Bicone

Author

Eric W. Weisstein
February 3, 2022

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Surfaces/Bicone.nb.

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

©2022 Wolfram Research, Inc. except for portions noted otherwise

Graphic

RegionPlot3D Semialgebraic

With[{h=2,r=1},
RegionPlot3D[((x^2+y^2)/(((r/h))^2))<(((Abs[z]-h))^2)&&-h<z<h,{x,-r,r},{y,-r,r},{z,-h,h}]
]

RegionPlot3D algebraic

With[{h=2,r=1,zz=5},
RegionPlot3D[((x^2+y^2)/(((r/h))^2))<(((Abs[z]-h))^2),{x,-r,r},{y,-r,r},{z,-zz,zz},MaxRecursion->4,PlotPoints->40]
]

RevolutionPlot

V5

bicone=SurfaceOfRevolution[{1-Abs[z],z},{z,-1,1},
PlotPoints->{15,25},Boxed->False,Axes->False,opts]

V6

RevolutionPlot3D[{1-Abs[z],z},{z,-1,1},Boxed->False,Axes->False]

Slices

<<Hidekazu`knife`

WriteLiveForm["biconesl.m",
Show[Graphics3D[{
knife[bicone,{1,0,0,0},{0,0,1},{1,.5,0}][[1]],
Polygon[{{0,-1,0},{0,0,1},{0,1,0},{0,0,-1}}],
Polygon[{1,.5,0}+{{Cos[1],-Sin[1],0},{Sin[1],Cos[1],0},{0,0,1}}.#&/@Reverse[{{0,-1,0},{0,0,1},{0,1,0},{0,0,-1}}]]
}],opts]]

Volume

Integrate[Boole[((x^2+y^2)/(((r/h))^2))<(((Abs[z]-h))^2)&&-h<z<h],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->r>0&&h>0]

Centroid

Integrate[{x,y,z}Boole[((x^2+y^2)/(((r/h))^2))<(((Abs[z]-h))^2)&&-h<z<h],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->r>0&&h>0]

Inertia Tensor

With[{ρ=(M/((2/3) h π (r^2)))},
ρ Integrate[{{y^2+z^2,-x y,-x z},
{-x y,z^2+x^2,-y z},
{-x z,-y z,x^2+y^2}}Boole[((x^2+y^2)/(((r/h))^2))<(((Abs[z]-h))^2)&&-h<z<h],
{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->r>0&&h>0]
]//Simplify//TraditionalForm