View source for Smallest enclosing circle problem

{{draft task}}
The '''smallest enclosing circle problem''' (aka '''minimum covering circle problem''', '''bounding circle problem''') is a mathematical problem of computing the smallest circle that contains all of a given set of points in the Euclidean plane.

Initially it was proposed by the English mathematician James Joseph Sylvester in 1857.

;Task:
Find circle of smallest radius containing all of given points.

* Circle is defined by it's center and radius;
* Points are defined by their coordinates in n-dimensional space;
* Circle (sphere) contains point when ''distance between point and circle center <= circle radius''.


=={{header|11l}}==
{{trans|Nim}}
{{trans|Go}}

<syntaxhighlight lang="11l">T Point = (Float, Float)

T Circle
   Point c
   Float r

   F (c, r)
      .c = c
      .r = r

   F String()
      R ‘Center ’(.c)‘, Radius ’(.r)

F getCircleCenter(Float bx, by, cx, cy)
   V b = bx * bx + by * by
   V c = cx * cx + cy * cy
   V d = bx * cy - by * cx
   R Point((cy * b - by * c) / (2 * d), (bx * c - cx * b) / (2 * d))

F contains(Circle ci, Point p)
   R sqlen(ci.c - p) <= ci.r ^ 2

F encloses(Circle ci, [Point] ps)
   L(p) ps
      I !contains(ci, p)
         R 0B
   R 1B

F circleFrom3(Point a, b, c)
   V i = getCircleCenter(b.x - a.x, b.y - a.y, c.x - a.x, c.y - a.y)
   i.x += a.x
   i.y += a.y
   R Circle(i, distance(i, a))

F circleFrom2(Point a, b)
   V c = Point((a.x + b.x) * 0.5, (a.y + b.y) * 0.5)
   R Circle(c, distance(a, b) / 2)

F secTrivial([Point] rs)
   I rs.empty
      R Circle(Point(0, 0), 0)
   I rs.len == 1
      R Circle(rs[0], 0)
   I rs.len == 2
      R circleFrom2(rs[0], rs[1])
   assert(rs.len == 3, ‘There shouldn't be more than three points.’)

   L(i) 2
      L(j) i + 1 .< 3
         V c = circleFrom2(rs[i], rs[j])
         I encloses(c, rs)
            R c
   R circleFrom3(rs[0], rs[1], rs[2])

F welzlHelper([Point] &ps, [Point] rs, Int n)
   V rc = copy(rs)
   I n == 0 | rc.len == 3
      R secTrivial(rc)
   V idx = random:(n)
   V p = ps[idx]
   swap(&ps[idx], &ps[n - 1])
   V d = welzlHelper(&ps, rc, n - 1)
   I contains(d, p)
      R d
   rc.append(p)
   R welzlHelper(&ps, rc, n - 1)

F welzl([Point] ps)
   V pc = copy(ps)
   random:shuffle(&pc)
   R welzlHelper(&pc, [Point](), pc.len)

V Tests = [[Point(0.0, 0.0), Point(0.0, 1.0), Point(1.0, 0.0)],
           [Point(5.0, -2.0), Point(-3.0, -2.0), Point(-2.0, 5.0), Point(1.0, 6.0), Point(0.0, 2.0)]]

L(test) Tests
   print(welzl(test))</syntaxhighlight>

{{out}}
<pre>
Center (0.5, 0.5), Radius 0.707106781
Center (1, 1), Radius 5
</pre>

=={{header|Go}}==
{{trans|Wren}}
<syntaxhighlight lang="go">package main

import (
    "fmt"
    "log"
    "math"
    "math/rand"
    "time"
)

type Point struct{ x, y float64 }

type Circle struct {
    c Point
    r float64
}

// string representation of a Point
func (p Point) String() string { return fmt.Sprintf("(%f, %f)", p.x, p.y) }

// returns the square of the distance between two points
func distSq(a, b Point) float64 {
    return (a.x-b.x)*(a.x-b.x) + (a.y-b.y)*(a.y-b.y)
}

// returns the center of a circle defined by 3 points
func getCircleCenter(bx, by, cx, cy float64) Point {
    b := bx*bx + by*by
    c := cx*cx + cy*cy
    d := bx*cy - by*cx
    return Point{(cy*b - by*c) / (2 * d), (bx*c - cx*b) / (2 * d)}
}

// returns whether a circle contains the point 'p'
func (ci Circle) contains(p Point) bool {
    return distSq(ci.c, p) <= ci.r*ci.r
}

// returns whether a circle contains a slice of points
func (ci Circle) encloses(ps []Point) bool {
    for _, p := range ps {
        if !ci.contains(p) {
            return false
        }
    }
    return true
}

// string representation of a Circle
func (ci Circle) String() string { return fmt.Sprintf("{%v, %f}", ci.c, ci.r) }

// returns a unique circle that intersects 3 points
func circleFrom3(a, b, c Point) Circle {
    i := getCircleCenter(b.x-a.x, b.y-a.y, c.x-a.x, c.y-a.y)
    i.x += a.x
    i.y += a.y
    return Circle{i, math.Sqrt(distSq(i, a))}
}

// returns smallest circle that intersects 2 points
func circleFrom2(a, b Point) Circle {
    c := Point{(a.x + b.x) / 2, (a.y + b.y) / 2}
    return Circle{c, math.Sqrt(distSq(a, b)) / 2}
}

// returns smallest enclosing circle for n <= 3
func secTrivial(rs []Point) Circle {
    size := len(rs)
    if size > 3 {
        log.Fatal("There shouldn't be more than 3 points.")
    }
    if size == 0 {
        return Circle{Point{0, 0}, 0}
    }
    if size == 1 {
        return Circle{rs[0], 0}
    }
    if size == 2 {
        return circleFrom2(rs[0], rs[1])
    }
    for i := 0; i < 2; i++ {
        for j := i + 1; j < 3; j++ {
            c := circleFrom2(rs[i], rs[j])
            if c.encloses(rs) {
                return c
            }
        }
    }
    return circleFrom3(rs[0], rs[1], rs[2])
}

// helper function for Welzl method
func welzlHelper(ps, rs []Point, n int) Circle {
    rc := make([]Point, len(rs))
    copy(rc, rs) // 'rs' passed by value so need to copy
    if n == 0 || len(rc) == 3 {
        return secTrivial(rc)
    }
    idx := rand.Intn(n)
    p := ps[idx]
    ps[idx], ps[n-1] = ps[n-1], p
    d := welzlHelper(ps, rc, n-1)
    if d.contains(p) {
        return d
    }
    rc = append(rc, p)
    return welzlHelper(ps, rc, n-1)
}

// applies the Welzl algorithm to find the SEC
func welzl(ps []Point) Circle {
    var pc = make([]Point, len(ps))
    copy(pc, ps)
    rand.Shuffle(len(pc), func(i, j int) {
        pc[i], pc[j] = pc[j], pc[i]
    })
    return welzlHelper(pc, []Point{}, len(pc))
}

func main() {
    rand.Seed(time.Now().UnixNano())
    tests := [][]Point{
        {Point{0, 0}, Point{0, 1}, Point{1, 0}},
        {Point{5, -2}, Point{-3, -2}, Point{-2, 5}, Point{1, 6}, Point{0, 2}},
    }
    for _, test := range tests {
        fmt.Println(welzl(test))
    }
}</syntaxhighlight>

{{out}}
<pre>
{(0.500000, 0.500000), 0.707107}
{(1.000000, 1.000000), 5.000000}
</pre>

=={{header|jq}}==
{{works with|jq}}
'''Works with gojq, the Go implementation of jq'''

'''Adapted from [[#Wren|Wren]]'''

In the following, a Point (x,y) is represented by a JSON array [x,y],
and a Circle by a JSON object of the form {center, radius}.
<syntaxhighlight lang="jq">
# Vector sum of the input array of Points
def sum: transpose | map(add);

# The square of the distance between two points
def distSq:
  def sq: . * .;
  . as [$a, $b]
  | (($a[0] - $b[0]) | sq) +(($a[1] - $b[1]) | sq) ;

