Hyperbolic Octahedron

Author

Eric W. Weisstein
June 10, 2014

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

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

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

Initialization

HyperbolicOctahedron[u_,v_]:=a{(Cos[u]Cos[v])^3,(Sin[u]Cos[v])^3,(Sin[v])^3}

Graphic

ParametricPlot

V5.2

ParametricPlot3D[Evaluate[{(a Cos[u]Cos[v])^3,(b Sin[u]Cos[v])^3,(c Sin[v])^3}/.{a->1,b->1,c->1}],{u,-Pi/2,Pi/2},{v,-Pi,Pi},PlotRange->All,Boxed->False,Axes->False]

V6

ParametricPlot3D[Evaluate[{(a Cos[u]Cos[v])^3,(b Sin[u]Cos[v])^3,(c Sin[v])^3}/.{a->1,b->1,c->1}],{u,-Pi/2,Pi/2},{v,-Pi,Pi},PlotRange->All,Boxed->False,Axes->False]

Trott

Graphics3D[HyperbolicPolyhedron["Octahedron",.5],Boxed->False]

Rivin

Cartesian Equation

eqns={(Cos[u]Cos[v])^3,(Sin[u]Cos[v])^3,(Sin[v])^3};

{eqn}=Factor/@GroebnerBasis[Join[Thread[{x,y,z}==eqns],Cos[#]^2+Sin[#]^2==1&/@{u,v}],{x,y,z},{Cos[u],Sin[u],Cos[v],Sin[v]},MonomialOrder->EliminationOrder]

