Problem:
Chain-Confined Path There is a chain consisting of multiple circles on a plane.
The first (last) circle of the chain only intersects with the next (previous) circle,
and each intermediate circle intersects only with the two neighboring circles. Your task is to find the shortest path that satisfies the following conditions.  The path connects the centers of the first circle and the last circle.  The path is confined in the chain, that is, all the points on the path are located within or on at least one of the circles.  Figure E-1 shows an example of such a chain and the corresponding shortest path.   Figure E-1: An  example chain and the corresponding shortest path   Input The input consists of multiple datasets.
Each dataset represents the shape of a chain in the following format.  n  x 1  y 1  r 1   x 2  y 2  r 2  ...  x n  y n  r n   The first line of a dataset contains an integer n (3 ≤ n ≤ 100) 
representing the number of the circles.
Each of the following n lines contains three integers separated by a single space.
( x i , y i ) 
and r i represent 
the center position and the radius of the i -th circle C i .
You can assume that 0 ≤ x i ≤ 1000, 
0 ≤ y i ≤ 1000, and
1 ≤ r i ≤ 25. You can assume that C i and C i +1 (1 ≤ i ≤ n −1) intersect at two separate points.
When j ≥ i +2, C i and C j are apart and either of them does not contain the other.
In addition, you can assume that any circle does not contain the center of any other circle. The end of the input is indicated by a line containing a zero. Figure E-1 corresponds to the first dataset of Sample Input below.
Figure E-2 shows the shortest paths for the subsequent datasets of Sample Input.   Figure E-2: Example chains and the corresponding shortest paths   Output For each dataset, output a single line containing the length of the shortest chain-confined path between the centers of the first circle and the last circle.
The value should not have an error greater than 0.001.
No extra characters should appear in the output. Sample Input 10
802 0 10
814 0 4
820 1 4
826 1 4
832 3 5
838 5 5
845 7 3
849 10 3
853 14 4
857 18 3
3
0 0 5
8 0 5
8 8 5
3
0 0 5
7 3 6
16 0 5
9
0 3 5
8 0 8
19 2 8
23 14 6
23 21 6
23 28 6
19 40 8
8 42 8
0 39 5
11
0 0 5
8 0 5
18 8 10
8 16 5
0 16 5
0 24 5
3 32 5
10 32 5
17 28 8
27 25 3
30 18 5
0 Output for the Sample Input 58.953437
11.414214
16.0
61.874812
63.195179

Solution (Python):
def solve():
    from math import acos
    from cmath import phase, rect, pi
    
    from sys import stdin
    file_input = stdin
    
    while True:
        n = int(file_input.readline())
        if n == 0:
            break
        
        C = (map(int, file_input.readline().split()) for i in range(n))
        P = []
        x, y, r1 = next(C)
        c1 = x + y * 1j
        P.append(c1)
        
        
        for x, y, r2 in C:
            c2 = x + y * 1j
            
            base = c2 - c1
            d = abs(base)
            a = acos((r1 ** 2 + d ** 2 - r2 ** 2) / (2 * r1 * d))
            t = phase(base)
            cp1 = c1 + rect(r1, t + a)
            cp2 = c1 + rect(r1, t - a)
            
            P.append(cp1)
            P.append(cp2)
            
            c1, r1 = c2, r2
        
        
        lim = 5000
        dist = [lim] * (2 * n)
        dist[0] = 0
        goal = c1
        g_idx = 2 * n - 1
        indices = ((i + (i % 2) + 1, i + (i % 2) + 2) for i in range(g_idx))
        
        for tpl_idx, cp, d in zip(indices, P, dist):
            j, k = tpl_idx
            s1 = None
            s2 = None
            p_s1 = None
            p_s2 = None
            for l, cp1, cp2 in zip(range(j, g_idx, 2), P[j::2], P[k::2]):
                t_s1 = cp1 - cp
                t_s2 = cp2 - cp
                if s1 is None or phase(s1 / t_s1) >= 0:
                    s1 = t_s1
                if s2 is None or phase(s2 / t_s2) <= 0:
                    s2 = t_s2
                if phase(s1 / s2) < 0:
                    break
                
