Disk

Author

Eric W. Weisstein
June 12, 2015

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

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

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

Definitions

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

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

implreg=ImplicitRegion[x^2+y^2≤a^2,{x,y}];

assum=a>0;

area=π (a^2);

ent="Disk";

Plots

Entity Diagram

LaminaData[ent,"Diagram"]

Entity ImplicitRegion

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

Entity ParametricRegion

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

bug(295139)

Entity Region

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

DiscretizeRegion

Block[{a=1},Show[DiscretizeRegion[reg],ImageSize->Small]]

RegionPlot

RegionPlot[x^2+y^2≤1,{x,-1,1},{y,-1,1}]

SemialgebraicRegionPlot

Region`SemialgebraicRegionPlot[ImplicitRegion[(x^2)+(y^2)≤1,{x,y}],{{-1.1,1.1},{-1.1,1.1}}]

Regions

LaminaData[ent,"ImplicitRegion"]

LaminaData[ent,"ParametricRegion"]

LaminaData[ent,"Region"]

Properties

Area

Entity

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

Integrate

Assuming[a>0,FullSimplify[Area[ineq[x,y],{x,y}]]]

Area

Area[reg]

Assuming[assum,Area[implreg]]

Assuming[assum,Area[EntityValue[Entity[ent,ent],"ImplicitRegion"][a]]]

Assuming[assum,Area[EntityValue[Entity[ent,ent],"ParametricRegion"][a]]]

Assuming[assum,Area[EntityValue[Entity[ent,ent],"Region"][a]]]

RegionMeasure

Assuming[a>1,RegionMeasure[ImplicitRegion[x^2+y^2≤a^2,{x,y}]]]//Timing

Assuming[a>1,RegionMeasure[ImplicitRegion[x^2+y^2≤a^2,{x,y}],2]]//Timing

Assuming[assum,RegionMeasure[reg]]

Boundary

LaminaData[ent,"BoundaryCurve"]

LaminaData[ent,"Boundary"]

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

LaminaData[ent,"BoundaryMeshRegion"]

LaminaData[ent,"MeshRegion"]

AreaInertiaTensor

Assuming[a>0,FullSimplify[AreaInertiaTensor[ineq[x,y],{x,y}]]]

Centroid

Package

Assuming[a>0,FullSimplify[Centroid[ineq[x,y],{x,y}]]]

RegionCentroid

RegionCentroid[R]//Timing

RegionCentroid[R,2]//Timing

RegionMeasure[R]

-------- Original Message --------
Subject: Re: RegionMeasure[x^2 + y^2 <= a^2]
Date: Thu, 24 Oct 2013 17:10:17 -0500
From: Adam Strzebonski <adams@wolfram.com>
To: Eric Weisstein <eww@wolfram.com>

The definition of RegionCentroid requires that the dimensions
of the measure and of the region agree, so it cannot take
a dimension argument. RegionCentroid will work for regions that
depend on symbolic parameters when we have code for computing
integrals over such regions. Currently RegionMeasure[R, n] defaults
to Integral[Boole[implicit description of R], vars] when n is equal
to the embedding dimension, and fails for other n.

Adam

Convex

Region`ConvexRegionQ[reg]//Timing

Region`ConvexRegionQ[implreg]//Timing

GeneralizedDiameter

ineqs=And@@ineq@@@{{x1,y1},{x2,y2}}/.a->1

Maximize[{(x1-x2)^2+(y1-y2)^2,ineqs},{x1,x2,y1,y2}]//Timing

RadiiOfGyration

Assuming[a>0,FullSimplify[RadiiOfGyration[ineq[x,y],{x,y}]]]

Regions

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

EntityValue[Entity[ent,ent],"ParametricRegion"][a]

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