Jan 14, 2006

### dagar

I'm trying to find the parameterization of the intersection of a cylinder x^2+y^2=1 and the plane x+y+z=1, but I'm not exactly sure how to go about it. Any guidance on how to find this intersection in a parameterized form would be most appreciated.

In general I don't know a great deal about finding intersections of various surfaces and shapes in r^3 or how to parameterize these things. Googling hasn't turned up anything particularly usefull, but a few scattered examples. I was also wondering if anyone knew of useful sites with this information.

Thanks

2. Jan 14, 2006

### benorin

parameterization of cylinder and plane intersection

The parameterization of the cylinder $x^2+y^2=1$ is standard:

Let x(t)=cos(t) and let y(t)=cos(t) for $0\leq t < 2\pi$.

We wish to parameterize the intersection of the above cylinder and the plane x+y+z=1, solving this for z gives z=1-x-y so we see that if we put

z(t) = 1-x(t)-y(t) = 1-cos(t)-sint(t) for $0\leq t < 2\pi$,

then the parameterization we seek is given by:

$$\vec{r}(t) = \left< x(t),y(t),z(t)\right> = \left< \cos (t),\sin (t), 1- \cos (t) - \sin (t)\right> ,\mbox{ for }0\leq t < 2\pi$$.

-Ben