Circular Sector

Author

Eric W. Weisstein
August 26, 2015

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Laminae/CircularSector.nb.

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

©2015 Wolfram Research, Inc. except for portions noted otherwise

Deifnitions

ineq=x^2+y^2≤a^2&&Tan[-q/2]≤x/y≤Tan[q/2]&&y≥0;

reg=Disk[{0,0},a,Pi/2+{-1,1} q/2];

assum=a>0&&0<q<π;

area=(((a^2) q)/2);

ent="CircularSector";

Plots

Entity Diagram

LaminaData[ent,"Diagram"]

Entity ImplicitRegion

Show[DiscretizeRegion@LaminaData[ent,"ImplicitRegion"][1,1],ImageSize->Small]

Entity ParametricRegion

LaminaData[ent,"ParametricRegion"]

...to be fixed...

Show[DiscretizeRegion@LaminaData[ent,"ParametricRegion"][1,1],ImageSize->Small]

bug(296038)

Entity Region

Show[DiscretizeRegion@LaminaData[ent,"Region"][1,1],ImageSize->Small]

InequalityGraphics

V5

<<Graphics`InequalityGraphics`

Sector[d_,R_:1]:=Module[{x,y,c=Sqrt[R^2-d^2]},
InequalityPlot[x^2+y^2<R^2&&Abs[x/y]<c/d&&y>0,{x},{y},Axes->False,
PlotRange->R{{-1,1},{-1,1}},
DisplayFunction->Identity][[1]]
]

With[{d=.5,R=1},
c=Sqrt[R^2-d^2];
Show[Graphics[{Sector[d,R],
Circle[{0,0},R],
PointSize[.02],Point[{0,0}],
Dashing[{.02}],
Line[{{-c,d},{c,d}}]}
],AspectRatio->Automatic,
PlotRange->All
]
]

With[{R=1,θ=1},
InequalityPlot[x^2+y^2<R^2&&Tan[-θ/2]<x/y<Tan[θ/2]&&y>0,{x,-∞,∞},{y,-∞,∞}]]

With[{R=1,θ=1},
InequalityPlot[-R Sin[(θ/2)]<x<R Sin[(θ/2)]&&Abs[x] Cot[(θ/2)]<y<(Sqrt[(R^2)-(x^2)]),{x,-∞,∞},{y,-∞,∞}]]

RegionPlot

Block[{a=1,q=3},Show[{RegionPlot[x^2+y^2<=a^2&&-q/2<=ArcTan[y,x]≤q/2&&y>0,{x,-1,1},{y,0,1}]}]]

Block[{a=1,q=1},Show[{RegionPlot[x^2+y^2<=a^2&&Tan[-q/2]≤x/y≤Tan[q/2]&&y>0,{x,-1,1},{y,0,1}]},AspectRatio->Automatic]]

Regions

LaminaData[ent,"ImplicitRegion"]

LaminaData[ent,"ParametricRegion"]

LaminaData[ent,"Region"]

Properties

Area

Entity

EntityValue[Entity["Lamina",ent],"Area"][q,a]

Area

Assuming[assum,Area[EntityValue[Entity["Lamina",ent],"ImplicitRegion"][q,a]]//Simplify]//Timing

Assuming[assum,Area[EntityValue[Entity["Lamina",ent],"Region"][q,a]]//Simplify]//Timing

Integrate

Integrate[Boole[x^2+y^2<R^2&&Abs[x/y]<Sqrt[R^2-d^2]/d&&y>0],{x,-∞,∞},{y,-∞,∞},Assumptions->0<d<R]//Timing

FullSimplify[(R^2) ArcTan[((Sqrt[-(d^2)+(R^2)])/d)]/.d->R Cos[(θ/2)],R>0&&0<θ<π/2]

Integrate[Boole[x^2+y^2<R^2&&-θ/2<ArcTan[x,y]<θ/2&&y>0],{x,-∞,∞},{y,-∞,∞},Assumptions->R>0&&0<θ<π/2]//Timing

Integrate[Boole[x^2+y^2<R^2&&Tan[-θ/2]<x/y<Tan[θ/2]&&y>0],{x,-∞,∞},{y,-∞,∞},Assumptions->R>0&&0<θ<π/2]//Timing

Integrate[1,{x,-R Sin[(θ/2)],R Sin[(θ/2)]},{y,Abs[x] Cot[(θ/2)],(Sqrt[(R^2)-(x^2)])},Assumptions->R>0&&0<θ<π/2]//Timing

Integrate[Boole[x^2+y^2<=a^2&&-Tan[q/2]≤x/y≤Tan[q/2]&&y>0],{x,-a,a},{y,0,a},Assumptions->a>0&&0<q<π]//Timing