                if p_s1 != s1:
                    dist[l] = min(dist[l], d + abs(s1))
                    p_s1 = s1
                if p_s2 != s2:
                    dist[l+1] = min(dist[l+1], d + abs(s2))
                    p_s2 = s2
            else:
                gs = goal - cp
                if (s1 is None and s2 is None) or \
                phase(s1 / gs) >= 0 and phase(s2 / gs) <= 0:
                    dist[g_idx] = min(dist[g_idx], d + abs(gs))
        
        print(dist[g_idx])

solve()

Solution (Python):
from math import sqrt
while 1:
    n = input()
    if n == 0:
        break
    C = [map(int, raw_input().split()) for i in xrange(n)]
    P = [[C[0][:2]]]
    for i in xrange(n-1):
        x0, y0, r0 = C[i]
        x1, y1, r1 = C[i+1]
        rr = (x1 - x0)**2 + (y1 - y0)**2; rd = sqrt(rr)
        rc = (r0**2 + rr - r1**2) / (2 * rd); rs = sqrt(4*r0**2*rr - (r0**2 + rr - r1**2)**2) / (2 * rd)
        ex = (x1 - x0) / rd; ey = (y1 - y0) / rd
        bx = x0 + ex * rc; by = y0 + ey * rc
        P.append([(bx + ey * rs, by - ex * rs), (bx - ey * rs, by + ex * rs)])
        
        
    P.append([C[-1][:2]])
    
    

    def calc((x0, y0), (x1, y1)):
        return sqrt((x1 - x0)**2 + (y1 - y0)**2)
    def cross((x0, y0), (x1, y1), (x2, y2)):
        return (x1-x0) * (y2-y0) - (x2-x0) * (y1-y0)
    INF = 10**18
    dist = {(0, 0): 0}
    for i in xrange(n):
        
        for j, (x, y) in enumerate(P[i]):
            left = p_left = None; right = p_right = None
            d = dist[i, j]
            for k in xrange(i+1, n):
                lp, rp = P[k]
                
                if left is None or cross((x, y), left, lp) >= 0:
                    left = lp
                if right is None or cross((x, y), rp, right) >= 0:
                    right = rp
                if cross((x, y), left, right) < 0:
                    break
                
                if p_left != left:
                    
                    dist[k, 0] = min(dist.get((k, 0), INF), d + calc((x, y), left))
                    p_left = left
                if p_right != right:
                    
                    dist[k, 1] = min(dist.get((k, 1), INF), d + calc((x, y), right))
                    p_right = right
            else:
                
                gp = P[-1][0]
                if (left is None and right is None) or (cross((x, y), left, gp) >= 0 and cross((x, y), gp, right) >= 0):
                    
                    dist[len(P)-1, 0] = min(dist.get((len(P)-1, 0), INF), d + calc((x, y), gp))
    print "%.06f" % dist[len(P)-1, 0]

Solution (Python):
from math import sqrt
while 1:
    n = input()
    if n == 0:
        break
    C = [map(int, raw_input().split()) for i in xrange(n)]
    P = [[C[0][:2]]]
    
    for i in xrange(n-1):
        x0, y0, r0 = C[i]
        x1, y1, r1 = C[i+1]
        rr = (x1 - x0)**2 + (y1 - y0)**2; rd = sqrt(rr)
        rc = (r0**2 + rr - r1**2) / (2 * rd); rs = sqrt(4*r0**2*rr - (r0**2 + rr - r1**2)**2) / (2 * rd)
        ex = (x1 - x0) / rd; ey = (y1 - y0) / rd
        bx = x0 + ex * rc; by = y0 + ey * rc
        P.append([(bx + ey * rs, by - ex * rs), (bx - ey * rs, by + ex * rs)])
    P.append([C[-1][:2]])

