Smallest enclosing circle problem
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.
11l
<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))</lang>
- Output:
Center (0.5, 0.5), Radius 0.707106781 Center (1, 1), Radius 5
Go
<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))
}
}</lang>
- Output:
{(0.500000, 0.500000), 0.707107}
{(1.000000, 1.000000), 5.000000}
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. <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()
</lang>
- Output:
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
Mathematica/Wolfram Language
<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`}}]</lang>
- Output:
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]
Nim
<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)</lang>
- Output:
Center (0.5, 0.5), Radius 0.7071067811865476 Center (1.0, 1.0), Radius 5.0
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.
with javascript_semantics type point(sequence s) return true end type enum CENTRE, RADIUS type circle(sequence s) return true end type function distance(point a, b) return sqrt(sum(sq_power(sq_sub(a,b),2))) end function function in_circle(point p, circle c) return distance(p,c[CENTRE]) <= c[RADIUS] end function function circle_from2(point a, b) -- return the smallest circle that intersects 2 points: point midpoint = sq_div(sq_add(a,b),2) atom halfdiameter = distance(a,b)/2 circle res = { midpoint, halfdiameter } return res end function function circle_from3(point a, b, c) -- return a unique circle that intersects three points point bma = sq_sub(b,a), cma = sq_sub(c,a) atom {{aX,aY},{bX,bY},{cX,cY}} = {a,bma,cma}, B = sum(sq_power(bma,2)), C = sum(sq_power(cma,2)), D = (bX*cY - bY*cX)*2 point centre = {(cY*B - bY*C)/D + aX, (bX*C - cX*B)/D + aY } atom radius = distance(centre,a) -- (=== b,c) circle res = { centre, radius } return res end function function trivial(sequence r) integer l = length(r) switch l do case 0: return {{0,0},0} case 1: return {r[1],0} case 2: return circle_from2(r[1],r[2]) case 3: return circle_from3(r[1],r[2],r[3]) end switch end function function welzlr(sequence p, r) if p={} or length(r)=3 then return trivial(r) end if point p1 = p[1] p = p[2..$] circle d = welzlr(p, r) if in_circle(p1,d) then return d end if return welzlr(p, append(deep_copy(r),p1)) end function procedure welzl(sequence p) circle c = welzlr(shuffle(p),{}) string s = sprintf("centre %v radius %.14g",c) printf(1,"Points %v ==> %s\n",{p,s}) end procedure constant tests = {{{0, 0},{ 0, 1},{ 1,0}}, {{5,-2},{-3,-2},{-2,5},{1,6},{0,2}}} papply(tests,welzl)
- Output:
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
n-dimensional
Uses the last test from Julia, however, since that's given more accurately.
with javascript_semantics constant inf = 1e300*1e300, nan = -(inf/inf), eps = 1e-8 function sqnorm(sequence p) return sum(sq_power(p,2)) end function function sqdist(sequence p1, p2) return sqnorm(sq_sub(p1,p2)) end function function projector_mult(sequence vs, v) sequence s = repeat(0,length(v)) for i=1 to length(vs) do sequence vi = vs[i] s = sq_add(s,sq_mul(vi,sum(sq_mul(vi, v)))) end for return s end function -- Gaertner Boundary: sequence empty_center, centers, square_radii, -- Stack of points that are shifted / projected to put first one at origin: projector function push_if_stable(sequence pt) if length(centers) == 0 then square_radii = append(square_radii, 0) centers = {pt} return true end if sequence q0 = centers[1], center = centers[$], C = sq_sub(center,q0), Qm = sq_sub(pt,q0), Qm_bar = projector_mult(projector,Qm), residue = sq_sub(Qm,Qm_bar) atom r2 = square_radii[$], e = sqdist(Qm, C) - r2, z = 2 * sqnorm(residue), tol = eps * max(r2, 1), isstable = abs(z) > tol if isstable then sequence center_new = sq_add(center,sq_mul(e/z,residue)) atom r2new = r2 + (e * e) / (2 * z) projector = append(projector,sq_div(residue,sqrt(sqnorm(residue)))) centers = append(centers, center_new) square_radii = append(square_radii, r2new) end if return isstable end function procedure pop_gb() integer n = length(centers) centers = centers[1..$-1] square_radii = square_radii[1..$-1] if n >= 2 then projector = projector[1..$-1] -- (pop/discard) end if end procedure function isinside(sequence pt, center, atom sqradius) atom r2 = sqdist(pt, center) return r2!=nan and (r2<=sqradius or abs(r2-sqradius)<eps) end function function move_to_front(sequence pts, integer i) sequence pt = pts[i] for j=1 to i do {pts[j], pt} = {pt, pts[j]} end for return pts end function function ismaxlength() return length(centers) == length(empty_center) + 1 end function function _welzl(sequence pts, integer pos) integer support_count = 0, s sequence center = iff(length(centers)?centers[$]:empty_center) atom sqradius = iff(length(centers)?square_radii[$]:0) if ismaxlength() then return {center, sqradius, 0} end if for i=1 to pos do if not isinside(pts[i], center, sqradius) then bool isstable = push_if_stable(pts[i]) if isstable then {center, sqradius, s} = _welzl(pts, i) pop_gb() pts = move_to_front(deep_copy(pts), i) support_count = s + 1 end if end if end for return {center, sqradius, support_count} end function function find_max_excess(sequence center, pts, atom sqradius, integer k1) atom err_max = -inf integer k_max = k1 - 1 for k=k_max to length(pts) do sequence pt = pts[k] atom err = sqdist(pt, center) - sqradius if err > err_max then err_max = err k_max = k + 1 end if end for return {err_max, k_max - 1} end function procedure welzl(sequence points, integer maxiterations=2000) sequence pts = points empty_center = repeat(nan,length(pts[1])) centers = {} square_radii = {} projector = {} integer t = 1, s_new {sequence center, atom sqradius, integer s} = _welzl(pts, t) for i=1 to maxiterations do atom {e, k} = find_max_excess(center, pts, sqradius, t + 1) if e <= eps then exit end if sequence pt = pts[k] {} = push_if_stable(pt) {center, sqradius, s_new} = _welzl(pts, s) pop_gb() pts = move_to_front(deep_copy(pts), k) t = s + 1 s = s_new + 1 end for printf(1,"For points: ") pp(points,{pp_Indent,12,pp_FltFmt,"%11.8f"}) printf(1," Center is at: %v\n", {center}) printf(1," Radius is: %.16g\n", {sqrt(sqradius)}) end procedure constant tests = {{{0, 0}, { 0, 1}, { 1, 0}}, {{5,-2}, {-3,-2}, {-2, 5}, {1, 6}, {0, 2}}, {{2, 4, -1}, {1, 5, -3}, {8, -4, 1}, {3, 9, -5}}, {{-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, 0.1486364315310990, 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}}} papply(tests,welzl)
- Output:
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
Python
<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")
</lang>
- Output:
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
Raku
<lang perl6>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;</lang>
- Output:
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
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 here. <lang ecmascript>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))</lang>
- Output:
Center (0.5, 0.5), Radius 0.70710678118655 Center (1, 1), Radius 5