Is there a workable numerical method for determining the center of a circle through three points?

Question

I'm a 73-year-old engineer working on a kinematic linkage project that generates motion paths (as long sequences of x,y coordinates) of the end points of linkages and dumps the data into a file to be imported into Excel for analysis.

The analysis involves moving along the linkage motion path data taking three consecutive data points and calculating the radius (curvature) and circle center point (x,y) coordinates. This all needs to be automated in Excel calculations for hundreds of triads of points.

It's not a difficult problem to do on paper with the usual equations on a single data set, but it gets excessively complicated as Excel equations.

Excel is very limited with determinants, so that approach is probably out.

I'm thinking that the most straightforward approach would be to find the intersection of the two perpendicular bisector lines of the two secant lines created by the three consecutive data points.

Or, is there a better way of computationally solving this?

Answer 1

From Socratic.org:

Question:

How to find the center of a circle given the three points (-3,5); (3,3); (11,19)?

Answer:

(3,13)



The center (a,b) is the same distance = r from each of the three points on the circle.

This gives a = 3 and b = 13..

Explanation:

The points are equidistant from the center (a, b). Equating squares of the distances,

(a+3)²+(b−5)² = (a−3)²+(b−3)² = (a−11)²+(b−19)²,

From the first two,

3a−b = 4, and from first and the third,

a+b = 16.

solving, (a, b) = (3, 13)...