Equation of circle in 3 dimensions

[davon806]

Homework Statement

Hi,
I am trying to find the general formula of a circle in 3D
Let's consider a sphere centred at (x1,0,0),with radius x1
It's equation is (x-x1)^2 + y^2 + z^2 = (x1)^2
If there is a plane x = x1 intersects with the sphere
A equation of a circle is formed,which is y^2 + z^2 = (x1)^2
This give the equation of circle at x = x1,with radius x1,in the 3-dimensional space.

My question is:
Is there a general formula of equation of circle in 3 dimensional retangular coordinates system,which involves 3 independent variables?So that we don't have to specify "The equation of circle at x = x1.",we could just include the information of x-coordinate in the equation?

I hope someone can understand what I mean..
If there is any mistake please point out,thx very much.


If there is a plane x=x1 intersects with the sphere

Homework Equations

The Attempt at a Solution

 

[phinds]

I seem to remember from WAY back that you can transform 3D equations from one 3D coordinate system to another (you'll need trig), so you could write the general equation of a 2D circle in the XY plane and then add a Z component to move it up/down, then transform it to a system with the same origin but with each axis at a different angle. This, assuming that it can even be done, would give you the general equation you want but I would expect it to be REALLY ugly.

 

[LCKurtz]

One way to specify a 3D circle is to give the plane of the circle, the center of the circle, and its radius. So you have 7 parameters with which to generate a parametric curve. It's an easy calculation for Maple, but the final equation is very long and complicated in the general case. If you have specific numbers, it's not so bad, especially if you let Maple do the grunt work.

 

[HallsofIvy]

A single equation in 3 dimensions describes a 3-1= 2 dimensional surface. In order to describe a one dimensional curve, such as a circle, in three dimensions you must use 2 equations: 3- 2= 1. For example, the two equations, $(x- a)^2+ (y- b)^2= R^2$, describe the circle with center at (a, b, c), radius R, lying in the z= c plane.

Another way to describe a one dimensional figure, such as a circle, in three dimensions is to write the three coordinates, x, y, and z, in terms of one parameter. For example, the three equations x= R cos(t), y= R sin(t), z= c also describe the circle with center at (a, b, c), radius R, lying in the z= c plane.