Centre and radius of a circle passing through 3 points in a plane

Centre and radius of a circle passing through 3 points in a plane is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Write a function which returns the centre and radius of a circle passing through three point in a plane. Demonstrate the function using the points (22.83,2.07) (14.39,30.24) and (33.65,17.31)

F#

// Centre and radius of a circle passing through 3 points in a plane. Nigel Galloway: February 20th., 2024
let c (a,b) (c,d) (e,f)=(0.5*((a*a+b*b)*(f-d)+(c*c+d*d)*(b-f)+(e*e+f*f)*(d-b))/(a*(f-d)+c*(b-f)+e*(d-b)),
                         0.5*((a*a+b*b)*(e-c)+(c*c+d*d)*(a-e)+(e*e+f*f)*(c-a))/(b*(e-c)+d*(a-e)+f*(c-a)))
let d n g = let n,g=fst n-fst g,snd n-snd g in sqrt(n*n+g*g)/2.0
let circ P1 P2 P3 = let c=c P1 P2 P3 in (c,d c P1)

let centre,radius=circ (22.83, 2.07) (14.39, 30.24) (33.65, 17.31)
printfn $"Centre = %A{centre}, radius = %f{radius}"

Output:

Centre = (18.97851566, 16.2654108), radius = 7.354312

Phix

Translation of: F#

with javascript_semantics
function circle(sequence p1, p2, p3)
    atom {a,b} = p1, {c,d} = p2, {e,f} = p3,
         a2b2 = a*a+b*b,
         c2d2 = c*c+d*d,
         e2f2 = e*e+f*f,
         ae = a-e,
         bf = b-f,
         ca = c-a,
         db = d-b,
         ec = e-c,
         fd = f-d,
         cx = 0.5*(a2b2*fd+c2d2*bf+e2f2*db)/(a*fd+c*bf+e*db),
         cy = 0.5*(a2b2*ec+c2d2*ae+e2f2*ca)/(b*ec+d*ae+f*ca),
         r = sqrt(power(cx-e,2)+power(cy-f,2))
    return {{cx,cy},r}
end function

atom {{cx,cy},r} = circle({22.83,2.07},{14.39,30.24},{33.65,17.31})
printf(1,"Centre = %.8f, %.7f, radius = %.6f\n",{cx,cy,r})

Output:

Centre = 18.97851566, 16.2654108, radius = 14.708624

Wren

This follows the lines of the C++ code here.

I've used [22.83, 2.07] for the first point, rather than [22.82, 2.07] as stated in the task description, to give agreement on the centre with the F# and Phix examples. However, my radius is twice theirs and looks right to me.

var findCircle = Fn.new { |x1, y1, x2, y2, x3, y3|
    var x12 = x1 - x2
    var x13 = x1 - x3
    var y12 = y1 - y2
    var y13 = y1 - y3
    var y31 = y3 - y1
    var y21 = y2 - y1
    var x31 = x3 - x1
    var x21 = x2 - x1

    var sx13 = x1 * x1 - x3 * x3
    var sy13 = y1 * y1 - y3 * y3
    var sx21 = x2 * x2 - x1 * x1
    var sy21 = y2 * y2 - y1 * y1

    var f = sx13 * x12 + sy13 * x12 + sx21 * x13 + sy21 * x13
    f = f / (y31 * x12 - y21 * x13) / 2

    var g = sx13 * y12 + sy13 * y12 + sx21 * y13 + sy21 * y13
    g = g / (x31 * y12 - x21 * y13) / 2

    var c = -x1 * x1 - y1 * y1 - 2 * g * x1 - 2 * f * y1
    var h = -g
    var k = -f
    var r = (h * h + k * k - c).sqrt
    return [h, k, r]
}

var hkr = findCircle.call(22.83, 2.07, 14.39, 30.24, 33.65, 17.31)
System.print("Centre is at %([hkr[0], hkr[1]])")
System.print("Radius is %(hkr[2])")

System.print("\nCheck radius as the distance between the centre and the first point:")
System.print(((22.83 - hkr[0]).pow(2) + (2.07 - hkr[1]).pow(2)).sqrt)

Output:

Centre is at [18.978515660149, 16.265410797716]
Radius is 14.708623978334

Check radius as the distance between the centre and the first point:
14.708623978334