Mohammed Sign

Author

Eric W. Weisstein
November 16, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/MohammedSign.nb.

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Plot

RegionPlot[x^2+y^2<1&&(x-.5)^2+y^2<1.05^2,{x,-1,1},{y,-1,1}]

RegionPlot[x^2+y^2<1&&(x-.5)^2+y^2>1.05^2,{x,-1,1},{y,-1,1},MaxRecursion->7]

Integrate[Boole[x^2+y^2<a^2&&(x-c)^2+y^2>b^2],{y,-∞,∞},{x,-∞,∞},Assumptions->a>0&&b>0&&c<a+b]//Timing

FullSimplify[Solve[x^2+y^2==a^2&&(x-c)^2+y^2==b^2,{x,y}],a>0&&b>0&&c>0]

Mean[{-a,c-b}]

Simplify[(a^2)-(((((a^2)-(b^2)+(c^2)))^2)/(4 (c^2)))]

Manipulate[
Show[{
RegionPlot[(x^2+y^2<a^2&&(x-c)^2+y^2>b^2)||((x-(c-b-a))^2+y^2<a^2&&(x-(-a-b))^2+y^2>b^2),{x,(-a-b+c)-(((a^2)-(b^2)+(c^2))/(2 c)),(((a^2)-(b^2)+(c^2))/(2 c))},{y,-Max[a,b],Max[a,b]},
MaxRecursion->4,PlotStyle->If[2a c>a^2-b^2+c^2,Yellow,Red]],
Graphics[{
Red,Circle[{0,0},a],Circle[{c,0},b],
Blue,Circle[{c-b-a,0},a],Circle[{-a-b,0},b]
(*
Blue,Point[{c-b,0}],
Red,Point[{-a,0}]
*)
}]
},PlotRange->All,AspectRatio->Automatic,Method->{Antialiasing->True}],
{{a,1},1,5},{{b,1.212},1,5},{{c,.5},.1,5}]

With[{a=1,b=1.212,c=.5},
Show[{
RegionPlot[(x^2+y^2<a^2&&(x-c)^2+y^2>b^2)||((x-(c-b-a))^2+y^2<a^2&&(x-(-a-b))^2+y^2>b^2),{x,(-a-b+c)-(((a^2)-(b^2)+(c^2))/(2 c)),(((a^2)-(b^2)+(c^2))/(2 c))},{y,-Max[a,b],Max[a,b]},
MaxRecursion->4],
Graphics[{
PointSize[.02],EdgeForm[Black],
Red,Circle[{0,0},a],Circle[{c,0},b],
Blue,Circle[{c-b-a,0},a],Circle[{-a-b,0},b],
Green,Point[{0,0}],
Purple,Point[{c,0}],
Blue,Point[{c-b,0}],
Red,Point[{-a,0}],
Orange,Point[{-a-b+c,0}],
Black,Point[{a,0}],
Yellow,Point[{(((a^2)-(b^2)+(c^2))/(2 c)),(Sqrt[(a^2)-(((((a^2)-(b^2)+(c^2)))^2)/(4 (c^2)))])}]
}]
},PlotRange->All,AspectRatio->Automatic,Method->{Antialiasing->True}]
]

2Mean[{-a,c-b}]

With[{a=1,b=1.212,c=.5},-a-b+c]

{A1,A2}={(a-(((a^2)-(b^2)+(c^2))/(2 c)))(Sqrt[(a^2)-(((((a^2)-(b^2)+(c^2)))^2)/(4 (c^2)))]),(2(R^2)ArcCos[(d/(2R))]-(1/2)d(Sqrt[4(R^2)-(d^2)]))/.{d->-(-a-b+c),R->a}}/.{a->1,b->1.212,c->.5}

Area

LuneArea[{aa_,bb_},c_]:=With[{fourd=(Sqrt[-(aa-bb-c) (aa+bb-c) (aa-bb+c) (aa+bb+c)]),a=Min[{aa,bb}],b=Max[{aa,bb}]},(fourd/2)-(b^2) ((π/2)+ArcTan[(((a^2)-(b^2)-(c^2))/fourd)])+(a^2) (ArcCos[((b-c)/a)]+ArcTan[((b-c)/(Sqrt[(a+b-c) (a-b+c)]))]+ArcTan[(((a^2)-(b^2)+(c^2))/fourd)])]

2LuneArea[{a,b},c]-(2(a^2)ArcCos[(d/(2a))]-(1/2)d(Sqrt[4(a^2)-(d^2)]))/.d->-(-a-b+c)/.{a->1.,b->1.212,c->.5}

FullSimplify[2LuneArea[{a,b},c]-(2(a^2)ArcCos[(d/(2a))]-(1/2)d(Sqrt[4(a^2)-(d^2)]))/.d->-(-a-b+c),0<a<b&&c>0]

%/.{a->1.,b->1.212,c->.5}

FullSimplify[2LuneArea[{a,b},c]-(2(b^2)ArcCos[(d/(2b))]-(1/2)d(Sqrt[4(b^2)-(d^2)]))/.d->-(-a-b+c),0<b<a&&c>0]

%/.{b->1.,a->1.212,c->.5}

FullSimplify[2LuneArea[{a,b},c]-(2((Min[a,b])^2)ArcCos[(d/(2Min[a,b]))]-(1/2)d(Sqrt[4((Min[a,b])^2)-(d^2)]))/.d->-(-a-b+c),0<a<b&&c>0]

%/.{b->1.,a->1.212,c->.5}

FullSimplify[2LuneArea[{a,b},c]-(2((Min[a,b])^2)ArcCos[(d/(2Min[a,b]))]-(1/2)d(Sqrt[4((Min[a,b])^2)-(d^2)]))/.d->-(-a-b+c),0<b<a&&c>0]

%/.{a->1.,b->1.212,c->.5}

With[{a=1,b=1.212,c=.5},NIntegrate[Boole[(x^2+y^2<a^2&&(x-c)^2+y^2>b^2)||((x-(c-b-a))^2+y^2<a^2&&(x-(-a-b))^2+y^2>b^2)],{x,(-a-b+c)-(((a^2)-(b^2)+(c^2))/(2 c)),(((a^2)-(b^2)+(c^2))/(2 c))},{y,-Max[a,b],Max[a,b]}]
]//Chop