Lozenge

Author

Eric W. Weisstein
July 30, 2014

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

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

©2014 Wolfram Research, Inc. except for portions noted otherwise

Definitions

ineq=y≥0&&a+y≥x&&Sqrt[2] a≥2 y&&x≥y;

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

reg=Polygon[a{{0,0},{1,0},{1+1/Sqrt[2],1/Sqrt[2]},{1/Sqrt[2],1/Sqrt[2]}}];

reg2=Parallelogram[{0,0},a{{1,0},{(1/(Sqrt[2])),(1/(Sqrt[2]))}}];

assum=a>0;

Plots

Diagram

LaminaData["FilledLozenge","Diagram"]

DiscretizeRegion

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

Block[{a=1},Show[DiscretizeRegion[reg2//N],ImageSize->Small,Axes->True]]

Block[{a=1},Show[DiscretizeRegion[implreg],ImageSize->Small,Axes->True]]

V10.0.0

Block[{a=1},Show[DiscretizeRegion[reg2],ImageSize->Small,Axes->True]]

V10.0.1

Block[{a=1},Show[DiscretizeRegion[reg2],ImageSize->Small,Axes->True]]

RegionPlot

Block[{a=1},RegionPlot[ineq,{x,0,1+1/(Sqrt[2])},{y,0,1/(Sqrt[2])},AspectRatio->Automatic,ImageSize->Small,Frame->False]]

Polygon

Through[{Cos,Sin}[45°]]

p=Polygon[a{{0,0},{1,0},{1+1/(Sqrt[2]),1/(Sqrt[2])},{1/(Sqrt[2]),1/(Sqrt[2])}}];

Graphics[Block[{a=1},p]]

Equations

Inequalities

FullSimplify[Inequalities[p,{x,y}],a>0]

ineq[x_,y_]:=y≥0&&a+y≥x&&(Sqrt[2]) a≥2 y&&x≥y

Block[{a=1},RegionPlot[y≥0&&a+y≥x&&(Sqrt[2]) a≥2 y&&x≥y,{x,0,1+1/(Sqrt[2])},{y,0,1/(Sqrt[2])},AspectRatio->Automatic]]

Propeties

Area

Area

Assuming[assum,Simplify@Area[reg]]

Assuming[assum,Simplify@Area[reg2]]

Assuming[assum,Simplify@Area[implreg]]

Integrate

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

RegionMeasure

Assuming[assum,RegionMeasure[reg]]

Assuming[assum,RegionMeasure[reg/.a->1]]

Assuming[assum,Simplify@RegionMeasure[reg2]]

Assuming[assum,Simplify@RegionMeasure[implreg]]

Centroid

Integrate

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

RegionCentroid

Assuming[assum,RegionCentroid[reg]]

Assuming[assum,Simplify@RegionCentroid[reg2]]

Assuming[assum,Simplify@RegionCentroid[implreg]]

Diagonals

a Norm/@{{1+1/(Sqrt[2]),1/(Sqrt[2])},{1,0}-{1/(Sqrt[2]),1/(Sqrt[2])}}//FullSimplify

Inradius

FullSimplify[((p q)/(2(Sqrt[(p^2)+(q^2)])))/.Thread[{p,q}->{(Sqrt[2+(Sqrt[2])]) a,(Sqrt[2-(Sqrt[2])]) a}],a>0]

InertiaTensor

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

RadiiOfGyration

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