    def calc((x0, y0), (x1, y1)):
        return sqrt((x1 - x0)**2 + (y1 - y0)**2)
    def cross((x0, y0), (x1, y1), (x2, y2)):
        return (x1-x0) * (y2-y0) - (x2-x0) * (y1-y0)
    INF = 10**18
    dist = {(0, 0): 0}
    for i in xrange(n):
        for j, (x, y) in enumerate(P[i]):
            left = p_left = None; right = p_right = None
            d = dist[i, j]
            for k in xrange(i+1, n):
                lp, rp = P[k]
                
                if left is None or cross((x, y), left, lp) >= 0:
                    left = lp
                if right is None or cross((x, y), rp, right) >= 0:
                    right = rp
                
                if cross((x, y), left, right) < 0:
                    break

                
                if p_left != left:
                    dist[k, 0] = min(dist.get((k, 0), INF), d + calc((x, y), left))
                    p_left = left
                if p_right != right:
                    dist[k, 1] = min(dist.get((k, 1), INF), d + calc((x, y), right))
                    p_right = right
            else:
                

                gp = P[-1][0]
                
                if (left is None and right is None) or (cross((x, y), left, gp) >= 0 and cross((x, y), gp, right) >= 0):
                    dist[len(P)-1, 0] = min(dist.get((len(P)-1, 0), INF), d + calc((x, y), gp))
    print "%.06f" % dist[len(P)-1, 0]

Solution (Python):
from math import sqrt
from sys import stdin, stdout
readline = stdin.readline
write = stdout.write
while 1:
    n = int(readline())
    if n == 0:
        break
    C = [map(int, readline().split()) for i in xrange(n)]
    P = [[C[0][:2]]]
    
    for i in xrange(n-1):
        x0, y0, r0 = C[i]
        x1, y1, r1 = C[i+1]
        rr = (x1 - x0)**2 + (y1 - y0)**2; rd = sqrt(rr)
        rc = (r0**2 + rr - r1**2) / (2 * rd); rs = sqrt(4*r0**2*rr - (r0**2 + rr - r1**2)**2) / (2 * rd)
        ex = (x1 - x0) / rd; ey = (y1 - y0) / rd
        bx = x0 + ex * rc; by = y0 + ey * rc
        P.append([(bx + ey * rs, by - ex * rs), (bx - ey * rs, by + ex * rs)])
    P.append([C[-1][:2]])

    def calc((x0, y0), (x1, y1)):
        return sqrt((x1 - x0)**2 + (y1 - y0)**2)
    def cross((x0, y0), (x1, y1), (x2, y2)):
        return (x1-x0) * (y2-y0) - (x2-x0) * (y1-y0)
    INF = 10**18
    dist = {(0, 0): 0}
    for i in xrange(n):
        for j, (x, y) in enumerate(P[i]):
            left = p_left = None; right = p_right = None
            d = dist[i, j]
            for k in xrange(i+1, n):
                lp, rp = P[k]
                
                if left is None or cross((x, y), left, lp) >= 0:
                    left = lp
                if right is None or cross((x, y), rp, right) >= 0:
                    right = rp
                
                if cross((x, y), left, right) < 0:
                    break

                
                if p_left != left:
                    dist[k, 0] = min(dist.get((k, 0), INF), d + calc((x, y), left))
                    p_left = left
                if p_right != right:
                    dist[k, 1] = min(dist.get((k, 1), INF), d + calc((x, y), right))
                    p_right = right
            else:
                
                gp = P[n][0]
                
                if (left is None and right is None) or (cross((x, y), left, gp) >= 0 and cross((x, y), gp, right) >= 0):
                    dist[n, 0] = min(dist.get((n, 0), INF), d + calc((x, y), gp))
    write("%.06f\n" % dist[n, 0])

