Kite

Author

Eric W. Weisstein
July 23, 2014

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

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

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

Definitions

ineq=h (Sqrt[a^2-h^2]+x)+Sqrt[a^2-h^2] y≥0&&Sqrt[(b-h) (b+h)] (h+y)≥h x&&h Sqrt[b^2-h^2]≥h x+Sqrt[b^2-h^2] y&&h (Sqrt[a^2-h^2]+x)≥Sqrt[a^2-h^2] y;

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

reg=With[{x1=-Sqrt[a^2-h^2],x2=Sqrt[b^2-h^2]},Polygon[{{x1,0},{0,-h},{x2,0},{0,h}}]];

assum=0<h<a&&0<h<a;

Construction

Solve[x1^2+h^2==b^2&&x2^2+h^2==a^2,{x1,x2}]

Plots

Diagram

LaminaData["FilledKite","Diagram"]

DiscretizeRegion

Block[{a=1,b=.7,h=.5},Show[DiscretizeRegion[reg],ImageSize->Small]]

RegionPlot

Block[{a=1,b=.7,h=.5},RegionPlot[ineq,{x,-Sqrt[a^2-h^2],Sqrt[a^2-h^2]},{y,-h,h},AspectRatio->Automatic,ImageSize->Small,Frame->False]]

Polygon

poly=Polygon[{{-(Sqrt[(a^2)-(h^2)]),0},{0,-h},{(Sqrt[(b^2)-(h^2)]),0},{0,h}}];

Block[{a=1,b=.7,h=.4},Graphics[poly]]

Equations

Inequality

FullSimplify[Inequalities[poly,{x,y}],0<h<a&&0<h<b]

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

Block[{a=1,b=.7,h=.4},RegionPlot[ineq[x,y],{x,-Sqrt[a^2-h^2],Sqrt[b^2-h^2]},{y,-h,h}]]

Block[{a=1,b=.7,h=.4},RegionPlot[-(( (Sqrt[(b^2)-(h^2)])-x)/(Sqrt[(b^2)-(h^2)]))h<= y<=(( (Sqrt[(b^2)-(h^2)])- x)/(Sqrt[(b^2)-(h^2)]))h,{x,-Sqrt[a^2-h^2],Sqrt[b^2-h^2]},{y,-h,h}]]

Block[{a=1,b=.7,h=.4},RegionPlot[-((h ((Sqrt[(a^2)-(h^2)])+x))/(Sqrt[(a^2)-(h^2)]))<= y<=((h ((Sqrt[(a^2)-(h^2)])+x))/(Sqrt[(a^2)-(h^2)])),{x,-Sqrt[a^2-h^2],Sqrt[b^2-h^2]},{y,-h,h}]]

Properties

Area

Area

Assuming[assum,Simplify@Area[reg]]

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

Integrate

h((Sqrt[(a^2)-(h^2)])+(Sqrt[(b^2)-(h^2)]))

(area=Integrate[Boole[ineq[x,y]],{x,-∞,∞},{y,-∞,∞},Assumptions->0<h<a&&0<h<b])//Timing

...hangs...

(a1=Assuming[0<h<a&&0<h<b,FullSimplify@Integrate[1,{x,-Sqrt[a^2-h^2],0},{y,-((h ((Sqrt[(a^2)-(h^2)])+x))/(Sqrt[(a^2)-(h^2)])),((h ((Sqrt[(a^2)-(h^2)])+x))/(Sqrt[(a^2)-(h^2)]))}]])//Timing

(a2=Assuming[0<h<a&&0<h<b,FullSimplify@Integrate[1,{x,0,Sqrt[b^2-h^2]},{y,-(( (Sqrt[(b^2)-(h^2)])-x)/(Sqrt[(b^2)-(h^2)]))h,(( (Sqrt[(b^2)-(h^2)])- x)/(Sqrt[(b^2)-(h^2)]))h}]])//Timing

FullSimplify[a2+a1]

Numerical Check

Block[{a=1,b=.7,h=.4},NIntegrate[Boole[ineq[x,y]],{x,-Sqrt[a^2-h^2],Sqrt[b^2-h^2]},{y,-h,h}]]

Block[{a=1,b=.7,h=.4},h ((Sqrt[(a^2)-(h^2)])+(Sqrt[(b^2)-(h^2)]))]

Centroid

(centroid=(1/area)Integrate[{x,y}Boole[ineq[x,y]],{x,-∞,∞},{y,-∞,∞},Assumptions->0<h<a&&0<h<b])//Timing

x

(x1=Assuming[0<h<a&&0<h<b,FullSimplify@Integrate[x,{x,-Sqrt[a^2-h^2],0},{y,-((h ((Sqrt[(a^2)-(h^2)])+x))/(Sqrt[(a^2)-(h^2)])),((h ((Sqrt[(a^2)-(h^2)])+x))/(Sqrt[(a^2)-(h^2)]))}]])//Timing