eqn=-1+9 (x^2)-36 (x^4)+84 (x^6)-126 (x^8)+126 (x^10)-84 (x^12)+36 (x^14)-9 (x^16)+(x^18)+9 (y^2)+9 (x^2) (y^2)-234 (x^4) (y^2)+711 (x^6) (y^2)-990 (x^8) (y^2)+711 (x^10) (y^2)-234 (x^12) (y^2)+9 (x^14) (y^2)+9 (x^16) (y^2)-36 (y^4)-234 (x^2) (y^4)-513 (x^4) (y^4)+2961 (x^6) (y^4)-2961 (x^8) (y^4)+513 (x^10) (y^4)+234 (x^12) (y^4)+36 (x^14) (y^4)+84 (y^6)+711 (x^2) (y^6)+2961 (x^4) (y^6)+12171 (x^6) (y^6)+2961 (x^8) (y^6)+711 (x^10) (y^6)+84 (x^12) (y^6)-126 (y^8)-990 (x^2) (y^8)-2961 (x^4) (y^8)+2961 (x^6) (y^8)+990 (x^8) (y^8)+126 (x^10) (y^8)+126 (y^10)+711 (x^2) (y^10)+513 (x^4) (y^10)+711 (x^6) (y^10)+126 (x^8) (y^10)-84 (y^12)-234 (x^2) (y^12)+234 (x^4) (y^12)+84 (x^6) (y^12)+36 (y^14)+9 (x^2) (y^14)+36 (x^4) (y^14)-9 (y^16)+9 (x^2) (y^16)+(y^18)+9 (z^2)+9 (x^2) (z^2)-234 (x^4) (z^2)+711 (x^6) (z^2)-990 (x^8) (z^2)+711 (x^10) (z^2)-234 (x^12) (z^2)+9 (x^14) (z^2)+9 (x^16) (z^2)+9 (y^2) (z^2)+3339 (x^2) (y^2) (z^2)-1512 (x^4) (y^2) (z^2)-12222 (x^6) (y^2) (z^2)+12222 (x^8) (y^2) (z^2)+1512 (x^10) (y^2) (z^2)-3339 (x^12) (y^2) (z^2)-9 (x^14) (y^2) (z^2)-234 (y^4) (z^2)-1512 (x^2) (y^4) (z^2)+27837 (x^4) (y^4) (z^2)+6867 (x^6) (y^4) (z^2)+27837 (x^8) (y^4) (z^2)-1512 (x^10) (y^4) (z^2)-234 (x^12) (y^4) (z^2)+711 (y^6) (z^2)-12222 (x^2) (y^6) (z^2)+6867 (x^4) (y^6) (z^2)-6867 (x^6) (y^6) (z^2)+12222 (x^8) (y^6) (z^2)-711 (x^10) (y^6) (z^2)-990 (y^8) (z^2)+12222 (x^2) (y^8) (z^2)+27837 (x^4) (y^8) (z^2)+12222 (x^6) (y^8) (z^2)-990 (x^8) (y^8) (z^2)+711 (y^10) (z^2)+1512 (x^2) (y^10) (z^2)-1512 (x^4) (y^10) (z^2)-711 (x^6) (y^10) (z^2)-234 (y^12) (z^2)-3339 (x^2) (y^12) (z^2)-234 (x^4) (y^12) (z^2)+9 (y^14) (z^2)-9 (x^2) (y^14) (z^2)+9 (y^16) (z^2)-36 (z^4)-234 (x^2) (z^4)-513 (x^4) (z^4)+2961 (x^6) (z^4)-2961 (x^8) (z^4)+513 (x^10) (z^4)+234 (x^12) (z^4)+36 (x^14) (z^4)-234 (y^2) (z^4)-1512 (x^2) (y^2) (z^4)+27837 (x^4) (y^2) (z^4)+6867 (x^6) (y^2) (z^4)+27837 (x^8) (y^2) (z^4)-1512 (x^10) (y^2) (z^4)-234 (x^12) (y^2) (z^4)-513 (y^4) (z^4)+27837 (x^2) (y^4) (z^4)-120312 (x^4) (y^4) (z^4)+120312 (x^6) (y^4) (z^4)-27837 (x^8) (y^4) (z^4)+513 (x^10) (y^4) (z^4)+2961 (y^6) (z^4)+6867 (x^2) (y^6) (z^4)+120312 (x^4) (y^6) (z^4)+6867 (x^6) (y^6) (z^4)+2961 (x^8) (y^6) (z^4)-2961 (y^8) (z^4)+27837 (x^2) (y^8) (z^4)-27837 (x^4) (y^8) (z^4)+2961 (x^6) (y^8) (z^4)+513 (y^10) (z^4)-1512 (x^2) (y^10) (z^4)+513 (x^4) (y^10) (z^4)+234 (y^12) (z^4)-234 (x^2) (y^12) (z^4)+36 (y^14) (z^4)+84 (z^6)+711 (x^2) (z^6)+2961 (x^4) (z^6)+12171 (x^6) (z^6)+2961 (x^8) (z^6)+711 (x^10) (z^6)+84 (x^12) (z^6)+711 (y^2) (z^6)-12222 (x^2) (y^2) (z^6)+6867 (x^4) (y^2) (z^6)-6867 (x^6) (y^2) (z^6)+12222 (x^8) (y^2) (z^6)-711 (x^10) (y^2) (z^6)+2961 (y^4) (z^6)+6867 (x^2) (y^4) (z^6)+120312 (x^4) (y^4) (z^6)+6867 (x^6) (y^4) (z^6)+2961 (x^8) (y^4) (z^6)+12171 (y^6) (z^6)-6867 (x^2) (y^6) (z^6)+6867 (x^4) (y^6) (z^6)-12171 (x^6) (y^6) (z^6)+2961 (y^8) (z^6)+12222 (x^2) (y^8) (z^6)+2961 (x^4) (y^8) (z^6)+711 (y^10) (z^6)-711 (x^2) (y^10) (z^6)+84 (y^12) (z^6)-126 (z^8)-990 (x^2) (z^8)-2961 (x^4) (z^8)+2961 (x^6) (z^8)+990 (x^8) (z^8)+126 (x^10) (z^8)-990 (y^2) (z^8)+12222 (x^2) (y^2) (z^8)+27837 (x^4) (y^2) (z^8)+12222 (x^6) (y^2) (z^8)-990 (x^8) (y^2) (z^8)-2961 (y^4) (z^8)+27837 (x^2) (y^4) (z^8)-27837 (x^4) (y^4) (z^8)+2961 (x^6) (y^4) (z^8)+2961 (y^6) (z^8)+12222 (x^2) (y^6) (z^8)+2961 (x^4) (y^6) (z^8)+990 (y^8) (z^8)-990 (x^2) (y^8) (z^8)+126 (y^10) (z^8)+126 (z^10)+711 (x^2) (z^10)+513 (x^4) (z^10)+711 (x^6) (z^10)+126 (x^8) (z^10)+711 (y^2) (z^10)+1512 (x^2) (y^2) (z^10)-1512 (x^4) (y^2) (z^10)-711 (x^6) (y^2) (z^10)+513 (y^4) (z^10)-1512 (x^2) (y^4) (z^10)+513 (x^4) (y^4) (z^10)+711 (y^6) (z^10)-711 (x^2) (y^6) (z^10)+126 (y^8) (z^10)-84 (z^12)-234 (x^2) (z^12)+234 (x^4) (z^12)+84 (x^6) (z^12)-234 (y^2) (z^12)-3339 (x^2) (y^2) (z^12)-234 (x^4) (y^2) (z^12)+234 (y^4) (z^12)-234 (x^2) (y^4) (z^12)+84 (y^6) (z^12)+36 (z^14)+9 (x^2) (z^14)+36 (x^4) (z^14)+9 (y^2) (z^14)-9 (x^2) (y^2) (z^14)+36 (y^4) (z^14)-9 (z^16)+9 (x^2) (z^16)+9 (y^2) (z^16)+(z^18);

