# Parameterizing A Circle Projected onto a Plane

by TranscendArcu
Tags: circle, parameterizing, plane, projected
 P: 288 1. The problem statement, all variables and given/known data Find a vector function that parameterizes a curve C which lies in the plane x-y+z=2 and directly above the circle x2 + (y-1)2 = 9 3. The attempt at a solutionSo, in order to parameterize the circle, I simply use x=cos(t), y = sin(t) with some adjustments. Namely, I let x=3cos(t) and y=3sin(t)+1. Then, to determine how the z-coordinates of this curve change according to x,y I use the definition of the plane. So I have z=2-x+y, which I convert using my parametric defintions for x,y. I have, z=3-3cos(t)+3sin(t). Therefore, my parameterized curve is, <3cos(t),3sin(t)+1,3-3cos(t)+3sin(t)> Right?
 HW Helper P: 6,189 Looks good. But... is it always above the circle?
 P: 288 So I just drew myself a little sketch. Basically I have plane and, beneath it, a cylinder which extends upwards, whose base represents the circle. The intersection of the cylinder and the plane should be the projection of the circle onto the plane, right? In this case, I can't see how the projection could be anywhere else besides over the circle. As an aside, the vector projection of a onto b is given as follows: ((b [dot] a)/|a|) * a/|a|. Is it possible to use the idea of projections along with dot products to solve this problem?
HW Helper
P: 6,189
Parameterizing A Circle Projected onto a Plane

 Quote by TranscendArcu So I just drew myself a little sketch. Basically I have plane and, beneath it, a cylinder which extends upwards, whose base represents the circle. The intersection of the cylinder and the plane should be the projection of the circle onto the plane, right? In this case, I can't see how the projection could be anywhere else besides over the circle.
Consider t=-pi/4.
Is the corresponding point above or below the circle?

 Quote by TranscendArcu As an aside, the vector projection of a onto b is given as follows: ((b [dot] a)/|a|) * a/|a|. Is it possible to use the idea of projections along with dot products to solve this problem?
This is the formula for projection onto a line.
It doesn't work for projection onto a plane.

Anyway, you still need a parametrization.