How to calculate the centre point of a circle given three points?

Question

I am using Javascript and I know the positions of 3 points. I wish to use these to find out the center point of a circle.

I have found this logic (Not the chosen answer

Answer 1

My favorite resolution:

Translate the three points to bring one of them at the origin (subtract (X0,Y0)).

The equation of a circle through two points and the origin can be written

2X.Xc + 2Y.Yc = X² + Y²

Plugging the coordinates of the two points, you get an easy system of two equations in two unknowns, and by Cramer

Xc = (Z1.Y2 - Z2.Y1) / D
Yc = (X1.Z2 - X2.Z1) / D

D = 2(X1.Y2 - X2.Y1), Z1 = X1²+Y1², Z2 = X2²+Y2²

to be translated back (add (X0,Y0)).

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The formula fails when the three points are aligned, which is diagnosed by D = 0 (or small in comparison to the numerators).

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        X1-= X0; Y1-= Y0; X2-= X0; Y2-= Y0;

        double Z1= X1 * X1 + Y1 * Y1;
        double Z2= X2 * X2 + Y2 * Y2;
        double D= 2 * (X1 * Y2 - X2 * Y1);

        double Xc= (Z1 * Y2 - Z2 * Y1) / D + X0;
        double Yc= (X1 * Z2 - X2 * Z1) / D + Y0;

Answer 2

Thanks to @Gaurav Ojha in the comments I found this solution : What is the algorithm for finding the center of a circle from three points?

And changed it to work with Javascript :

function CalculateCircleCenter(A,B,C)
{
    var yDelta_a = B.y - A.y;
    var xDelta_a = B.x - A.x;
    var yDelta_b = C.y - B.y;
    var xDelta_b = C.x - B.x;

    center = [];

    var aSlope = yDelta_a / xDelta_a;
    var bSlope = yDelta_b / xDelta_b;

    center.x = (aSlope*bSlope*(A.y - C.y) + bSlope*(A.x + B.x) - aSlope*(B.x+C.x) )/(2* (bSlope-aSlope) );
    center.y = -1*(center.x - (A.x+B.x)/2)/aSlope +  (A.y+B.y)/2;
    return center;


}

All you need to do is pass it 3 points :

 var threePoints = [{x:1, y: 2},{x:4, y: 4},{x:6, y: 2} ]

console.log(CalculateCircleCenter(threePoints[0],threePoints[1],threePoints[2]))

To get this answer :

[x: 3.5, y: 1.5]

Hope this helps :)