V5.2

<<Graphics`ContourPlot3D`

ContourPlot3D[eqn,{x,-1,1},{y,-1,1},{z,-1,1},PlotPoints->7,Boxed->False]//Timing

ContourPlot3D[eqn==0,{x,-1,1},{y,-1,1},{z,-1,1},Boxed->False,Axes->None]

ContourPlot3D[((((x^2)))^(1/3))+((((y^2)))^(1/3))+((((z^2)))^(1/3))==1,{x,-1,1},{y,-1,1},{z,-1,1},Boxed->False,Axes->None]

RegionPlot3D[((((x^2)))^(1/3))+((((y^2)))^(1/3))+((((z^2)))^(1/3))<1,{x,-1,1},{y,-1,1},{z,-1,1},Boxed->False,Axes->None]

Bounding Box

ContourPlot3D[Evaluate[eqn==0],{x,-1,1},{y,-1,1},{z,-1,1}]

ContourPlot3D[Evaluate[eqn==0],{x,-2,2},{y,-2,2},{z,-2,2}]

RegionPlot3D[Evaluate[eqn<0],{x,-1,1},{y,-1,1},{z,-1,1}]

RegionPlot3D[Evaluate[eqn<0],{x,-1.5,1.5},{y,-1.5,1.5},{z,-1.5,1.5},MaxRecursion->6]

Maximize[{x,((((x^2)))^(1/3))+((((y^2)))^(1/3))+((((z^2)))^(1/3))==1},{x,y,z}]//Timing

...correct...

Maximize[{x,Evaluate[eqn==0]&&-1≤x≤1&&-1≤y≤1&&-1≤z≤1},{x,y,z},Reals]//Timing

Maximize[{x,Evaluate[eqn==0]},{x,y,z},Reals]

no constraints have been imposed that u,v are real-valued, so Cos and Sin are free to take on values outside [-1,1]

Fundamental Forms

FullSimplify[FirstFundamentalForm[HyperbolicOctahedron[u,v],{u,v}],-π/2≤u≤π/2&&-π≤v≤π]

%//ColumnForm//TraditionalForm

FullSimplify[SecondFundamentalForm[HyperbolicOctahedron[u,v],{u,v}],-π/2≤u≤π/2&&-π≤v≤π&&a>0]

%//ColumnForm//TraditionalForm

Curvatures

Gaussian

FullSimplify[GaussianCurvature[HyperbolicOctahedron[u,v],{u,v}],-π/2≤u≤π/2&&-π≤v≤π]

Mean

FullSimplify[MeanCurvature[HyperbolicOctahedron[u,v],{u,v}],-π/2≤u≤π/2&&-π≤v≤π,Assumptions->a>0]

TraditionalForm[%]

Principal

PrincipalCurvatures[HyperbolicOctahedron[u,v],{u,v}]

FullSimplify[%,a>0&&-π/2≤u≤π/2&&-π≤v≤π]

Surface Area

FullSimplify[AreaElement[HyperbolicOctahedron[u,v],{u,v}],-π/2≤u≤π/2&&-π≤v≤π]//Timing

Numeric

NIntegrate[(9/4) Abs[Sin[u]] Abs[Sin[v]] Cos[u] ((Cos[v])^4) (Sqrt[9-2 Cos[4 u] ((Cos[v])^2)-7 Cos[2 v]]),{u,-π/2,π/2},{v,-π,π}]//Timing