(x2=Assuming[0<h<a&&0<h<b,FullSimplify@Integrate[x,{x,0,Sqrt[b^2-h^2]},{y,-(( (Sqrt[(b^2)-(h^2)])-x)/(Sqrt[(b^2)-(h^2)]))h,(( (Sqrt[(b^2)-(h^2)])- x)/(Sqrt[(b^2)-(h^2)]))h}]])//Timing

With[{A=h ((Sqrt[(a^2)-(h^2)])+(Sqrt[(b^2)-(h^2)]))},FullSimplify[(1/A)(x1+x2)]]

y

(y1=Assuming[0<h<a&&0<h<b,FullSimplify@Integrate[y,{x,-Sqrt[a^2-h^2],0},{y,-((h ((Sqrt[(a^2)-(h^2)])+x))/(Sqrt[(a^2)-(h^2)])),((h ((Sqrt[(a^2)-(h^2)])+x))/(Sqrt[(a^2)-(h^2)]))}]])//Timing

(y2=Assuming[0<h<a&&0<h<b,FullSimplify@Integrate[y,{x,0,Sqrt[b^2-h^2]},{y,-(( (Sqrt[(b^2)-(h^2)])-x)/(Sqrt[(b^2)-(h^2)]))h,(( (Sqrt[(b^2)-(h^2)])- x)/(Sqrt[(b^2)-(h^2)]))h}]])//Timing

With[{A=h ((Sqrt[(a^2)-(h^2)])+(Sqrt[(b^2)-(h^2)]))},FullSimplify[(1/A)(y1+y2)]]

Numerical Check

Block[{a=1,b=.7,h=.4},(1/(h ((Sqrt[(a^2)-(h^2)])+(Sqrt[(b^2)-(h^2)]))))NIntegrate[{x,y}Boole[ineq[x,y]],{x,-Sqrt[a^2-h^2],Sqrt[b^2-h^2]},{y,-h,h}]]//Quiet//Chop

Block[{a=1,b=.7,h=.4},{(1/3) (-(Sqrt[(a^2)-(h^2)])+(Sqrt[(b^2)-(h^2)])),0}]

EdgeLengths

FullSimplify[Norm/@Subtract@@@Partition[poly[[1]],2,1,1],0<h<a&&0<h<b]

GeneralizedDiameter

FullSimplify[Max[a,b,Sqrt[a^2-h^2]+Sqrt[b^2-h^2],2h],0<h<a&&0<h<b]

Incircle

d2[{{x1_,y1_},{x2_,y2_}},{x0_,y0_}]:=(((((x2-x1)(y1-y0)-(x1-x0)(y2-y1)))^2)/((((x2-x1))^2)+(((y2-y1))^2)))

FullSimplify[Solve[d2[{{0,h},{-Sqrt[a^2-h^2],0}},{x,0}]==d2[{{0,h},{Sqrt[b^2-h^2],0}},{x,0}]==r^2,{x,r}],0<h<a&&0<h<b]

Manipulate[Show[{
RegionPlot[h ((Sqrt[(a^2)-(h^2)])+x)+(Sqrt[(a^2)-(h^2)]) y≥0&&(Sqrt[(b^2)-(h^2)]) (h+y)≥h x&&h (Sqrt[(b^2)-(h^2)])≥h x+(Sqrt[(b^2)-(h^2)]) y&&h ((Sqrt[(a^2)-(h^2)])+x)≥(Sqrt[(a^2)-(h^2)]) y,{x,-Sqrt[a^2-h^2],Sqrt[b^2-h^2]},{y,-h,h},AspectRatio->Automatic],
Graphics[Circle[{(((b^2) (Sqrt[(a^2)-(h^2)])+(a^2) (Sqrt[(b^2)-(h^2)])-a b (Sqrt[(a^2)+(b^2)-2 (h^2)+2 (Sqrt[((a^2)-(h^2)) ((b^2)-(h^2))])]))/((a^2)-(b^2))),0},(1/((a+b) Abs[a-b]))h √((a^4)+2 (a^2) ((b^2)-(h^2)+(Sqrt[(a-h) (b-h) (a+h) (b+h)]))+(b^2) ((b^2)-2 (h^2)+2 (Sqrt[(a-h) (b-h) (a+h) (b+h)]))-2 a b ((Sqrt[(a-h) (a+h) ((a^2)+(b^2)-2 (h^2)+2 (Sqrt[(a-h) (b-h) (a+h) (b+h)]))])+(Sqrt[(b-h) (b+h) ((a^2)+(b^2)-2 (h^2)+2 (Sqrt[(a-h) (b-h) (a+h) (b+h)]))])))]]
}],
{{a,1},.1,5},{{b,.5},.1,5},{{h,.1},.1,5}
]