# The distance between two points
def distance:
  distSq | sqrt;

# Does the input Circle contain the point $p?
def contains($p):
  ([.center, $p] | distSq) <= .radius * .radius;

# Does the input Circle contain the given array of points?
def encloses($ps):
  . as $c
  | all($ps[]; . as $p | $c | contains($p));

# The center of a circle defined by 3 points
def getCircleCenter(bx; by; cx; cy):
  (bx*bx + by*by) as $b
  | (cx*cx + cy*cy) as $c
  | (bx*cy - by*cx) as $d
  | [(cy*$b - by*$c) / (2 * $d), (bx*$c - cx*$b) / (2 * $d)];

# The (smallest) circle that intersects 1, 2 or 3 distinct points
def Circle:
  if length == 1 then {center: .[0], radius: 0}
  elif length == 2
  then {center: (sum | map(./2)),
        radius: (distance/2) } 

  elif length == 3
  then . as [$a, $b, $c]
  | ([getCircleCenter($b[0] - $a[0];
                      $b[1] - $a[1];
                      $c[0] - $a[0];
                      $c[1] - $a[1]), $a] | sum) as $i
  | {center: $i, radius: ([$i, $a]|distance)}
  else "Circle/0 expects an input array of from 1 to 3 Points" | error
  end;
  
# The smallest enclosing circle for n <= 3
def secTrivial:
  . as $rs
  | length as $length
  | if $length > 3 then "Internal error: secTrivial given more than 3 points." | error
    elif $length == 0 then [[0, 0]] | Circle
    elif $length <= 2 then Circle
    else first(
       range(0; 2) as $i
       | range($i+1; 3) as $j
       | ([$rs[$i], $rs[$j]] | Circle)
       | select(encloses($rs)) )
    // Circle
    end;

# Use the Welzl algorithm to find the SEC of the incoming array of points
def welzl:
  # Helper function:
  #   $rs is an array of points on the boundary;
  #   $n keeps track of how many points remain:
  def welzl_($rs; $n):
    if $n == 0 or ($rs|length) == 3
    then $rs | secTrivial
    else 0 as $idx     # or Rand($n)
    | .[$idx] as $p
    | .[$idx] = .[$n-1]
    | .[$n-1] = $p
    | welzl_($rs; $n-1) as $d
    | if ($d | contains($p)) then $d
      else welzl_($rs + [$p]; $n-1)
      end
    end ;
  
  welzl_([]; length);

def tests:
    [ [0, 0], [0, 1], [1, 0] ],
    [ [5, -2], [-3, -2], [-2, 5], [1, 6], [0, 2] ]
;

tests
| welzl
| "Circle(\(.center), \(.radius))"
</syntaxhighlight>
{{output}}
<pre>
Circle([0.5,0.5], 0.7071067811865476)
Circle([1,1], 5)
</pre>

=={{header|Julia}}==
N-dimensional solution using the Welzl algorithm. Derived from the BoundingSphere.jl module at https://github.com/JuliaFEM. See also Bernd Gärtner's paper at people.inf.ethz.ch/gaertner/subdir/texts/own_work/esa99_final.pdf.
<syntaxhighlight lang="julia">import Base.pop!, Base.push!, Base.length, Base.*
using LinearAlgebra, Random

struct ProjectorStack{P <: AbstractVector}
    vs::Vector{P}
end
Base.push!(p::ProjectorStack, v) = (push!(p.vs, v); p)
Base.pop!(p::ProjectorStack) = (pop!(p.vs); p)
Base.:*(p::ProjectorStack, v) = (s = zero(v); for vi in p.vs s += vi * dot(vi, v) end; s)

"""
    GärtnerBoundary

See the passage regarding M_B in Section 4 of Gärtner's paper.
"""
mutable struct GärtnerBoundary{P<:AbstractVector, F<:AbstractFloat}
    centers::Vector{P}
    square_radii::Vector{F}
    # projection onto of affine space spanned by points
    # shifted such that first point becomes origin
    projector::ProjectorStack{P}
    empty_center::P # center of nsphere spanned by empty boundary
end

function GärtnerBoundary(pts)
    P = eltype(pts)
    F = eltype(P)
    projector, centers, square_radii = ProjectorStack(P[]), P[], F[]
    empty_center = F(NaN) * first(pts)
    GärtnerBoundary(centers, square_radii, projector, empty_center)
end

function push_if_stable!(b::GärtnerBoundary, pt)
    if isempty(b)
        push!(b.square_radii, zero(eltype(pt)))
        push!(b.centers, pt)
        dim = length(pt)
        return true
    end
    q0, center = first(b.centers), b.centers[end]
    C, r2  = center - q0, b.square_radii[end]
    Qm, M = pt - q0, b.projector
    Qm_bar = M*Qm
    residue, e = Qm - Qm_bar, sqdist(Qm, C) - r2
    z, tol = 2*sqnorm(residue), eps(eltype(pt)) * max(r2, one(r2))
    isstable = abs(z) > tol
    if isstable
        center_new  = center + (e/z) * residue
        r2new = r2 + (e^2)/(2z)
        push!(b.projector, residue / norm(residue))
        push!(b.centers, center_new)
        push!(b.square_radii, r2new)
    end
    return isstable
end

function Base.pop!(b::GärtnerBoundary)
    n = length(b)
    pop!(b.centers)
    pop!(b.square_radii)
    if n >= 2
        pop!(b.projector)
    end
    return b
end

struct NSphere{P,F}
    center::P
    sqradius::F
end

function isinside(pt, nsphere::NSphere; atol=0, rtol=0)
    r2, R2 = sqdist(pt, center(nsphere)), sqradius(nsphere)
    return r2 <= R2 || isapprox(r2, R2;atol=atol^2,rtol=rtol^2)
end
allinside(pts, nsphere; kw...) = all(pt -> isinside(pt, nsphere; kw...), pts)

function move_to_front!(pts, i)
    pt = pts[i]
    for j in eachindex(pts)
        pts[j], pt = pt, pts[j]
        j == i && break
    end
    pts
end

prefix(pts, i) = view(pts, 1:i)
Base.length(b::GärtnerBoundary) = length(b.centers)
Base.isempty(b::GärtnerBoundary) = length(b) == 0
center(b::NSphere) = b.center
radius(b::NSphere) = sqrt(b.sqradius)
sqradius(b::NSphere) = b.sqradius
dist(p1,p2) = norm(p1-p2)
sqdist(p1::AbstractVector, p2::AbstractVector) = sqnorm(p1-p2)
sqnorm(p) = sum(abs2,p)
ismaxlength(b::GärtnerBoundary) = length(b) == length(b.empty_center) + 1

function NSphere(b::GärtnerBoundary)
    return isempty(b) ? NSphere(b.empty_center, 0.0) :
        NSphere(b.centers[end], b.square_radii[end])
end

function welzl!(pts, bdry)
    bdry_len, support_count, nsphere = length(bdry), 0, NSphere(bdry)
    ismaxlength(bdry) && return nsphere, 0
    for i in eachindex(pts)
        pt = pts[i]
        if !isinside(pt, nsphere)
            pts_i = prefix(pts, i-1)
            isstable = push_if_stable!(bdry, pt)
            if isstable
                nsphere, s = welzl!(pts_i, bdry)
                pop!(bdry)
                move_to_front!(pts, i)
                support_count = s + 1
            end
        end
    end
    return nsphere, support_count
end

function find_max_excess(nsphere, pts, k1)
    T = eltype(first(pts))
    e_max, k_max = T(-Inf), k1 -1
    for k in k1:length(pts)
        pt = pts[k]
        e = sqdist(pt, center(nsphere)) - sqradius(nsphere)
        if  e > e_max
            e_max = e
            k_max = k
        end
    end
    return e_max, k_max
end