Solution (Python):
from math import sqrt
from sys import stdin, stdout
readline = stdin.readline
write = stdout.write
while 1:
    n = int(readline())
    if n == 0:
        break
    C = [map(int, readline().split()) for i in xrange(n)]
    P = {(0, 0): C[0][:2], (n, 0): C[-1][:2]}
    
    for i in xrange(n-1):
        x0, y0, r0 = C[i]
        x1, y1, r1 = C[i+1]
        rr = (x1 - x0)**2 + (y1 - y0)**2; rd = sqrt(rr)
        rc = (r0**2 + rr - r1**2) / (2 * rd); rs = sqrt(4*r0**2*rr - (r0**2 + rr - r1**2)**2) / (2 * rd)
        ex = (x1 - x0) / rd; ey = (y1 - y0) / rd
        bx = x0 + ex * rc; by = y0 + ey * rc
        P[i+1, 0] = (bx + ey * rs, by - ex * rs)
        P[i+1, 1] = (bx - ey * rs, by + ex * rs)

    def calc((x0, y0), (x1, y1)):
        return sqrt((x1 - x0)**2 + (y1 - y0)**2)
    def cross((x0, y0), (x1, y1), (x2, y2)):
        return (x1-x0) * (y2-y0) - (x2-x0) * (y1-y0)
    INF = 10**18
    dist = {(0, 0): 0}
    gp = P[n, 0]
    for i, j in sorted(P):
        pos = P[i, j]
        left = p_left = right = p_right = None
        d = dist[i, j]
        for k in xrange(i+1, n):
            lp = P[k, 0]; rp = P[k, 1]
            
            if left is None or cross(pos, left, lp) >= 0:
                left = lp
            if right is None or cross(pos, rp, right) >= 0:
                right = rp
            
            if cross(pos, left, right) < 0:
                break

            
            if p_left != left:
                dist[k, 0] = min(dist.get((k, 0), INF), d + calc(pos, left))
                p_left = left
            if p_right != right:
                dist[k, 1] = min(dist.get((k, 1), INF), d + calc(pos, right))
                p_right = right
        else:
            
            
            if (left is None and right is None) or (cross(pos, left, gp) >= 0 and cross(pos, gp, right) >= 0):
                dist[n, 0] = min(dist.get((n, 0), INF), d + calc(pos, gp))
    write("%.06f\n" % dist[n, 0])

Solution (Python):
import math


class Vec:
    def __init__(self, x, y):
        self.x = x
        self.y = y

    def __mul__(self, other):
        return Vec(self.x * other, self.y * other)

    def length(self):
        return math.sqrt(self.x * self.x + self.y * self.y)

    def __str__(self):
        return "({},{})".format(self.x, self.y)

    def __sub__(self, other):
        return Vec(self.x - other.x, self.y - other.y)

    def __add__(self, other):
        return Vec(self.x + other.x, self.y + other.y)

    def __truediv__(self, other):
        return Vec(self.x / other, self.y / other)

    def turn90(self):
        return Vec(-self.y, self.x)

    def normalized(self):
        return self / self.length()

    def is_clock(self, other):
        return self.x * other.y - self.y * other.x < 0


class Circle:
    def __init__(self, x, y, r):
        self.r = r
        self.center = Vec(x, y)

    def intersection(self, other):
        d = (self.center - other.center).length()
        r1 = self.r
        r2 = other.r
        d1 = (d * d + r1 * r1 - r2 * r2) / 2 / d
        x = math.sqrt(r1 * r1 - d1 * d1)
        e1 = (other.center - self.center).normalized()
        e2 = e1.turn90()

        p1 = self.center + e1 * d1 + e2 * x
        p2 = self.center + e1 * d1 - e2 * x
        return p1, p2

    def __str__(self):
        return "({},{})".format(self.center, self.r)


def solve(ps: [Circle]):
    intersects = []
    for i in range(len(ps) - 1):
        ci = ps[i]
        ci_next = ps[i + 1]
        intersection1, intersection2 = ci.intersection(ci_next)
        intersects.append((intersection1, intersection2))

    res = [[ps[0].center, -1, 0]]  
    dists = [[2 ** 32, 2 ** 32] for _ in range(len(intersects))]  

    min_path_length = 2 ** 32

    while len(res) != 0:
        res = sorted(res, key=lambda a: a[1], reverse=True)

        c, depth, lr = res.pop()
        current_path_length = dists[depth][lr] if depth != -1 else 0
        if depth == len(intersects) - 1:
            last_path = ps[-1].center - c
            min_path_length = min(min_path_length, current_path_length + last_path.length())
            continue