N[17π/12]

v5.2

Integrate[(9/4) Abs[Sin[u]] Abs[Sin[v]] Cos[u] ((Cos[v])^4) (Sqrt[9-2 Cos[4 u] ((Cos[v])^2)-7 Cos[2 v]]),{u,-π/2,π/2},{v,-π,π}]//Timing

N[((17 π)/12)]

Integrate[(9/4) Abs[Sin[u]] Abs[Sin[v]] Cos[u] ((Cos[v])^4) (Sqrt[9-2 Cos[4 u] ((Cos[v])^2)-7 Cos[2 v]]),{v,-π,π},{u,-π/2,π/2}]//Timing

Integrate[(9/4) Abs[Sin[u]] Abs[Sin[v]] Cos[u] ((Cos[v])^4) (Sqrt[9-2 Cos[4 u] ((Cos[v])^2)-7 Cos[2 v]]),{v,-π,π},{u,-π/2,π/2},GenerateConditions->True]//Timing

...hang...

v6

Integrate[(9/4) Abs[Sin[u]] Abs[Sin[v]] Cos[u] ((Cos[v])^4) (Sqrt[9-2 Cos[4 u] ((Cos[v])^2)-7 Cos[2 v]]),{u,-π/2,π/2},{v,-π,π}]//Timing

Integrate[(9/4) Abs[Sin[u]] Abs[Sin[v]] Cos[u] ((Cos[v])^4) (Sqrt[9-2 Cos[4 u] ((Cos[v])^2)-7 Cos[2 v]]),{v,-π,π},{u,-π/2,π/2}]//Timing

Volume

0.359038

Astroid (2D)

4Integrate[Boole[(x^(2/3))+(y^(2/3))<1],{x,0,1},{y,0,1}]

Expand[(1-(y^(2/3)))^3]

Numeric

v5.2

NIntegrate[Boole[eqn<0],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]//Timing

...hangs...

NIntegrate[Boole[eqn<0],{x,-1,1},{y,-1,1},{z,-1,1}]//Timing

...hangs...

NIntegrate[Boole[eqn<0],{x,-1,1},{y,-1,1},{z,-1,1},Method->QuasiMonteCarlo,PrecisionGoal->3,MaxPoints->10^6]//Timing

8NIntegrate[Boole[eqn<0],{x,0,1},{y,0,1},{z,0,1},Method->QuasiMonteCarlo,PrecisionGoal->3,MaxPoints->10^6]//Timing

NIntegrate[Boole[eqn<0],{x,-1,1},{y,-1,1},{z,-1,1},Method->QuasiMonteCarlo,PrecisionGoal->3,MaxPoints->10^7]//Timing

v6

NIntegrate[Boole[eqn<0],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]//Timing

...hangs...

NIntegrate[Boole[eqn<0],{x,-1,1},{y,-1,1},{z,-1,1}]//Timing

...hangs...

NIntegrate[Boole[eqn<0],{x,-1,1},{y,-1,1},{z,-1,1},Method->QuasiMonteCarlo,PrecisionGoal->3,MaxPoints->10^6]//Timing

...hangs...

NIntegrate[Boole[eqn<0],{x,-1,1},{y,-1,1},{z,-1,1},Method->{"SymbolicPiecewiseSubdivision",Method->QuasiMonteCarlo,"ExpandSpecialPiecewise"->{Boole}}]//Timing

NIntegrate[Boole[eqn<0],{x,-1,1},{y,-1,1},{z,-1,1},SymbolicPreprocessing->None]//Timing

NIntegrate[Boole[eqn<0],{x,-1,1},{y,-1,1},{z,-1,1},SymbolicPreprocessing->None,Method->AdaptiveQuasiMonteCarlo]//Timing

V6.1

NIntegrate[Boole[eqn<0],{x,-1,1},{y,-1,1},{z,-1,1}]//Timing

NIntegrate[Boole[eqn<0],{x,-1,1},{y,-1,1},{z,-1,1},Method->"AdaptiveQuasiMonteCarlo",PrecisionGoal->2]//Timing

8NIntegrate[Boole[eqn<0],{x,0,1},{y,0,1},{z,0,1},Method->"AdaptiveQuasiMonteCarlo",PrecisionGoal->3]//Timing

8NIntegrate[Boole[(x^(2/3))+(y^(2/3))+(z^(2/3))<1],{x,0,1},{y,0,1},{z,0,1},Method->"AdaptiveQuasiMonteCarlo",PrecisionGoal->3,MaxPoints->10^7]//Timing

