Parametrizing Intersection of Cylinder and Plane

[dbkats]

Homework Statement

Let C be a cylinder of radius 1. It is cut by the x-y plane from below, and by the plane z-x=1 above. Parametrize all the surfaces of the cylinder. Find a unit normal (pointing outward) for each surface.

Homework Equations

Equation for a cylinder:
$$x^2+y^2=1$$
Equation of the plane:
$$z-x=1$$

The Attempt at a Solution

Bottom:
$$g(r,\theta)=(rcos(\theta),rsin(\theta), 0)$$
$$n=(0,0,-1)$$
Side:
I'm not so certain about this one... it should be a function of the height as well as the angle, but I'm not certain how to restrict the angle to depend on the height...
I guess something like $$x^2+y^2 \le z-x$$
I think this is the normal, though...
$$n=(x,y,0)$$
Top:
I should compute the intersection of the plane and the cylinder. So I get
$$x^2+y^2=z-x$$
$$z=(x-1/2)^2+y^2-1/4$$
And what now?

 

[LCKurtz]

Homework Statement

Let C be a cylinder of radius 1. It is cut by the x-y plane from below, and by the plane z-x=1 above. Parametrize all the surfaces of the cylinder. Find a unit normal (pointing outward) for each surface.

Homework Equations

Equation for a cylinder:
$$x^2+y^2=1$$
Equation of the plane:
$$z-x=1$$

The Attempt at a Solution

Bottom:
$$g(r,\theta)=(rcos(\theta),rsin(\theta), 0)$$
$$n=(0,0,-1)$$

Looks good for the bottom.

Side:
I'm not so certain about this one... it should be a function of the height as well as the angle, but I'm not certain how to restrict the angle to depend on the height...
I guess something like $$x^2+y^2 \le z-x$$
I think this is the normal, though...
$$n=(x,y,0)$$

Your original thought, height and angle are the natural parameters. So try using $z$ and $\theta$ as your parameters.

Top:
I should compute the intersection of the plane and the cylinder. So I get
$$x^2+y^2=z-x$$
$$z=(x-1/2)^2+y^2-1/4$$
And what now?

The top is the slanted plane $z=1+x$, not the "intersection of the plane and cylinder". You might try $x$ and $y$ as the parameters.