        l_limit_min, r_limit_min = [a - c for a in intersects[depth + 1]]  
        l_limit, r_limit = l_limit_min, r_limit_min  
        l_limit_d, r_limit_d = depth + 1, depth + 1

        flag = True
        for i in range(depth + 2, len(intersects)):
            l_limit2, r_limit2 = [a - c for a in intersects[i]]

            if l_limit_min.is_clock(l_limit2):  
                l_limit_min = l_limit2
                if l_limit2.is_clock(r_limit_min):  
                    l_limit = l_limit2
                    l_limit_d = i
            if r_limit2.is_clock(r_limit_min):  
                r_limit_min = r_limit2
                if l_limit_min.is_clock(r_limit2):  
                    r_limit = r_limit2
                    r_limit_d = i
            if r_limit_min.is_clock(l_limit_min):
                flag = False
                break
        last_path = ps[-1].center - c
        if flag and l_limit_min.is_clock(last_path) and last_path.is_clock(r_limit_min):  
            min_path_length = min(min_path_length, current_path_length + last_path.length())
            continue
        if dists[l_limit_d][0] > l_limit.length() + current_path_length:
            dists[l_limit_d][0] = l_limit.length() + current_path_length
            res.append([l_limit + c, l_limit_d, 0])
        if dists[r_limit_d][1] > r_limit.length() + current_path_length:
            dists[r_limit_d][1] = r_limit.length() + current_path_length
            res.append([r_limit + c, r_limit_d, 1])
    return min_path_length


def main():
    while True:
        n = int(input())
        if n == 0:
            break

        ps = [Circle(*[int(a) for a in input().split()]) for _ in range(n)]

        print(solve(ps))


if __name__ == '__main__':
    main()

Solution (Python):
from heapq import heappop, heappush

def cross(a,b):
    return a[0]*b[1]-a[1]*b[0]

def calc(r1,r2):
    a = ((r1[0]-r2[0])**2+(r1[1]-r2[1])**2)**0.5
    b = r1[2]
    c = r2[2]
    cos = (a**2+b**2-c**2)/(2*a*b)
    s = b*cos
    t = (b**2-s**2)**0.5
    e = ((r2[0]-r1[0])/a,(r2[1]-r1[1])/a)
    h = (-e[1],e[0])
    return [[r1[0]+s*e[0]+t*h[0], r1[1]+s*e[1]+t*h[1]],[r1[0]+s*e[0]-t*h[0], r1[1]+s*e[1]-t*h[1]]]

def check(a,b,s,t):
    v = (b[0]-a[0],b[1]-a[1])
    v1 = (s[0]-a[0],s[1]-a[1])
    v2 = (t[0]-a[0],t[1]-a[1])
    if cross(v,v1)*cross(v,v2) > 0:
        return 0
    v = (t[0]-s[0],t[1]-s[1])
    v1 = (a[0]-s[0],a[1]-s[1])
    v2 = (b[0]-s[0],b[1]-s[1])
    if cross(v,v1)*cross(v,v2) > 0:
        return 0
    return 1

def dijkstra(m,v):
    dist = [float("inf")]*m
    q = [(0,0)]
    dist[0] = 0
    while q:
        dx,x = heappop(q)
        for y,d in v[x]:
            if dist[y] > dx+d:
                dist[y] = dx+d
                heappush(q,(dx+d,y))
    return dist[m-1]
def solve(n):
    r = [[int(x) for x in input().split()] for i in range(n)]
    p = [[r[0][0],r[0][1]]]
    for i in range(n-1):
        p += calc(r[i],r[i+1])
    p.append([r[-1][0],r[-1][1]])
    m = 2*n
    v = [[] for i in range(m)]
    gate = []
    for i in range(1,n):
        gate.append([p[2*i-1],p[2*i]])

    for i in range(m):
        x = p[i]
        for j in range(i+1,m):
            y = p[j]
            for g1,g2 in gate[(i-1)//2+1:(j-1)//2]:
                if not check(x,y,g1,g2):
                    break
            else:
                d = ((x[0]-y[0])**2+(x[1]-y[1])**2)**0.5
                v[i].append((j,d))
                v[j].append((i,d))
    print(dijkstra(m,v))

while 1:
    n = int(input())
    if n == 0:
        break
    solve(n)