function welzl(points, maxiterations=2000)
    pts = deepcopy(points)
    bdry, t = GärtnerBoundary(pts), 1
    nsphere, s = welzl!(prefix(pts, t), bdry)
    for i in 1:maxiterations
        e, k = find_max_excess(nsphere, pts, t + 1)
        P = eltype(pts)
        F = eltype(P)
        e <= eps(F) && break
        pt = pts[k]
        push_if_stable!(bdry, pt)
        nsphere_new, s_new = welzl!(prefix(pts, s), bdry)
        pop!(bdry)
        move_to_front!(pts, k)
        nsphere = nsphere_new
        t, s = s + 1, s_new + 1
    end
    return nsphere
end
 
function testwelzl()
    testdata =[
        [[0.0, 0.0], [0.0, 1.0], [1.0, 0.0]],
        [[5.0, -2.0], [-3.0, -2.0], [-2.0, 5.0], [1.0, 6.0], [0.0, 2.0]],
        [[2.0, 4.0, -1.0], [1.0, 5.0, -3.0], [8.0, -4.0, 1.0], [3.0, 9.0, -5.0]],
        [randn(5) for _ in 1:8]
    ]
    for test in testdata
        nsphere = welzl(test)
        println("For points: ", test)
        println("    Center is at: ", nsphere.center)
        println("    Radius is: ", radius(nsphere), "\n")
    end
end

testwelzl()
</syntaxhighlight>{{out}}
<pre>
For points: [[0.0, 0.0], [0.0, 1.0], [1.0, 0.0]]
    Center is at: [0.5, 0.5]
    Radius is: 0.7071067811865476

For points: [[5.0, -2.0], [-3.0, -2.0], [-2.0, 5.0], [1.0, 6.0], [0.0, 2.0]]
    Center is at: [1.0, 1.0]
    Radius is: 5.0

For points: [[2.0, 4.0, -1.0], [1.0, 5.0, -3.0], [8.0, -4.0, 1.0], [3.0, 9.0, -5.0]]
    Center is at: [5.5, 2.5, -2.0]
    Radius is: 7.582875444051551

For points: [[-0.6400900432782123, 2.643703255134232, 0.4016122094706093, 1.8959601399652273, -1.1624046608380516], [0.5632393652621324, -0.015981105190064373, -2.193725940351997, -0.9094586577358262, 0.7165036664470906], [-1.7704367632976061, 0.2518283698686299, -1.3810444289625348, -0.597516704360172, 1.089645656962647], [1.3448578652803103, -0.18140877132223002, -0.4288734015080915, 1.53271973321691, 0.41896461833399573], [0.2132336397213029, 0.07659442168765788, 0.148636431531099, 2.3386893481333795, -2.3000459709300927], [0.6023153188617328, 0.3051735340025314, 1.0732600437151525, 1.1148388039984898, 0.047605838564167786], [1.3655523661720959, 0.5461416420929995, -0.09321951900362889, -1.004771137760985, 1.6532914656050117], [-0.14974382165751837, -0.5375672589202939, -0.15845596754003466, -0.2750720340454088, -0.441247015836271]]
    Center is at: [0.0405866077655439, 0.5556683897981481, -0.2313678300507679, 0.6074066023194586, -0.2003463948612026]
    Radius is: 2.7946020036995454
</pre>

=={{header|Mathematica}}/{{header|Wolfram Language}}==
<syntaxhighlight lang="mathematica">f=BoundingRegion[#,"MinBall"]&;
f[{{0,0},{0,1},{1,0}}]
f[{{5,-2},{-3,-2},{-2,5},{1,6},{0,2}}]
f[{{2,4,-1},{1,5,-3},{8,-4,1},{3,9,-5}}]
f[{{-0.6400900432782123`,2.643703255134232`,0.4016122094706093`,1.8959601399652273`,-1.1624046608380516`},{0.5632393652621324`,-0.015981105190064373`,-2.193725940351997`,-0.9094586577358262`,0.7165036664470906`},{-1.7704367632976061`,0.2518283698686299`,-1.3810444289625348`,-0.597516704360172`,1.089645656962647`},{1.3448578652803103`,-0.18140877132223002`,-0.4288734015080915`,1.53271973321691`,0.41896461833399573`},{0.2132336397213029`,0.07659442168765788`,0.148636431531099`,2.3386893481333795`,-2.3000459709300927`},{0.6023153188617328`,0.3051735340025314`,1.0732600437151525`,1.1148388039984898`,0.047605838564167786`},{1.3655523661720959`,0.5461416420929995`,-0.09321951900362889`,-1.004771137760985`,1.6532914656050117`},{-0.14974382165751837`,-0.5375672589202939`,-0.15845596754003466`,-0.2750720340454088`,-0.441247015836271`}}]</syntaxhighlight>
{{out}}
<pre>Ball[{0.5,0.5},0.707107]
Ball[{1.,1.},5.]
Disk[{11/2,5/2,-2},Sqrt[115/2]]
Ball[{0.0405866,0.555668,-0.231368,0.607407,-0.200346},2.7946]</pre>

=={{header|Nim}}==
{{trans|Go}}
{{trans|Wren}}
<syntaxhighlight lang="nim">import math, random, strutils

type
  Point = tuple[x, y: float]
  Circle = tuple[c: Point; r: float]


func `$`(p: Point): string =
  ## Return the string representation of a point.
  "($1, $2)".format(p.x, p.y)


func dist(a, b: Point): float =
  ## Return the distance between two points.
  hypot(a.x - b.x, a.y - b.y)


func getCircleCenter(bx, by, cx, cy: float): Point =
  ## Return the center of a circle defined by three points.
  let
    b = bx * bx + by * by
    c = cx * cx + cy * cy
    d = bx * cy - by * cx
  result = ((cy * b - by * c) / (2 * d), (bx * c - cx * b) / (2 * d))

func contains(ci: Circle; p: Point): bool =
  ## Return whether a circle contains the point 'p'.
  ## Used by 'in' and 'notin'.
  dist(ci.c, p) <= ci.r


func contains(ci: Circle; ps: seq[Point]): bool =
  ## Return whether a circle contains a slice of points.
  ## Used by 'in'.
  for p in ps:
    if p notin ci: return false
  result = true


func `$`(ci: Circle): string =
  ## Return the string representation of a circle.
  "Center $1, Radius $2".format(ci.c, ci.r)


func circleFrom3(a, b, c: Point): Circle =
  ## Return the smallest circle that intersects three points.
  var i = getCircleCenter(b.x - a.x, b.y - a.y, c.x - a.x, c.y - a.y)
  i.x += a.x
  i.y += a.y
  result = (c: i, r: dist(i, a))


func circleFrom2(a, b: Point): Circle =
  ## Return the smallest circle that intersects two points.
  let c = ((a.x + b.x) * 0.5, (a.y + b.y) * 0.5)
  result = (c: c, r: dist(a, b) * 0.5)


func secTrivial(rs: seq[Point]): Circle =
  ## Return the smallest enclosing circle for n <= 3.
  case rs.len
  of 0: return ((0.0, 0.0), 0.0)
  of 1: return (rs[0], 0.0)
  of 2: return circleFrom2(rs[0], rs[1])
  of 3: discard
  else: raise newException(ValueError, "There shouldn't be more than three points.")

