Annulus

Author

Eric W. Weisstein
March 22, 2017

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

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

©2017 Wolfram Research, Inc. except for portions noted otherwise

Definitions

ineq=b^2<x^2+y^2<a^2;

implreg=ImplicitRegion[ineq,{x,y}];

reg=RegionDifference[Disk[{0,0},a],Disk[{0,0},b]];

assum=0<b<a;

Plots

Entity Diagram

Entity["Lamina","Annulus"]["Diagram"]

Entity Image

Entity["Lamina","Annulus"]["Image"]

Entity ImplicitRegion

Entity["Lamina","Annulus"]["ImplicitRegion"]

Show[DiscretizeRegion[Entity["Lamina","Annulus"]["ImplicitRegion"][1,.5]],ImageSize->Small]

Entity ParametricRegion

Show[DiscretizeRegion[Entity["Lamina","Annulus"]["ParametricRegion"][1,.5]],ImageSize->Small]

Entity Region

Show[DiscretizeRegion[Entity["Lamina","Annulus"]["Region"][1,.5]],ImageSize->Small]

DiscretizeRegion

With[{a=1,b=.5},Show[DiscretizeRegion[RegionDifference[Disk[{0,0},a],Disk[{0,0},b]]],ImageSize->Small]]

Plot

RegionPlot[.5<x^2+y^2<1,{x,-1,1},{y,-1,1},Frame->False,ImageSize->Small]

Properties

Area

FullSimplify[Subtract@@{(a-b) (a+b) π,(a^2-b^2)π}]

Simplify`SimplifyCount/@{(a-b) (a+b) π,(a^2-b^2)π}

ByteCount/@{(a-b) (a+b) π,((a^2)-(b^2)) π}

Area

area=Assuming[assum,FullSimplify[Area[ineq[x,y],{x,y}]]]

Assuming[assum,Area[Annulus[{0,0},{b,a}]]//Simplify]//Timing

Assuming[assum,Area[EntityValue[Entity["Lamina","Annulus"],"ImplicitRegion"][a,b]]//Simplify]//Timing

Assuming[assum,Area[EntityValue[Entity["Lamina","Annulus"],"ParametricRegion"][a,b]]//Simplify]//Timing

Assuming[assum,Area[EntityValue[Entity["Lamina","Annulus"],"Region"][a,b]]//Simplify]//Timing

Integrate

Assuming[assum,FullSimplify@Integrate[1,{x,y}∈reg]]//Timing

Assuming[assum,FullSimplify@Integrate[1,{x,y}∈implreg]]//Timing

RegionMeasure

Assuming[assum,FullSimplify@RegionMeasure[reg]]//Timing

Assuming[assum,FullSimplify@RegionMeasure[implreg]]//Timing

Assuming[assum,RegionMeasure[EntityValue[Entity["Lamina","Annulus"],"ImplicitRegion"][a,b]]//Simplify]//Timing

Assuming[assum,RegionMeasure[EntityValue[Entity["Lamina","Annulus"],"ParametricRegion"][a,b]]//FullSimplify]//Timing

Assuming[assum,RegionMeasure[EntityValue[Entity["Lamina","Annulus"],"Region"][a,b]]//Simplify]//Timing

Area Moment of Inertia

(i=Assuming[0<b<a,FullSimplify[AreaInertiaTensor[ineq,{x,y}]]])//Timing

Centroid

Integrate

Assuming[assum,FullSimplify[Centroid[ineq[x,y],{x,y}]]]

RegionCentroid

Assuming[assum,RegionCentroid[Annulus[{0,0},{b,a}]]]

Assuming[assum,RegionCentroid[Annulus[{x,y},{b,a}]]]

Assuming[assum,RegionCentroid[reg]]

Assuming[assum,RegionCentroid[implreg]]

Assuming[assum,RegionCentroid[EntityValue[Entity["Lamina","Annulus"],"ImplicitRegion"][a,b]]//Simplify]//Timing

Assuming[assum,RegionCentroid[EntityValue[Entity["Lamina","Annulus"],"ParametricRegion"][a,b]]//Simplify]//Timing

Assuming[assum,RegionCentroid[EntityValue[Entity["Lamina","Annulus"],"Region"][a,b]]//Simplify]//Timing

Convex

TimeConstrained[Region`ConvexRegionQ[reg],600]//Timing

TimeConstrained[Region`ConvexRegionQ[implreg],600]//Timing

TimeConstrained[Region`ConvexRegionQ[Annulus[{0,0},{b,a}]],600]//Timing

TimeConstrained[Region`ConvexRegionQ[EntityValue[Entity["Lamina","Annulus"],"ImplicitRegion"][a,b]],600]//Timing

TimeConstrained[Region`ConvexRegionQ[EntityValue[Entity["Lamina","Annulus"],"ParametricRegion"][a,b]],600]//Timing

TimeConstrained[Region`ConvexRegionQ[EntityValue[Entity["Lamina","Annulus"],"Region"][a,b]],600]//Timing

RadiiOfGyration

Assuming[0<b<a,FullSimplify[RadiiOfGyration[ineq[x,y],{x,y}]]]//Timing

Regions

EntityValue[Entity["Lamina","Annulus"],"ImplicitRegion"][a,b]

EntityValue[Entity["Lamina","Annulus"],"ParametricRegion"][a,b]

EntityValue[Entity["Lamina","Annulus"],"Region"][a,b]