Sphere

Author

Eric W. Weisstein
September 17, 2014

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

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

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

Graphics

Single Sphere

V5.2

<<Graphics`Shapes`

Show[Graphics3D[Sphere[]],Boxed->False]

V6

Graphics3D[Sphere[],Boxed->False,Method->{"ShrinkWrap"->True}]

Graphics3D[Sphere[],Boxed->False,ImageSize->Small]

LiveGraphics3D

<<Utilities`LiveGraphics3D`

WriteLiveForm["sphere.m",Graphics3D[Sphere[],Boxed->False]]

Multiple Spheres

V5

Show[Graphics3D[{
Sphere[{0,0,0},1],
Sphere[{1.5,0,0},1]
}]]

Show[Graphics3D[Table[Sphere[Table[Random[],{3}],Random[]],{5}]]]

V6

Graphics3D[{Opacity[.8],Sphere[{0,0,0},1],Sphere[{1.5,0,0},1]}]

Graphics3D[{Opacity[.8],Table[Sphere[Table[Random[],{3}],Random[]],{5}]}]

Properties

Normal Vector

Head[x^2+y^2==z^2]

Parametric

FullSimplify[NormalVector[a{Cos[u] Sin[v],Sin[u] Sin[v],Cos[v]},{u,v}],a>0&&0<u<2π&&0<v<π]

Manipulate[
Graphics3D[{
{Opacity[.8],Sphere[{0,0,0}]},
Thickness[.02],PointSize[.02],
Point[{Cos[u] Sin[v],Sin[u] Sin[v],Cos[v]}],
Arrow[{
{Cos[u] Sin[v],Sin[u] Sin[v],Cos[v]},
{Cos[u] Sin[v],Sin[u] Sin[v],Cos[v]}+{Cos[u] Sin[v],Sin[u] Sin[v],Cos[v]}
}]
}],
{{u,0},0,2π},
{{v,0},0,π}
]

Implicit

Clear[NormalVector]

FullSimplify[NormalVector[x^2+y^2+z^2==a^2,{x,y,z}],a>0&&(x|y|z)∈Reals]

Manipulate[
With[{z=Sqrt[1-x^2-y^2]},
Graphics3D[{
{Opacity[.8],Sphere[{0,0,0}]},
Thickness[.02],PointSize[.02],
Point[{(x/(Sqrt[(x^2)+(y^2)+(z^2)])),(y/(Sqrt[(x^2)+(y^2)+(z^2)])),(z/(Sqrt[(x^2)+(y^2)+(z^2)]))}],
Arrow[{
{x,y,z},
{x,y,z}+.2{(x/(Sqrt[(x^2)+(y^2)+(z^2)])),(y/(Sqrt[(x^2)+(y^2)+(z^2)])),(z/(Sqrt[(x^2)+(y^2)+(z^2)]))}
}]
}]],
{{x,1},-1,1},
{{y,0},-1,1}
]

Vector Length

FullSimplify[Norm[a{Cos[u] Sin[v],Sin[u] Sin[v],Cos[v]}],a>0&&0<u<2π&&0<v<π]

Metric Properties

Metric Tensor

MetricTensor[a{Cos[u] Sin[v],Sin[u] Sin[v],Cos[v]},{u,v}]//FullSimplify

Christoffel Symbol of the Second Kind

?ChristoffelSymbols

?ChristoffelSymbols2

FullSimplify[ChristoffelSymbols2[a{Cos[u]Sin[v],Sin[u]Sin[v],Cos[v]},{a,u,v}],a>0]

FullSimplify[ChristoffelSymbols[a{Cos[u]Sin[v],Sin[u]Sin[v],Cos[v]},{a,u,v}],a>0]

TraditionalForm[%]

FullSimplify[ChristoffelSymbols[a{Cos[u]Sin[v],Sin[u]Sin[v],Cos[v]},{u,v}],a>0]

TraditionalForm[%]

FullSimplify[ChristoffelSymbols2[a{Cos[u]Sin[v],Sin[u]Sin[v],Cos[v]},{u,v}],a>0]

TraditionalForm[%]

Fundamental Forms

Gray's Convention

Curvatures[a{Cos[v]Cos[u],Cos[v]Sin[u],Sin[v]},{u,v}]

Eric's Convention

eqn=x^2+y^2+z^2-a^2;
sphere=a{Cos[u]Sin[v],Sin[u]Sin[v],Cos[v]};

Curvatures[a{Cos[u]Sin[v],Sin[u]Sin[v],Cos[v]},{u,v},Simplify[#,a>0&&0<v<π]&]

{ℰ,ℱ,𝒢}=Simplify[FirstFundamentalForm[sphere,{u,v}]]

GaussianCurvatue

GaussianCurvature[eqn,{x,y,z}]

Mean Curvature

Simplify[MeanCurvature[eqn,{x,y,z}]]

Simplify[MeanCurvature[eqn,{x,y,z}],z<0]

Simplify[MeanCurvature[eqn,{x,y,z}],a>0&&z∈Reals]

Principal Curvature

FullSimplify[PrincipalCurvatures[sphere,{u,v}],a>0&&0<v<2π]

PiecewiseExpand[%,0<v<π]//FullSimplify[#,0<v<π]&

Surface Area

dA

Integrate[a^2 Sin[v],{u,0,2Pi},{v,0,Pi}]

First Fundamental Form

Simplify[(Sqrt[ℰ 𝒢-(ℱ^2)]),a>0&&0<v<π]

Integrate[(a^2) Sin[v],{u,0,2Pi},{v,0,Pi}]