Torispherical Dome

Author

Eric W. Weisstein
August 17, 2010

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SolidGeometry/TorisphericalDome.nb.

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

©2010 Wolfram Research, Inc. except for portions noted otherwise

Plot

V5

<<Graphics`InequalityGraphics`

With[{a=1,c=2},InequalityPlot3D[(((c-(Sqrt[(x^2)+(y^2)])))^2)<(a^2)-(((z-1))^2),{x},{y},{z},PlotPoints->20]]

FullSimplify[Solve[c^2+(R-h)^2==(R-a)^2,h]]

V6

With[{a=1,c=2},RegionPlot3D[(((c-(Sqrt[(x^2)+(y^2)])))^2)<(a^2)-(((z-1))^2),{x,-3,3},{y,-3,3},{z,0,2},BoxRatios->Automatic]]

Computation

V5

With[{R=1,c=.6,a=.3},
With[{h=R-(Sqrt[(a-c-R) (a+c-R)])},
InequalityPlot3D[((x^2)+(y^2)<(((c(1+(a/(R-a)))))^2)&&x^2+y^2+z^2<R^2||((x^2)+(y^2)>(((c(1+(a/(R-a)))))^2)&&(((c-(Sqrt[(x^2)+(y^2)])))^2)<(a^2)-(((z-(R-h)))^2)))&&z>R-h,{x},{y},{z},Boxed->False,PlotPoints->{20,10},PlotRange->All,Axes->False,ViewPoint->{2,2,2}]
]]

V6

With[{R=1,c=.6,a=.3},
With[{h=R-(Sqrt[(a-c-R) (a+c-R)])},
RegionPlot3D[((x^2)+(y^2)<(((c(1+(a/(R-a)))))^2)&&x^2+y^2+z^2<R^2||((x^2)+(y^2)>(((c(1+(a/(R-a)))))^2)&&(((c-(Sqrt[(x^2)+(y^2)])))^2)<(a^2)-(((z-(R-h)))^2)))&&z>R-h,{x,-1,1},{y,-1,1},{z,.4,1.1},Boxed->True,BoxRatios->Automatic,Axes->True,PlotPoints->20]
]]

With[{R=1,c=.2,a=.4},
With[{h=R-(Sqrt[(a-c-R) (a+c-R)])},
RegionPlot3D[((x^2)+(y^2)<(((c(1+(a/(R-a)))))^2)&&x^2+y^2+z^2<R^2||((x^2)+(y^2)>(((c(1+(a/(R-a)))))^2)&&(((c-(Sqrt[(x^2)+(y^2)])))^2)<(a^2)-(((z-(R-h)))^2)))&&z>R-h,{x,-.6,.6},{y,-.6,.6},{z,.4,1.1},Boxed->True,BoxRatios->Automatic,Axes->True,PlotPoints->30]
]]

With[{R=1,c=.5,a=.1},
With[{h=R-(Sqrt[(a-c-R) (a+c-R)])},
RegionPlot3D[((x^2)+(y^2)<(((c(1+(a/(R-a)))))^2)&&x^2+y^2+z^2<R^2||((x^2)+(y^2)>(((c(1+(a/(R-a)))))^2)&&(((c-(Sqrt[(x^2)+(y^2)])))^2)<(a^2)-(((z-(R-h)))^2)))&&z>R-h,{x,-.6,.6},{y,-.6,.6},{z,.4,1.1},Boxed->True,BoxRatios->Automatic,Axes->True,PlotPoints->30]
]]

Volume

c(1+(1/((R/a)-1)))==c(1+((((R/a)-1))^(-1)))

Compute as a surface of revolution

R-(R-(Sqrt[(a-c-R) (a+c-R)]))

With[{h=R-(Sqrt[(a-c-R) (a+c-R)])},Solve[(((c-r))^2)==a^2,r]
]//Simplify

V5.2

(v=With[{h=R-(Sqrt[(a-c-R) (a+c-R)])},
Integrate[Boole[((x^2)+(y^2)<(((c(1+(a/(R-a)))))^2)&&x^2+y^2+z^2<R^2||((x^2)+(y^2)>(((c(1+(a/(R-a)))))^2)&&(((c-(Sqrt[(x^2)+(y^2)])))^2)<(a^2)-(((z-(R-h)))^2)))],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a+c<R&&0<a<c]
])//Timing

V6

(v=With[{h=R-(Sqrt[(a-c-R) (a+c-R)])},
Integrate[Boole[((x^2)+(y^2)<(((c(1+(a/(R-a)))))^2)&&x^2+y^2+z^2<R^2||((x^2)+(y^2)>(((c(1+(a/(R-a)))))^2)&&(((c-(Sqrt[(x^2)+(y^2)])))^2)<(a^2)-(((z-(R-h)))^2)))],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a+c<R&&0<a<c]
])//Timing

V8

$Version

(v=Integrate[Boole[((x^2)+(y^2)<(((c(1+(a/(R-a)))))^2)&&x^2+y^2+z^2<R^2||((x^2)+(y^2)>(((c(1+(a/(R-a)))))^2)&&(((c-(Sqrt[(x^2)+(y^2)])))^2)<(a^2)-(((z-(R-h)))^2)))],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a+c<R&&0<a<c&&0<h<R])//Timing