Refine[Circle[{(((b^2) (Sqrt[(a^2)-(h^2)])+(a^2) (Sqrt[(b^2)-(h^2)])-a b (Sqrt[(a^2)+(b^2)-2 (h^2)+2 (Sqrt[((a^2)-(h^2)) ((b^2)-(h^2))])]))/((a^2)-(b^2))),0},(1/((a+b) Abs[a-b]))h √((a^4)+2 (a^2) ((b^2)-(h^2)+(Sqrt[(a-h) (b-h) (a+h) (b+h)]))+(b^2) ((b^2)-2 (h^2)+2 (Sqrt[(a-h) (b-h) (a+h) (b+h)]))-2 a b ((Sqrt[(a-h) (a+h) ((a^2)+(b^2)-2 (h^2)+2 (Sqrt[(a-h) (b-h) (a+h) (b+h)]))])+(Sqrt[(b-h) (b+h) ((a^2)+(b^2)-2 (h^2)+2 (Sqrt[(a-h) (b-h) (a+h) (b+h)]))])))]/.{(a-h) (a+h) ->r,((a^2)-(h^2)) ->(r^2),(b-h) (b+h)->s,((b^2)-(h^2))->(s^2),(Sqrt[((a-h) (b-h) (a+h) (b+h))])->(Sqrt[r s])},r>0&&s>0]

Circle[{(((b^2) r+(a^2) s-a b (Sqrt[(a^2)+(b^2)-2 (h^2)+2 r s]))/((a^2)-(b^2))),0},(1/((a+b) Abs[a-b]))h √((a^4)-2 a b ((Sqrt[(a^2)+(b^2)-2 (h^2)+2 (Sqrt[r s])]) ((Sqrt[r])+ (Sqrt[s])))+2 (a^2) ((Sqrt[r s])+(s^2))+(b^2) ((b^2)-2 (h^2)+2 (Sqrt[r s])))]//TraditionalForm

InertiaTensor

AreaInertiaTensor[{x,y}]

(i=Integrate[AreaInertiaTensor[{x,y}]Boole[ineq[x,y]],{x,-∞,∞},{y,-∞,∞},Assumptions->0<h<a&&0<h<b])//Timing

yy

(yy1=Assuming[0<h<a&&0<h<b,FullSimplify@Integrate[y^2,{x,-Sqrt[a^2-h^2],0},{y,-((h ((Sqrt[(a^2)-(h^2)])+x))/(Sqrt[(a^2)-(h^2)])),((h ((Sqrt[(a^2)-(h^2)])+x))/(Sqrt[(a^2)-(h^2)]))}]])//Timing

(yy2=Assuming[0<h<a&&0<h<b,FullSimplify@Integrate[y^2,{x,0,Sqrt[b^2-h^2]},{y,-(( (Sqrt[(b^2)-(h^2)])-x)/(Sqrt[(b^2)-(h^2)]))h,(( (Sqrt[(b^2)-(h^2)])- x)/(Sqrt[(b^2)-(h^2)]))h}]])//Timing

yy=FullSimplify[(yy1+yy2)]

x y

(xy1=Assuming[0<h<a&&0<h<b,FullSimplify@Integrate[-x y,{x,-Sqrt[a^2-h^2],0},{y,-((h ((Sqrt[(a^2)-(h^2)])+x))/(Sqrt[(a^2)-(h^2)])),((h ((Sqrt[(a^2)-(h^2)])+x))/(Sqrt[(a^2)-(h^2)]))}]])//Timing

(xy2=Assuming[0<h<a&&0<h<b,FullSimplify@Integrate[-x y,{x,0,Sqrt[b^2-h^2]},{y,-(( (Sqrt[(b^2)-(h^2)])-x)/(Sqrt[(b^2)-(h^2)]))h,(( (Sqrt[(b^2)-(h^2)])- x)/(Sqrt[(b^2)-(h^2)]))h}]])//Timing

xy=FullSimplify[(xy1+xy2)]

x x

(xx1=Assuming[0<h<a&&0<h<b,FullSimplify@Integrate[x^2,{x,-Sqrt[a^2-h^2],0},{y,-((h ((Sqrt[(a^2)-(h^2)])+x))/(Sqrt[(a^2)-(h^2)])),((h ((Sqrt[(a^2)-(h^2)])+x))/(Sqrt[(a^2)-(h^2)]))}]])//Timing

(xx2=Assuming[0<h<a&&0<h<b,FullSimplify@Integrate[x^2,{x,0,Sqrt[b^2-h^2]},{y,-(( (Sqrt[(b^2)-(h^2)])-x)/(Sqrt[(b^2)-(h^2)]))h,(( (Sqrt[(b^2)-(h^2)])- x)/(Sqrt[(b^2)-(h^2)]))h}]])//Timing

xx=FullSimplify[(xx1+xx2)]

Numerical Check

Block[{a=1,b=.7,h=.4},NIntegrate[{{(y^2),-x y},{-x y,(x^2)}}Boole[ineq[x,y]],{x,-Sqrt[a^2-h^2],Sqrt[b^2-h^2]},{y,-h,h}]]//Quiet//Chop

Block[{a=1,b=.7,h=.4},{{yy,xy},{xy,xx}}]

{{yy,xy},{xy,xx}}

{{(1/6) (h^3) ((Sqrt[(a^2)-(h^2)])+(Sqrt[(b^2)-(h^2)])),0},{0,(1/6) h (((((a^2)-(h^2)))^(3/2))+((((b^2)-(h^2)))^(3/2)))}}