RegionMeasure

Assuming[a>0&&0<q<π/2,Simplify@RegionMeasure[Disk[{0,0},a,Pi/2+{-1,1} q/2]]]

Centroid

EntityVaue

EntityValue[Entity["Lamina",ent],"Centroid"][q,a]

Integrate

Integrate[x Boole[x^2+y^2<R^2&&Abs[x/y]<Sqrt[R^2-d^2]/d&&y>0],{x,-∞,∞},{y,-∞,∞},Assumptions->0<d<R]//Timing

Integrate[y Boole[x^2+y^2<R^2&&Abs[x/y]<Sqrt[R^2-d^2]/d&&y>0],{x,-∞,∞},{y,-∞,∞},Assumptions->0<d<R]//Timing

FullSimplify[(2/3) (R^2) (Sqrt[-(d^2)+(R^2)])/.d->R Cos[(θ/2)],R>0&&0<θ<π/2]

Integrate[y,{x,-R Sin[(θ/2)],R Sin[(θ/2)]},{y,Abs[x] Cot[(θ/2)],(Sqrt[(R^2)-(x^2)])},Assumptions->R>0&&0<θ<π/2]//Timing

Integrate[y Boole[-R Sin[(θ/2)]<x<R Sin[(θ/2)]&&Abs[x] Cot[(θ/2)]<y<(Sqrt[(R^2)-(x^2)])],{x,-∞,∞},{y,-∞,∞},Assumptions->R>0&&0<θ<π/2]//Timing

ybar=(((2/3) (R^3) Sin[(θ/2)])/(((R^2) θ)/2))

Limit[ybar,θ->#]&/@{0,π}

RegionCentroid

Assuming[assum,RegionCentroid[EntityValue[Entity["Lamina",ent],"ImplicitRegion"][q,a]]//FullSimplify]//Timing

ByteCount/@{((4 a Sin[(q/2)])/(3 q)),(2/3) a Sinc[(q/2)]}

FullSimplify[Subtract@@{((4 a Sin[(q/2)])/(3 q)),(2/3) a Sinc[(q/2)]}]

Assuming[assum,RegionCentroid[EntityValue[Entity["Lamina",ent],"Region"][q,a]]]//Timing

Circumcircle

FullSimplify[Circumcircle[Triangle[{{0,0},{a Sin[q/2],a Cos[q/2]},{-a Sin[q/2],a Cos[q/2]}}]],a>0&&0<q<π]

Block[{a=1,q=π/2-.01},Show[{
RegionPlot[x^2+y^2<=a^2&&-q/2<=ArcTan[y,x]≤q/2&&y>0,{x,-1,1},{y,-1,1}],
Graphics[{
Circle[{0,a Csc[q] Sin[(q/2)]},a Csc[q] Sin[(q/2)]],
Point[{{0,0},{a Sin[q/2],a Cos[q/2]},{-a Sin[q/2],a Cos[q/2]}}]
}]}]]

MomentOfInertiaTensor

AreaInertiaTensor[{x,y},"CoordinateSystem"->"Cartesian"]

(i=Integrate[{{(y^2),-x y},{-x y,(x^2)}}Boole[x^2+y^2<=a^2&&-Tan[q/2]≤x/y≤Tan[q/2]&&y>0],{x,-a,a},{y,0,a},Assumptions->a>0&&0<q<π])//Timing

Limit[{{(1/8) (a^4) (q+Sin[q]),0},{0,(1/8) (a^4) (q-Sin[q])}},q->π]

Perimeter

a(q+2)/.{q->π}

RadiiOfGyration

Assuming[a>0&&0<q<π,FullSimplify[RadiiOfGyration[x^2+y^2<=a^2&&-Tan[q/2]≤x/y≤Tan[q/2]&&y>0,{x,y}]]]

Regions

EntityValue[Entity["Lamina",ent],"ImplicitRegion"][θ,a]

EntityValue[Entity["Lamina",ent],"Region"][θ,a]

SectionModuli

Assuming[a>0&&0<q<π,FullSimplify[Maximize[{#,x^2+y^2<=a^2&&-Tan[q/2]≤x/y≤Tan[q/2]&&y>0},{x,y}]&/@{x,y}]]

Assuming[a>0&&0<q<π,FullSimplify[Minimize[{#,x^2+y^2<=a^2&&-Tan[q/2]≤x/y≤Tan[q/2]&&y>0},{x,y}]&/@{x,y}]]

FullSimplify[Tr[i,List]/{a,a Sin[(q/2)]}]