  # Three points.
  for i in 0..1:
    for j in (i + 1)..2:
      let c = circleFrom2(rs[i], rs[j])
      if rs in c: return c
  result = circleFrom3(rs[0], rs[1], rs[2])


proc welzl(ps: var seq[Point]; rs: seq[Point]; n: int): Circle =
  ## Helper function for Welzl method.
  var rc = rs
  if n == 0 or rc.len == 3: return secTrivial(rc)
  let idx = rand(n-1)
  let p = ps[idx]
  swap ps[idx], ps[n-1]
  let d = welzl(ps, rc, n-1)
  if p in d: return d
  rc.add p
  result = welzl(ps, rc, n-1)


proc welzl(ps: seq[Point]): Circle =
  # Applies the Welzl algorithm to find the SEC.
  var pc = ps
  pc.shuffle()
  result = welzl(pc, @[], pc.len)


when isMainModule:
  randomize()
  const Tests = [@[(0.0, 0.0), (0.0, 1.0), (1.0, 0.0)],
                 @[(5.0, -2.0), (-3.0, -2.0), (-2.0, 5.0), (1.0, 6.0), (0.0, 2.0)]]

  for test in Tests:
    echo welzl(test)</syntaxhighlight>

{{out}}
<pre>Center (0.5, 0.5), Radius 0.7071067811865476
Center (1.0, 1.0), Radius 5.0</pre>

=={{header|Phix}}==
Based on the same code as Wren, and likewise limited to 2D circles - I believe (but cannot prove) the main barrier to coping with more than two dimensions lies wholly within the circle_from3() routine.
<!--<syntaxhighlight lang="phix">(phixonline)-->
 <span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
 <span style="color: #008080;">type</span> <span style="color: #000000;">point</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">return</span> <span style="color: #004600;">true</span> <span style="color: #008080;">end</span> <span style="color: #008080;">type</span>
 <span style="color: #008080;">enum</span> <span style="color: #000000;">CENTRE</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">RADIUS</span>
 <span style="color: #008080;">type</span> <span style="color: #000000;">circle</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">return</span> <span style="color: #004600;">true</span> <span style="color: #008080;">end</span> <span style="color: #008080;">type</span>
 
 <span style="color: #008080;">function</span> <span style="color: #000000;">distance</span><span style="color: #0000FF;">(</span><span style="color: #000000;">point</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">return</span> <span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sum</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_power</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">),</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)))</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
 
 <span style="color: #008080;">function</span> <span style="color: #000000;">in_circle</span><span style="color: #0000FF;">(</span><span style="color: #000000;">point</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">circle</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">return</span> <span style="color: #000000;">distance</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">CENTRE</span><span style="color: #0000FF;">])</span> <span style="color: #0000FF;"><=</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">[</span><span style="color: #000000;">RADIUS</span><span style="color: #0000FF;">]</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
 
 <span style="color: #008080;">function</span> <span style="color: #000000;">circle_from2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">point</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
     <span style="color: #000080;font-style:italic;">-- return the smallest circle that intersects 2 points:</span>
     <span style="color: #000000;">point</span> <span style="color: #000000;">midpoint</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sq_div</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">),</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">atom</span> <span style="color: #000000;">halfdiameter</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">distance</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">2</span>
     <span style="color: #000000;">circle</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span> <span style="color: #000000;">midpoint</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">halfdiameter</span> <span style="color: #0000FF;">}</span>
     <span style="color: #008080;">return</span> <span style="color: #000000;">res</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
 
 <span style="color: #008080;">function</span> <span style="color: #000000;">circle_from3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">point</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">)</span>
     <span style="color: #000080;font-style:italic;">-- return a unique circle that intersects three points </span>
     <span style="color: #000000;">point</span> <span style="color: #000000;">bma</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">),</span>
           <span style="color: #000000;">cma</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{{</span><span style="color: #000000;">aX</span><span style="color: #0000FF;">,</span><span style="color: #000000;">aY</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">bX</span><span style="color: #0000FF;">,</span><span style="color: #000000;">bY</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">cX</span><span style="color: #0000FF;">,</span><span style="color: #000000;">cY</span><span style="color: #0000FF;">}}</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">bma</span><span style="color: #0000FF;">,</span><span style="color: #000000;">cma</span><span style="color: #0000FF;">},</span>
          <span style="color: #000000;">B</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sum</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">bma</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)),</span>
          <span style="color: #000000;">C</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sum</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">cma</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)),</span>
          <span style="color: #000000;">D</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">bX</span><span style="color: #0000FF;">*</span><span style="color: #000000;">cY</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">bY</span><span style="color: #0000FF;">*</span><span style="color: #000000;">cX</span><span style="color: #0000FF;">)*</span><span style="color: #000000;">2</span> 
     <span style="color: #000000;">point</span> <span style="color: #000000;">centre</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{(</span><span style="color: #000000;">cY</span><span style="color: #0000FF;">*</span><span style="color: #000000;">B</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">bY</span><span style="color: #0000FF;">*</span><span style="color: #000000;">C</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">D</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">aX</span><span style="color: #0000FF;">,</span> 
                     <span style="color: #0000FF;">(</span><span style="color: #000000;">bX</span><span style="color: #0000FF;">*</span><span style="color: #000000;">C</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">cX</span><span style="color: #0000FF;">*</span><span style="color: #000000;">B</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">D</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">aY</span> <span style="color: #0000FF;">}</span>
     <span style="color: #004080;">atom</span> <span style="color: #000000;">radius</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">distance</span><span style="color: #0000FF;">(</span><span style="color: #000000;">centre</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- (=== b,c)</span>
     <span style="color: #000000;">circle</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span> <span style="color: #000000;">centre</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">radius</span> <span style="color: #0000FF;">}</span>
     <span style="color: #008080;">return</span> <span style="color: #000000;">res</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
   
 <span style="color: #008080;">function</span> <span style="color: #000000;">trivial</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">integer</span> <span style="color: #000000;">l</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">switch</span> <span style="color: #000000;">l</span> <span style="color: #008080;">do</span>
         <span style="color: #008080;">case</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">:</span> <span style="color: #008080;">return</span> <span style="color: #0000FF;">{{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0</span><span style="color: #0000FF;">},</span><span style="color: #000000;">0</span><span style="color: #0000FF;">}</span>
         <span style="color: #008080;">case</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">:</span> <span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">r</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">],</span><span style="color: #000000;">0</span><span style="color: #0000FF;">}</span>
         <span style="color: #008080;">case</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">:</span> <span style="color: #008080;">return</span> <span style="color: #000000;">circle_from2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">],</span><span style="color: #000000;">r</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">])</span>
         <span style="color: #008080;">case</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">:</span> <span style="color: #008080;">return</span> <span style="color: #000000;">circle_from3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">],</span><span style="color: #000000;">r</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">],</span><span style="color: #000000;">r</span><span style="color: #0000FF;">[</span><span style="color: #000000;">3</span><span style="color: #0000FF;">])</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">switch</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
 
 <span style="color: #008080;">function</span> <span style="color: #000000;">welzlr</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">if</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">={}</span> <span style="color: #008080;">or</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">)=</span><span style="color: #000000;">3</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">trivial</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
     <span style="color: #000000;">point</span> <span style="color: #000000;">p1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
     <span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">..$]</span>
     <span style="color: #000000;">circle</span> <span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">welzlr</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">if</span> <span style="color: #000000;">in_circle</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">d</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
     <span style="color: #008080;">return</span> <span style="color: #000000;">welzlr</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">deep_copy</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">),</span><span style="color: #000000;">p1</span><span style="color: #0000FF;">))</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
 
 <span style="color: #008080;">procedure</span> <span style="color: #000000;">welzl</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">)</span>
     <span style="color: #000000;">circle</span> <span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">welzlr</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">shuffle</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">),{})</span>
     <span style="color: #004080;">string</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"centre %v radius %.14g"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">c</span><span style="color: #0000FF;">)</span>
     <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Points %v ==&gt; %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">s</span><span style="color: #0000FF;">})</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
 
 <span style="color: #008080;">constant</span> <span style="color: #000000;">tests</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},{</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">},{</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0</span><span style="color: #0000FF;">}},</span>
                   <span style="color: #0000FF;">{{</span><span style="color: #000000;">5</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">},{-</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">},{-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">5</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">6</span><span style="color: #0000FF;">},{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">}}}</span>
 <span style="color: #7060A8;">papply</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tests</span><span style="color: #0000FF;">,</span><span style="color: #000000;">welzl</span><span style="color: #0000FF;">)</span>
<!--</syntaxhighlight>-->
{{out}}
<pre>
Points {{0,0},{0,1},{1,0}} ==> centre {0.5,0.5} radius 0.70710678118655
Points {{5,-2},{-3,-2},{-2,5},{1,6},{0,2}} ==> centre {1,1} radius 5
</pre>

