How to Calculate a Circle's Equation in 3D Space?

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To calculate the equation of a circle in 3D space centered at the origin of a sphere, one must recognize that a single point on the sphere's surface does not uniquely define the circle; a direction is also needed. The discussion emphasizes that a curve in three dimensions cannot be represented by a single equation, as it typically requires parametric equations or the intersection of two surfaces. To define a unique great circle, two points on the sphere must be specified. The use of spherical coordinates, including radius, theta, and phi angles, is suggested for calculating the circle's coordinates. Understanding these concepts is crucial for accurately determining the circle's equation in 3D space.
dehseth
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Hey everyone,
I would like to find out equation of circle in 3D space.
My problem's details are:

I have a sphere, and I need to calculate circle equation which its center is on sphere's center.
To be clear, think about a sphere in x,y,z coordinate system where it's origin is at 0,0,0. And select a dot on the sphere surface. Now draw a circle which it's origin is at 0,0,0 (same as sphere) and also selected dot proves the circle (circle's at selected dot)...
Well how can I calculate this circle's x,y,z values. :confused:

It would be better if think this sphere at spherical system which is represented by radius r, teta angle, and phi angle. After finding these values I can calculate x,y,z values.

I hope it was clear.. Thank you everybody, if you cannot understand let me explain a lil bit more.. :smile:
 
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Yeap I saw that message before but it's kinda confusing... I am looking for a lil bit more clear answer...
 
A curve in 3 dimensions is not given by a single equation. A single equation in 3 dimensions always gives a 3- 1= 2 dimensional surface. For a curve you need either two equations (thinking of the curve as the intersection of two surfaces) or parametric equations: x, y, and z as functions of the parameter t, say.

However, a single point on the surface of a sphere does NOT define a unique circle, even requiring the circle to have the same center as the sphere. You also need to specify a direction for the circle.

Suppose you are given two points on the sphere. Then you have a unique "great circle" through those two points.
 

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