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

In summary: The center of the sphere will be the midpoint of the line segment between those two points. The "equator" of the sphere is a "great circle" which goes through the two points (1, 0, 0) and (-1, 0, 0). The "prime meridian" goes through the two points (0, 1, 0) and (0, -1, 0).In summary, the conversation is about finding the equation of a circle in 3D space, specifically on a sphere with its center at the origin. The person is looking for a clear explanation of how to calculate the x, y, and z values of the circle, and someone suggests using param
  • #1
dehseth
25
4
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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  • #3
Yeap I saw that message before but it's kinda confusing... I am looking for a lil bit more clear answer...
 
  • #4
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.
 

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

1. What is the equation for a circle in 3D space?

The equation for a circle in 3D space is (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2, where (a,b,c) is the center point and r is the radius. This equation is derived from the Pythagorean theorem.

2. How is the circle equation in 3D space different from the 2D equation?

The 2D equation for a circle is (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center point and r is the radius. In 3D space, there is an additional variable z, making the equation (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2. This allows the circle to exist in three dimensions rather than just two.

3. Can the circle equation in 3D space be used to model real-world objects?

Yes, the circle equation in 3D space can be used to model real-world objects such as spheres, cylinders, and any other shape that can be created by rotating a circle around an axis. It is a fundamental equation in geometry and is used in various fields such as engineering, physics, and architecture.

4. How is the radius of a circle in 3D space determined using the equation?

The radius of a circle in 3D space is determined by taking the square root of the number on the right side of the equation. This number represents the distance from the center point to any point on the circle. Therefore, the square root of this number gives us the radius of the circle.

5. Can the circle equation in 3D space be generalized to other shapes?

Yes, the circle equation in 3D space can be generalized to other shapes by replacing the squared terms with other equations that represent those shapes. For example, a cylinder can be represented by (x-a)^2 + (y-b)^2 = r^2 and a sphere can be represented by (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2. This allows for the creation of various geometric shapes in 3D space.

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