===n-dimensional===
{{trans|Python}}
Uses the last test from Julia, however, since that's given more accurately.
<!--<syntaxhighlight lang="phix">(phixonline)-->
 <span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
 <span style="color: #008080;">constant</span> <span style="color: #000000;">inf</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1e300</span><span style="color: #0000FF;">*</span><span style="color: #000000;">1e300</span><span style="color: #0000FF;">,</span>
          <span style="color: #000000;">nan</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">-(</span><span style="color: #000000;">inf</span><span style="color: #0000FF;">/</span><span style="color: #000000;">inf</span><span style="color: #0000FF;">),</span>
          <span style="color: #000000;">eps</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1e-8</span>
  
 <span style="color: #008080;">function</span> <span style="color: #000000;">sqnorm</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">return</span> <span style="color: #7060A8;">sum</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">,</span><span style="color: #000000;">2</span><span style="color: #0000FF;">))</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
  
 <span style="color: #008080;">function</span> <span style="color: #000000;">sqdist</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">p1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">p2</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">return</span> <span style="color: #000000;">sqnorm</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p2</span><span style="color: #0000FF;">))</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
  
 <span style="color: #008080;">function</span> <span style="color: #000000;">projector_mult</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">vs</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">sequence</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">v</span><span style="color: #0000FF;">))</span>
     <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">vs</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
         <span style="color: #004080;">sequence</span> <span style="color: #000000;">vi</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">vs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span>
         <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sq_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">sq_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">vi</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">sum</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">vi</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">))))</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #008080;">return</span> <span style="color: #000000;">s</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
  
 <span style="color: #000080;font-style:italic;">-- Gaertner Boundary:</span>
 <span style="color: #004080;">sequence</span> <span style="color: #000000;">empty_center</span><span style="color: #0000FF;">,</span>
          <span style="color: #000000;">centers</span><span style="color: #0000FF;">,</span>
          <span style="color: #000000;">square_radii</span><span style="color: #0000FF;">,</span>
          <span style="color: #000080;font-style:italic;">-- Stack of points that are shifted / projected to put first one at origin:</span>
          <span style="color: #000000;">projector</span>
 
 <span style="color: #008080;">function</span> <span style="color: #000000;">push_if_stable</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">pt</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">if</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">centers</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">==</span> <span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
         <span style="color: #000000;">square_radii</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #000000;">square_radii</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">)</span>
         <span style="color: #000000;">centers</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">pt</span><span style="color: #0000FF;">}</span>
         <span style="color: #008080;">return</span> <span style="color: #004600;">true</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
     <span style="color: #004080;">sequence</span> <span style="color: #000000;">q0</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">centers</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">],</span>
              <span style="color: #000000;">center</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">centers</span><span style="color: #0000FF;">[$],</span>
              <span style="color: #000000;">C</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">center</span><span style="color: #0000FF;">,</span><span style="color: #000000;">q0</span><span style="color: #0000FF;">),</span>
              <span style="color: #000000;">Qm</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pt</span><span style="color: #0000FF;">,</span><span style="color: #000000;">q0</span><span style="color: #0000FF;">),</span>
              <span style="color: #000000;">Qm_bar</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">projector_mult</span><span style="color: #0000FF;">(</span><span style="color: #000000;">projector</span><span style="color: #0000FF;">,</span><span style="color: #000000;">Qm</span><span style="color: #0000FF;">),</span>
              <span style="color: #000000;">residue</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">Qm</span><span style="color: #0000FF;">,</span><span style="color: #000000;">Qm_bar</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">atom</span> <span style="color: #000000;">r2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">square_radii</span><span style="color: #0000FF;">[$],</span>
          <span style="color: #000000;">e</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">sqdist</span><span style="color: #0000FF;">(</span><span style="color: #000000;">Qm</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">C</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">r2</span><span style="color: #0000FF;">,</span>
          <span style="color: #000000;">z</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">2</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">sqnorm</span><span style="color: #0000FF;">(</span><span style="color: #000000;">residue</span><span style="color: #0000FF;">),</span>
          <span style="color: #000000;">tol</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">eps</span> <span style="color: #0000FF;">*</span> <span style="color: #7060A8;">max</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">),</span>
          <span style="color: #000000;">isstable</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">abs</span><span style="color: #0000FF;">(</span><span style="color: #000000;">z</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">></span> <span style="color: #000000;">tol</span>
     <span style="color: #008080;">if</span> <span style="color: #000000;">isstable</span> <span style="color: #008080;">then</span>
         <span style="color: #004080;">sequence</span> <span style="color: #000000;">center_new</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sq_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">center</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">sq_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">e</span><span style="color: #0000FF;">/</span><span style="color: #000000;">z</span><span style="color: #0000FF;">,</span><span style="color: #000000;">residue</span><span style="color: #0000FF;">))</span>
         <span style="color: #004080;">atom</span> <span style="color: #000000;">r2new</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">r2</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">e</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">e</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">/</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">2</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">z</span><span style="color: #0000FF;">)</span>
         <span style="color: #000000;">projector</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #000000;">projector</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">sq_div</span><span style="color: #0000FF;">(</span><span style="color: #000000;">residue</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">sqnorm</span><span style="color: #0000FF;">(</span><span style="color: #000000;">residue</span><span style="color: #0000FF;">))))</span>
         <span style="color: #000000;">centers</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #000000;">centers</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">center_new</span><span style="color: #0000FF;">)</span>
         <span style="color: #000000;">square_radii</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #000000;">square_radii</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">r2new</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
     <span style="color: #008080;">return</span> <span style="color: #000000;">isstable</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
  
 <span style="color: #008080;">procedure</span> <span style="color: #000000;">pop_gb</span><span style="color: #0000FF;">()</span>
     <span style="color: #004080;">integer</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">centers</span><span style="color: #0000FF;">)</span>
     <span style="color: #000000;">centers</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">centers</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..$-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
     <span style="color: #000000;">square_radii</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">square_radii</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..$-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
     <span style="color: #008080;">if</span> <span style="color: #000000;">n</span> <span style="color: #0000FF;">>=</span> <span style="color: #000000;">2</span> <span style="color: #008080;">then</span>
         <span style="color: #000000;">projector</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">projector</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..$-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #000080;font-style:italic;">-- (pop/discard)</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
  
 <span style="color: #008080;">function</span> <span style="color: #000000;">isinside</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">pt</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">center</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">sqradius</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">atom</span> <span style="color: #000000;">r2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">sqdist</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pt</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">center</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">return</span> <span style="color: #000000;">r2</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">nan</span> <span style="color: #008080;">and</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">r2</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">sqradius</span> <span style="color: #008080;">or</span> <span style="color: #7060A8;">abs</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r2</span><span style="color: #0000FF;">-</span><span style="color: #000000;">sqradius</span><span style="color: #0000FF;">)<</span><span style="color: #000000;">eps</span><span style="color: #0000FF;">)</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
  
 <span style="color: #008080;">function</span> <span style="color: #000000;">move_to_front</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">pts</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">sequence</span> <span style="color: #000000;">pt</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">pts</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span>
     <span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">i</span> <span style="color: #008080;">do</span>
         <span style="color: #0000FF;">{</span><span style="color: #000000;">pts</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">pt</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">pt</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">pts</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]}</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #008080;">return</span> <span style="color: #000000;">pts</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
  
 <span style="color: #008080;">function</span> <span style="color: #000000;">ismaxlength</span><span style="color: #0000FF;">()</span>
     <span style="color: #008080;">return</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">centers</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">==</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">empty_center</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">1</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
  