Solution (Python):
import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,random,time,copy,functools

sys.setrecursionlimit(10**7)
inf = 10**20
eps = 1.0 / 10**13
mod = 10**9+7
dd = [(-1,0),(0,1),(1,0),(0,-1)]
ddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]

def LI(): return [int(x) for x in sys.stdin.readline().split()]
def LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]
def LF(): return [float(x) for x in sys.stdin.readline().split()]
def LS(): return sys.stdin.readline().split()
def I(): return int(sys.stdin.readline())
def F(): return float(sys.stdin.readline())
def S(): return input()
def pf(s): return print(s, flush=True)

def _kosa(a1, a2, b1, b2):
    x1,y1 = a1
    x2,y2 = a2
    x3,y3 = b1
    x4,y4 = b2

    tc = (x1-x2)*(y3-y1)+(y1-y2)*(x1-x3)
    td = (x1-x2)*(y4-y1)+(y1-y2)*(x1-x4)
    return tc*td < 0

def kosa(a1, a2, b1, b2):
    return _kosa(a1,a2,b1,b2) and _kosa(b1,b2,a1,a2)

def distance(x1, y1, x2, y2):
    return math.sqrt((x1-x2)**2 + (y1-y2)**2)

def distance_p(a, b):
    return distance(a[0], a[1], b[0], b[1])

def polar(a, r):
    return (math.cos(r) * a, math.sin(r) * a)

def intersection_c(ap, bp):
    ax,ay,ar = ap
    bx,by,br = bp
    d = distance(ax,ay,bx,by)
    if d > ar + br or d < abs(ar - br):
        return None

    ac = math.acos((ar**2 + d**2 - br**2) / (2 * ar * d))
    t = math.atan2(by - ay, bx - ax)
    r1 = polar(ar, t + ac)
    r2 = polar(ar, t - ac)
    return [(ax+r1[0], ay+r1[1]), (ax+r2[0], ay+r2[1])]

def main():
    rr = []

    def f(n):
        a = [LI() for _ in range(n)]
        inters = []
        for i in range(n-1):
            inters.append(intersection_c(a[i], a[i+1]))
        dp = [[inf]*2 for _ in range(n-1)]
        s = a[0][:2]
        rt = a[-1][:2]
        rf = True
        for k in range(n-1):
            if not kosa(s,rt,inters[k][0],inters[k][1]):
                rf = False
                break
        if rf:
            return distance_p(s,rt)

        for i in range(n-1):
            for j in range(2):
                t = inters[i][j]
                f = True
                for k in range(i):
                    if not kosa(s,t, inters[k][0], inters[k][1]):
                        f = False
                        break
                if f:
                    dp[i][j] = distance_p(s,t)

        r = inf
        
        for i in range(n-1):
            for j in range(2):
                s = inters[i][j]
                sd = dp[i][j]
                for k in range(i+1,n-1):
                    for l in range(2):
                        t = inters[k][l]
                        f = True
                        for m in range(i+1,k):
                            if not kosa(s,t,inters[m][0], inters[m][1]):
                                f = False
                                break
                        
                        if f:
                            nd = sd + distance_p(s,t)
                            if dp[k][l] > nd:
                                dp[k][l] = nd
                t = a[-1][:2]
                f = True
                for k in range(i+1,n-1):
                    if not kosa(s,t,inters[k][0], inters[k][1]):
                        f = False
                        break
                if f:
                    nd = sd + distance_p(s,t)
                    if r > nd:
                        r = nd
            
            
            
        return '{:0.5f}'.format(r)

    while 1:
        n = I()
        if n == 0:
            break
        rr.append(f(n))

    return '\n'.join(map(str,rr))


print(main())