Circle Covering by Arcs

Author

Eric W. Weisstein
January 16, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/ComputationalGeometry/CircleCoveringByArcs.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/CircleCoveringByArcs.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Probability of Complete Coverage

(∑_(k=0)^(Floor[1/a]))(((-1))^k) (((1-a k))^(n-1)) (Binomial[n, k])

V5

Sum[(-1)^k Binomial[n,k](1-k a)^(n-1),{k,0,Floor[1/a]}]

V7

Sum[(-1)^k Binomial[n,k](1-k a)^(n-1),{k,0,Floor[1/a]}]

(∑_(k=0)^m)(((-1))^k) (((1-a k))^(n-1)) (Binomial[n, i])

V5

Sum[(-1)^k Binomial[n,i](1-k a)^(n-1),{k,0,m}]

V7

Sum[(-1)^k Binomial[n,i](1-k a)^(n-1),{k,0,m}]

Probability of l gaps

(Binomial[n, l]) (∑_(j=l)^k)(((-1))^(j-l)) (((1-a j))^(n-1)) (Binomial[n-l, j-l])

V5

Binomial[n,l]Sum[(-1)^(j-l)Binomial[n-l,j-l](1-j a)^(n-1),{j,l,k}]

V7

Binomial[n,l]Sum[(-1)^(j-l)Binomial[n-l,j-l](1-j a)^(n-1),{j,l,k}]

Binomial[n,l]Sum[(-1)^(j-l)Binomial[n-l,j-l](1-j a)^(n-1),{j,l,n}]

Plots

RandomArcs[a_,n_]:=Table[{Hue[i/n],Circle[{0,0},1/(2Pi)(1-0i/n),{#,#+a}&@Random[Real,{0,2Pi}]]},{i,n}]

l=RandomArcs[.4,30];
movie=Table[Graphics[{Circle[{0,0},1/(2Pi)],Thickness[.018],Take[l,n]},AspectRatio->Automatic],{n,0,Length[l]}];

Export[gifs<>"circarcs.gif",Show/@movie]

Show[Graphics[{Circle[{0,0},1/(2Pi)],Thickness[.03],l},AspectRatio->Automatic]]