 <span style="color: #008080;">function</span> <span style="color: #000000;">_welzl</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">pts</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">pos</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">integer</span> <span style="color: #000000;">support_count</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s</span>
     <span style="color: #004080;">sequence</span> <span style="color: #000000;">center</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">centers</span><span style="color: #0000FF;">)?</span><span style="color: #000000;">centers</span><span style="color: #0000FF;">[$]:</span><span style="color: #000000;">empty_center</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">atom</span> <span style="color: #000000;">sqradius</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">centers</span><span style="color: #0000FF;">)?</span><span style="color: #000000;">square_radii</span><span style="color: #0000FF;">[$]:</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">if</span> <span style="color: #000000;">ismaxlength</span><span style="color: #0000FF;">()</span> <span style="color: #008080;">then</span>
         <span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">center</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">sqradius</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">}</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
     <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">pos</span> <span style="color: #008080;">do</span>
         <span style="color: #008080;">if</span> <span style="color: #008080;">not</span> <span style="color: #000000;">isinside</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pts</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">center</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">sqradius</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
             <span style="color: #004080;">bool</span> <span style="color: #000000;">isstable</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">push_if_stable</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pts</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">])</span>
             <span style="color: #008080;">if</span> <span style="color: #000000;">isstable</span> <span style="color: #008080;">then</span>
                 <span style="color: #0000FF;">{</span><span style="color: #000000;">center</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">sqradius</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">_welzl</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pts</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
                 <span style="color: #000000;">pop_gb</span><span style="color: #0000FF;">()</span>
                 <span style="color: #000000;">pts</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">move_to_front</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">deep_copy</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pts</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
                 <span style="color: #000000;">support_count</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">1</span>
             <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
         <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">center</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">sqradius</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">support_count</span><span style="color: #0000FF;">}</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
  
 <span style="color: #008080;">function</span> <span style="color: #000000;">find_max_excess</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">center</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">pts</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">sqradius</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">k1</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">atom</span> <span style="color: #000000;">err_max</span>  <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">inf</span>
     <span style="color: #004080;">integer</span> <span style="color: #000000;">k_max</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">k1</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">1</span>
     <span style="color: #008080;">for</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">=</span><span style="color: #000000;">k_max</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pts</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
         <span style="color: #004080;">sequence</span> <span style="color: #000000;">pt</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">pts</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]</span>
         <span style="color: #004080;">atom</span> <span style="color: #000000;">err</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">sqdist</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pt</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">center</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">sqradius</span>
         <span style="color: #008080;">if</span> <span style="color: #000000;">err</span> <span style="color: #0000FF;">></span> <span style="color: #000000;">err_max</span> <span style="color: #008080;">then</span>
             <span style="color: #000000;">err_max</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">err</span>
             <span style="color: #000000;">k_max</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">k</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">1</span>
         <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">err_max</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">k_max</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">}</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
  
 <span style="color: #008080;">procedure</span> <span style="color: #000000;">welzl</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">points</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">maxiterations</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2000</span><span style="color: #0000FF;">)</span>
     <span style="color: #004080;">sequence</span> <span style="color: #000000;">pts</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">points</span>
     <span style="color: #000000;">empty_center</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">nan</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pts</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]))</span>
     <span style="color: #000000;">centers</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span>
     <span style="color: #000000;">square_radii</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span>
     <span style="color: #000000;">projector</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span>
 
     <span style="color: #004080;">integer</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s_new</span>
     <span style="color: #0000FF;">{</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">center</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">sqradius</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">integer</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">_welzl</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pts</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">)</span>
     <span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">maxiterations</span> <span style="color: #008080;">do</span>
         <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">e</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">find_max_excess</span><span style="color: #0000FF;">(</span><span style="color: #000000;">center</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">pts</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">sqradius</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
         <span style="color: #008080;">if</span> <span style="color: #000000;">e</span> <span style="color: #0000FF;"><=</span> <span style="color: #000000;">eps</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
         <span style="color: #004080;">sequence</span> <span style="color: #000000;">pt</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">pts</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]</span>
         <span style="color: #0000FF;">{}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">push_if_stable</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pt</span><span style="color: #0000FF;">)</span>
         <span style="color: #0000FF;">{</span><span style="color: #000000;">center</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">sqradius</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s_new</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">_welzl</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pts</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">)</span>
         <span style="color: #000000;">pop_gb</span><span style="color: #0000FF;">()</span>
         <span style="color: #000000;">pts</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">move_to_front</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">deep_copy</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pts</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">)</span>
         <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">1</span>
         <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">s_new</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">1</span>
     <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
     <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"For points: "</span><span style="color: #0000FF;">)</span> <span style="color: #7060A8;">pp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">points</span><span style="color: #0000FF;">,{</span><span style="color: #004600;">pp_Indent</span><span style="color: #0000FF;">,</span><span style="color: #000000;">12</span><span style="color: #0000FF;">,</span><span style="color: #004600;">pp_FltFmt</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%11.8f"</span><span style="color: #0000FF;">})</span>
     <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"    Center is at: %v\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">center</span><span style="color: #0000FF;">})</span>
     <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"    Radius is: %.16g\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">sqradius</span><span style="color: #0000FF;">)})</span>
 <span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
  
 <span style="color: #008080;">constant</span> <span style="color: #000000;">tests</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},</span> <span style="color: #0000FF;">{</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">},</span> <span style="color: #0000FF;">{</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">}},</span>
                   <span style="color: #0000FF;">{{</span><span style="color: #000000;">5</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">},</span> <span style="color: #0000FF;">{-</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">},</span> <span style="color: #0000FF;">{-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">},</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">6</span><span style="color: #0000FF;">},</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">}},</span>
                   <span style="color: #0000FF;">{{</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">3</span><span style="color: #0000FF;">},</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">8</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">4</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">},</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">9</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">5</span><span style="color: #0000FF;">}},</span>
                   <span style="color: #0000FF;">{{-</span><span style="color: #000000;">0.6400900432782123</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2.643703255134232</span><span style="color: #0000FF;">,</span>    <span style="color: #000000;">0.4016122094706093</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1.8959601399652273</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">1.1624046608380516</span><span style="color: #0000FF;">},</span>
                    <span style="color: #0000FF;">{</span> <span style="color: #000000;">0.5632393652621324</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">0.015981105190064373</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">2.193725940351997</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">0.9094586577358262</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0.7165036664470906</span><span style="color: #0000FF;">},</span>
                    <span style="color: #0000FF;">{-</span><span style="color: #000000;">1.7704367632976061</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0.2518283698686299</span><span style="color: #0000FF;">,</span>  <span style="color: #0000FF;">-</span><span style="color: #000000;">1.3810444289625348</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">0.597516704360172</span><span style="color: #0000FF;">,</span>  <span style="color: #000000;">1.089645656962647</span><span style="color: #0000FF;">},</span>
                    <span style="color: #0000FF;">{</span> <span style="color: #000000;">1.3448578652803103</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">0.18140877132223002</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">0.4288734015080915</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1.53271973321691</span><span style="color: #0000FF;">,</span>   <span style="color: #000000;">0.41896461833399573</span><span style="color: #0000FF;">},</span>
                    <span style="color: #0000FF;">{</span> <span style="color: #000000;">0.2132336397213029</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0.07659442168765788</span><span style="color: #0000FF;">,</span>  <span style="color: #000000;">0.1486364315310990</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2.3386893481333795</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">2.3000459709300927</span><span style="color: #0000FF;">},</span>
                    <span style="color: #0000FF;">{</span> <span style="color: #000000;">0.6023153188617328</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0.3051735340025314</span><span style="color: #0000FF;">,</span>   <span style="color: #000000;">1.0732600437151525</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1.1148388039984898</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0.047605838564167786</span><span style="color: #0000FF;">},</span>
                    <span style="color: #0000FF;">{</span> <span style="color: #000000;">1.3655523661720959</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0.5461416420929995</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">0.09321951900362889</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">1.004771137760985</span><span style="color: #0000FF;">,</span>  <span style="color: #000000;">1.6532914656050117</span><span style="color: #0000FF;">},</span>
                    <span style="color: #0000FF;">{-</span><span style="color: #000000;">0.14974382165751837</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">0.5375672589202939</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">0.15845596754003466</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">0.2750720340454088</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">0.441247015836271</span><span style="color: #0000FF;">}}}</span>
 <span style="color: #7060A8;">papply</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tests</span><span style="color: #0000FF;">,</span><span style="color: #000000;">welzl</span><span style="color: #0000FF;">)</span>
<!--</syntaxhighlight>-->
{{out}}
<pre>
For points: {{0,0}, {0,1}, {1,0}}
    Center is at: {0.5,0.5}
    Radius is: 0.7071067811865476
For points: {{5,-2}, {-3,-2}, {-2,5}, {1,6}, {0,2}}
    Center is at: {1,1}
    Radius is: 5
For points: {{2,4,-1}, {1,5,-3}, {8,-4,1}, {3,9,-5}}
    Center is at: {5.5,2.5,-2}
    Radius is: 7.582875444051551
For points: {{-0.64009004, 2.64370326, 0.40161221, 1.89596014,-1.16240466},
             { 0.56323937,-0.01598111,-2.19372594,-0.90945866, 0.71650367},
             {-1.77043676, 0.25182837,-1.38104443,-0.59751670, 1.08964566},
             { 1.34485787,-0.18140877,-0.42887340, 1.53271973, 0.41896462},
             { 0.21323364, 0.07659442, 0.14863643, 2.33868935,-2.30004597},
             { 0.60231532, 0.30517353, 1.07326004, 1.11483880, 0.04760584},
             { 1.36555237, 0.54614164,-0.09321952,-1.00477114, 1.65329147},
             {-0.14974382,-0.53756726,-0.15845597,-0.27507203,-0.44124702}}
    Center is at: {0.0405866078,0.5556683898,-0.2313678301,0.6074066023,-0.2003463949}
    Radius is: 2.794602003699545
</pre>

