# Equation for circle points in 3D

• I
• Nikkki
In summary, the conversation involved a person seeking help with creating an algebraic formula to calculate the coordinates for 10 points on a circle composed of three points in 3D space. The first idea was to create a triangle inscribed in the circle and then calculate the radius. The person was directed to a website for further assistance and was advised to think about the bisector for a non-collinear solution.
Nikkki
TL;DR Summary
Calculate the coordinates of consecutive points based on ABC points lying on a circle in 3D space
Hello,

I am trying to solve a problem and I would like to ask for help.

I have 3 points (A, B, C) in 3D space that are assumed to be on a circle.

EXAMPLE 1
EXAMPLE 2

My goal is to create an algebraic formula to calculate the coordinates for 10 points on a circle composed of ABC points at any distance from each other.

MY IDEA

My first idea was to create a triangle inscribed in a circle from the ABC points and then the radius of the circle.
First, I calculate the lengths of the triangle's legs by recalculating the lengths of |AB| |AC| and |CB| vectors.

I calculate the radius length using the formula (https://www.physicsforums.com/threads/equation-of-a-circle-through-3-points-in-3d-space.173847/):

And at this step, I have now stopped

The center has a distance r to all these points. Can you find its coordinates as function of e.g. a, (b-a) and (c-a), e.g. the position of a and two sides? This is completely analogous to the two-dimensional problem.

jim mcnamara
mfb said:
The center has a distance r to all these points. Can you find its coordinates as function of e.g. a, (b-a) and (c-a), e.g. the position of a and two sides? This is completely analogous to the two-dimensional problem.
Thank you for your answer.
I am testing the formula for the circle center coordinates in the "Cartesian coordinates from cross- and dot-products" section on the website : https://en.wikipedia.org/wiki/Circu...sian_coordinates_from_cross-_and_dot-products

Hint: think bisector - the problem says 'lies on a circle' which has to imply that the points are not collinear.

## 1. What is the equation for finding the points of a circle in 3D?

The equation for finding the points of a circle in 3D is (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2, where (a,b,c) represents the center of the circle and r represents the radius.

## 2. How is this equation different from the equation for a circle in 2D?

The equation for a circle in 2D is (x-a)^2 + (y-b)^2 = r^2, which only includes the x and y coordinates. In 3D, the equation includes an additional term for the z coordinate, making it a more complex equation.

## 3. Can this equation be used for any type of circle in 3D?

Yes, this equation can be used for any type of circle in 3D, including circles that are tilted or positioned at different angles.

## 4. How can this equation be applied in real-world situations?

The equation for a circle in 3D can be used in various fields such as engineering, physics, and computer graphics. It can be used to calculate the coordinates of points on a circular object, determine the trajectory of a moving object, or create 3D models of circular shapes.

## 5. Are there any limitations to using this equation for circles in 3D?

One limitation of this equation is that it only works for perfect circles, meaning that all points on the circle are equidistant from the center. It may not accurately represent irregular or distorted circular shapes. Additionally, the equation assumes a Euclidean space, so it may not be applicable in non-Euclidean spaces.

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