Equation for circle points in 3D

Nikkki · Nov 26, 2021

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

GeoGebra 3D view of a red x-axis, green y-axis and blue z-axis with three black points labelled A = (-9.86, 2.3, 0.42), B = (-7.33, -1.69, 1.26) and C = (-3.47, -3.55, 0.96).

3D plot with red, green and blue coordinate axes on a grey plane, showing a dashed-and-solid blue closed loop with points labelled A, B and C on the solid arc.

EXAMPLE 2

3D coordinate axes with a grey plane, showing points A = (-0.82, 2.96, 2.45), B = (2.89, 1.67, 5) and C = (3.05, 0.47, 1.36) plotted in space.

3D coordinate plot with red x-axis, green y-axis and blue z-axis, showing a blue circle with points labeled A, B and C on it, above a grey plane.

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.

3D coordinate axes with a dashed blue circle-like curve passing through labeled points A to M, with points B, C, D, E, F, G, H, I, J, K, L and M marked on it.

3D plot of a closed dashed circle with red points E, F, G, H, J, K, L, M, D, and black points A, B, C, on axes with a grey plane and green and red axis lines.

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.

Triangle with vertices A, B and C in 3D, labelled A = (-0.82, 2.96, 2.45), B = (2.89, 1.67, 5), C = (3.05, 0.47, 1.36); side lengths AB = 4.69, CB = 3.84, AC = 4.73.

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

Text reading that the distances between three points in 3D space give the triangle side lengths a, b, c, followed by the circumradius formula R = abc over the square root of (2a²b² + 2b²c² + 2c²a² − a⁴ − b⁴ − c⁴), and a note that the radius becomes infinite when the points are collinear.

And at this step, I have now stopped

mfb · Nov 26, 2021

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.

Nikkki · Nov 29, 2021

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

jim mcnamara · Nov 29, 2021

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

Equation for circle points in 3D

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