=={{header|Python}}==
{{trans|Julia}}
<syntaxhighlight lang="python">import numpy as np

class ProjectorStack:
    """
    Stack of points that are shifted / projected to put first one at origin.
    """
    def __init__(self, vec):
        self.vs = np.array(vec)
        
    def push(self, v):
        if len(self.vs) == 0:
            self.vs = np.array([v])
        else:
            self.vs = np.append(self.vs, [v], axis=0)
        return self
    
    def pop(self):
        if len(self.vs) > 0:
            ret, self.vs = self.vs[-1], self.vs[:-1]
            return ret
    
    def __mul__(self, v):
        s = np.zeros(len(v))
        for vi in self.vs:
            s = s + vi * np.dot(vi, v)
        return s
    
class GaertnerBoundary:
    """
        GärtnerBoundary

    See the passage regarding M_B in Section 4 of Gärtner's paper.
    """
    def __init__(self, pts):
        self.projector = ProjectorStack([])
        self.centers, self.square_radii = np.array([]), np.array([])
        self.empty_center = np.array([np.NaN for _ in pts[0]])


def push_if_stable(bound, pt):
    if len(bound.centers) == 0:
        bound.square_radii = np.append(bound.square_radii, 0.0)
        bound.centers = np.array([pt])
        return True
    q0, center = bound.centers[0], bound.centers[-1]
    C, r2  = center - q0, bound.square_radii[-1]
    Qm, M = pt - q0, bound.projector
    Qm_bar = M * Qm
    residue, e = Qm - Qm_bar, sqdist(Qm, C) - r2
    z, tol = 2 * sqnorm(residue), np.finfo(float).eps * max(r2, 1.0)
    isstable = np.abs(z) > tol
    if isstable:
        center_new  = center + (e / z) * residue
        r2new = r2 + (e * e) / (2 * z)
        bound.projector.push(residue / np.linalg.norm(residue))
        bound.centers = np.append(bound.centers, np.array([center_new]), axis=0)
        bound.square_radii = np.append(bound.square_radii, r2new)
    return isstable

def pop(bound):
    n = len(bound.centers)
    bound.centers = bound.centers[:-1]
    bound.square_radii = bound.square_radii[:-1]
    if n >= 2:
        bound.projector.pop()
    return bound


class NSphere:
    def __init__(self, c, sqr):
        self.center = np.array(c)
        self.sqradius = sqr

def isinside(pt, nsphere, atol=1e-6, rtol=0.0):
    r2, R2 = sqdist(pt, nsphere.center), nsphere.sqradius
    return r2 <= R2 or np.isclose(r2, R2, atol=atol**2,rtol=rtol**2)

def allinside(pts, nsphere, atol=1e-6, rtol=0.0):
    for p in pts:
        if not isinside(p, nsphere, atol, rtol):
            return False
    return True

def move_to_front(pts, i):
    pt = pts[i]
    for j in range(len(pts)):
        pts[j], pt = pt, np.array(pts[j])
        if j == i:
            break
    return pts

def dist(p1, p2):
    return np.linalg.norm(p1 - p2)

def sqdist(p1, p2):
    return sqnorm(p1 - p2)

def sqnorm(p):
    return np.sum(np.array([x * x for x in p]))

def ismaxlength(bound):
    len(bound.centers) == len(bound.empty_center) + 1

def makeNSphere(bound):
    if len(bound.centers) == 0: 
        return NSphere(bound.empty_center, 0.0)
    return NSphere(bound.centers[-1], bound.square_radii[-1])

def _welzl(pts, pos, bdry):
    support_count, nsphere = 0, makeNSphere(bdry)
    if ismaxlength(bdry):
        return nsphere, 0
    for i in range(pos):
        if not isinside(pts[i], nsphere):
            isstable = push_if_stable(bdry, pts[i])
            if isstable:
                nsphere, s = _welzl(pts, i, bdry)
                pop(bdry)
                move_to_front(pts, i)
                support_count = s + 1
    return nsphere, support_count

def find_max_excess(nsphere, pts, k1):
    err_max, k_max = -np.Inf, k1 - 1
    for (k, pt) in enumerate(pts[k_max:]):
        err = sqdist(pt, nsphere.center) - nsphere.sqradius
        if  err > err_max:
            err_max, k_max = err, k + k1
    return err_max, k_max - 1

def welzl(points, maxiterations=2000):
    pts, eps = np.array(points, copy=True), np.finfo(float).eps
    bdry, t = GaertnerBoundary(pts), 1
    nsphere, s = _welzl(pts, t, bdry)
    for i in range(maxiterations):
        e, k = find_max_excess(nsphere, pts, t + 1)
        if e <= eps:
            break
        pt = pts[k]
        push_if_stable(bdry, pt)
        nsphere_new, s_new = _welzl(pts, s, bdry)
        pop(bdry)
        move_to_front(pts, k)
        nsphere = nsphere_new
        t, s = s + 1, s_new + 1
    return nsphere

if __name__ == '__main__':
    TESTDATA =[
        np.array([[0.0, 0.0], [0.0, 1.0], [1.0, 0.0]]),
        np.array([[5.0, -2.0], [-3.0, -2.0], [-2.0, 5.0], [1.0, 6.0], [0.0, 2.0]]),
        np.array([[2.0, 4.0, -1.0], [1.0, 5.0, -3.0], [8.0, -4.0, 1.0], [3.0, 9.0, -5.0]]),
        np.random.normal(size=(8, 5))
    ]
    for test in TESTDATA:
        nsphere = welzl(test)
        print("For points: ", test)
        print("    Center is at: ", nsphere.center)
        print("    Radius is: ", np.sqrt(nsphere.sqradius), "\n")
</syntaxhighlight>{{out}}
<pre>
For points:  [[0. 0.]
 [0. 1.]
 [1. 0.]]
    Center is at:  [0.5 0.5]
    Radius is:  0.7071067811865476 

For points:  [[ 5. -2.]
 [-3. -2.]
 [-2.  5.]
 [ 1.  6.]
 [ 0.  2.]]
    Center is at:  [1. 1.]
    Radius is:  5.0 

For points:  [[ 2.  4. -1.]
 [ 1.  5. -3.]
 [ 8. -4.  1.]
 [ 3.  9. -5.]]
    Center is at:  [ 5.5  2.5 -2. ]
    Radius is:  7.582875444051551 

For points:  [[-0.28550471 -0.12787614 -0.65666165 -0.57319826 -0.09253894]
 [ 0.08947276  0.64778564 -0.54953015  0.05332253 -0.93055218]
 [ 1.26721546 -1.26410984 -0.79937865 -0.45096792 -1.23946668]
 [-0.62653779  0.56561466  0.62403237  0.23226903 -0.95820264]
 [ 0.90973949 -0.9821474   0.41400032 -1.11268937  0.19568717]
 [-0.4931165   0.94778404 -0.10534124  0.96431358  0.36504087]
 [ 1.43895269 -0.69957774 -0.31486014 -1.14980913  0.42550193]
 [ 1.4722404   1.19711551 -0.18735466  1.69208268  1.04594225]]
    Center is at:  [ 1.00038333  0.08534583 -0.22414667  0.50927606 -0.15936026]
    Radius is:  2.076492284522762 
</pre>

=={{header|Raku}}==
{{trans|Go}}
<syntaxhighlight lang="raku" line>class P { has ($.x, $.y) is rw;                # Point
   method gist { "({$.x~", "~$.y})" }
}

class C { has (P $.c, Numeric $.r);            # Circle
   method gist { "Center = " ~$.c.gist ~ " and Radius = $.r\n" }

   # tests whether a circle contains the point 'p'
   method contains(P \p --> Bool) { distSq($.c, p) ≤ $.r² }

   # tests whether a circle contains a slice of point
   method encloses(@ps --> Bool) { [and] @ps.map: { $.contains($_) } } 
}

sub infix:<−> (P \a, P \b) { a.x - b.x, a.y - b.y } # note: Unicode 'minus'

# returns the square of the distance between two points
sub distSq (P \a, P \b) { [+] (a − b)»² }

sub getCenter (\bx, \by, \cx, \cy --> P) {
   my (\b,\c,\d) = bx²+by², cx²+cy², bx*cy - by*cx;
   P.new: x => (cy*b - by*c) / (2 * d), y => (bx*c - cx*b) / (2 * d)
} # returns the center of a circle defined by 3 points

sub circleFrom3 (P \a, P \b, P \c --> C)  {
   my \k = $ = getCenter |(b − a), |(c − a);
   k.x, k.y Z[+=] a.x, a.y;
   C.new: c => k, r => distSq(k, a).sqrt
} # returns a unique circle that intersects 3 points

sub circleFrom2 (P \a, P \b --> C ) {
   my \center = P.new: x => ((a.x + b.x) / 2), y => ((a.y + b.y) / 2) ;
   C.new: c => center, r => (distSq(a, b).sqrt / 2)
} # returns smallest circle that intersects 2 points

sub secTrivial( @rs --> C ) {
   given @rs {
      when * == 0 { return C.new: c => (P.new: x => 0, y => 0), r => 0 }
      when * == 1 { return C.new: c => @rs[0], r => 0 }
      when * == 2 { return circleFrom2 |@rs }
      when * == 3 { #`{ no-op } }
      when *  > 3 { die "There shouldn't be more than 3 points." }
   }
   for 0, 1 X 1, 2 -> ( \i, \j ) {
      return $_ if .encloses(@rs) given circleFrom2 |@rs[i,j]
   }
   circleFrom3 |@rs
} # returns smallest enclosing circle for n ≤ 3

sub Welzl-helper( @ps is copy, @rs is copy , \n --> C ) {
   return secTrivial(@rs) if n == 0 or @rs == 3;
   my \p = @ps.shift;
   return $_ if .contains(p) given Welzl-helper @ps, @rs, n-1;
   Welzl-helper @ps, @rs.append(p), n-1
} # helper function for Welzl method

# applies the Welzl algorithm to find the SEC
sub welzl(@ps --> C) { Welzl-helper @ps.pick(*), [], @ps }

my @tests = (
   [ (0,0), (0,1), (1,0) ],
   [ (5,-2), (-3,-2), (-2,5), (1,6), (0,2) ]
).map: {
   @_.map: { P.new: x => $_[0], y => $_[1] }
}

say "Solution for smallest circle enclosing {$_.gist} :\n", welzl $_ for @tests;</syntaxhighlight>
{{out}}
<pre>Solution for smallest circle enclosing ((0, 0) (0, 1) (1, 0)) :
Center = (0.5, 0.5) and Radius = 0.7071067811865476

Solution for smallest circle enclosing ((5, -2) (-3, -2) (-2, 5) (1, 6) (0, 2)) :
Center = (1, 1) and Radius = 5</pre>

=={{header|Wren}}==
Well a circle is a two dimensional figure and so, despite any contradictory indications in the task description, that's what this solution provides.

It is based on Welzl's algorithm and follows closely the C++ code [https://www.geeksforgeeks.org/minimum-enclosing-circle-set-2-welzls-algorithm/?ref=rp here].
<syntaxhighlight lang="wren">import "random" for Random

var Rand = Random.new()

class Point {
    // returns the square of the distance between two points
    static distSq(a, b) { (a.x - b.x)*(a.x - b.x) + (a.y - b.y)*(a.y - b.y) }

    // returns the center of a circle defined by 3 points
    static getCircleCenter(bx, by, cx, cy) {
        var b = bx*bx + by*by
        var c = cx*cx + cy*cy
        var d = bx*cy - by*cx
        return Point.new((cy*b - by*c) / (2 * d), (bx*c - cx*b) / (2 * d))
    }

    // constructs a new Point object
    construct new(x, y) {
        _x = x
        _y = y
    }

    // basic properties
    x { _x }
    x=(o) { _x = o }
    y { _y }
    y=(o) { _y = o }

    // returns a copy of the current instance
    copy() { Point.new(_x, _y) }

    // string representation
    toString { "(%(_x), %(_y))" }
}

class Circle {
    // returns a unique circle that intersects 3 points
    static from(a, b, c) {
        var i = Point.getCircleCenter(b.x - a.x, b.y - a.y, c.x - a.x, c.y - a.y)
        i.x = i.x + a.x
        i.y = i.y + a.y
        return Circle.new(i, Point.distSq(i, a).sqrt)
    }

    // returns smallest circle that intersects 2 points
    static from(a, b) {
        var c = Point.new((a.x + b.x) / 2, (a.y + b.y) / 2)
        return Circle.new(c, Point.distSq(a, b).sqrt/2)
    }

    // returns smallest enclosing circle for n <= 3
    static secTrivial(rs) {
        var size = rs.count
        if (size > 3) Fiber.abort("There shouldn't be more than 3 points.")
        if (size == 0) return Circle.new(Point.new(0, 0), 0)
        if (size == 1) return Circle.new(rs[0], 0)
        if (size == 2) return Circle.from(rs[0], rs[1])
        for (i in 0..1) {
            for (j in i+1..2) {
                var c = Circle.from(rs[i], rs[j])
                if (c.encloses(rs)) return c
            }
        }
        return Circle.from(rs[0], rs[1], rs[2])
    }

    // private helper method for Welzl method
    static welzl_(ps, rs, n) {
        rs = rs.toList // passed by value so need to copy
        if (n == 0 || rs.count == 3) return secTrivial(rs)
        var idx = Rand.int(n)
        var p = ps[idx]
        ps[idx] = ps[n-1]
        ps[n-1] = p
        var d = welzl_(ps, rs, n-1)
        if (d.contains(p)) return d
        rs.add(p)
        return welzl_(ps, rs, n-1)
    }

    // applies the Welzl algorithm to find the SEC
    static welzl(ps) {
        var pc = List.filled(ps.count, null)
        for (i in 0...pc.count) pc[i] = ps[i].copy()
        Rand.shuffle(pc)
        return welzl_(pc, [], pc.count)
    }

    // constructs a new Circle object from its center point and its radius
    construct new(c, r) {
        _c = c.copy()
        _r = r
    }

    // basic properties
    c { _c }
    r { _r }

    // returns whether the current instance contains the point 'p'
    contains(p) { Point.distSq(_c, p) <= _r * _r }

    // returns whether the current instance contains a list of points
    encloses(ps) {
        for (p in ps) if (!contains(p)) return false
        return true
    }

    // string representation
    toString { "Center %(_c), Radius %(_r)" }
}

var tests = [
    [Point.new(0, 0), Point.new(0, 1), Point.new(1, 0)],
    [Point.new(5, -2), Point.new(-3, -2), Point.new(-2, 5), Point.new(1, 6), Point.new(0, 2)]
]

for (test in tests) System.print(Circle.welzl(test))</syntaxhighlight>

{{out}}
<pre>
Center (0.5, 0.5), Radius 0.70710678118655
Center (1, 1), Radius 5
</pre>