Numeric Due to Michael Trott

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

f[x_, y_, z_]  = eqn;

ContourPlot[Evaluate[f[x, y, 0]], {x, -1, 1}, {y, -1, 1}, 
            Contours -> {0}, PlotPoints -> 120]

f[x, y, z] - (f @@@ Permutations[{x, y, z}])

Plot[Evaluate[Sign@(f[x, x, x])], {x, 0, 1}]

fr[r_, ϑ_, φ_] = Collect[f0 = f[r Cos[φ] Sin[ϑ], r Sin[φ] Sin[ϑ], r Cos[ϑ]], r, Simplify]

Clear[rad];
rad[ϑ1_?NumberQ, φ1_?NumberQ] := 
With[{ϑ = SetPrecision[SetPrecision[ϑ1, Infinity], 40], 
      φ = SetPrecision[SetPrecision[φ1, Infinity], 40]},  
      FindRoot[fr[r, ϑ, φ] == 0, {r, 1/6, 5/4},
              WorkingPrecision -> 25, PrecisionGoal -> 15 , Method -> Brent][[1, 2]]]

ParametricPlot3D[rad[ϑ, φ] {Cos[φ] Sin[ϑ], Sin[φ] Sin[ϑ], Cos[ϑ]}, {φ, 0, Pi/2}, {ϑ, 0, Pi/2},
                 PlotRange -> All, ViewPoint -> 3 {1, 1, 1}]

Gauss-Kronrod

16 NIntegrate[rad[ϑ, φ]^3/3 Sin[ϑ], {φ, 0, Pi/4}, {ϑ, 0, Pi/2}, PrecisionGoal -> 4, Method -> GaussKronrod] // Timing

16 NIntegrate[rad[ϑ,φ]^3/3 Sin[ϑ],{φ,0,Pi/4},{ϑ,0,Pi/2},PrecisionGoal->6,Method->GaussKronrod]//Timing

Multidimensional

16 NIntegrate[rad[ϑ,φ]^3/3 Sin[ϑ],{φ,0,Pi/4},{ϑ,0,Pi/2},PrecisionGoal->4,Method->MultiDimensional]//InputForm//Timing

16 NIntegrate[rad[ϑ,φ]^3/3 Sin[ϑ],{φ,0,Pi/4},{ϑ,0,Pi/2},PrecisionGoal->6,Method->MultiDimensional]//InputForm//Timing

16 NIntegrate[rad[ϑ,φ]^3/3 Sin[ϑ],{φ,0,Pi/4},{ϑ,0,Pi/2},PrecisionGoal->7,Method->MultiDimensional]//InputForm//Timing

16 NIntegrate[rad[ϑ,φ]^3/3 Sin[ϑ],{φ,0,Pi/4},{ϑ,0,Pi/2},PrecisionGoal->7,Method->MultiDimensional,WorkingPrecision->20]//InputForm//Timing

16 NIntegrate[rad[ϑ,φ]^3/3 Sin[ϑ],{φ,0,Pi/4},{ϑ,0,Pi/2},PrecisionGoal->10,Method->MultiDimensional,WorkingPrecision->30]//InputForm//Timing

16 NIntegrate[rad[ϑ,φ]^3/3 Sin[ϑ],{φ,0,Pi/4},{ϑ,0,Pi/2},PrecisionGoal->10,Method->MultiDimensional,MaxRecursion->20,WorkingPrecision->40]//InputForm//Timing

Monte Carlo

NIntegrate[Boole[eqn<0],{x,-1,1},{y,-1,1},{z,-1,1},Method -> MonteCarlo, PrecisionGoal->3]//Timing

Cartesian

V6?

Integrate[Boole[eqn<0],{x,-1,1},{y,-1,1},{z,-1,1}]//Timing

Integrate[Boole[eqn<0],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]//Timing

V6.1

8Integrate[Boole[(x^(2/3))+(y^(2/3))+(z^(2/3))<1],{x,0,1},Assumptions->0<y<1&&0<z<1]//Timing

8Integrate[Boole[(x^2)<(((1-((y^(2/3))+(z^(2/3)))))^3)],{x,0,1},Assumptions->0<y<1&&0<z<1]//Timing

...hangs...

Cylindrical Algebraic

NIntegrate[Boole[Evaluate[eqn/.{x->r Cos[θ],y->r Sin[θ]}]<0]r,{r,0,1},{θ,0,2π},{z,-1,1}]//Timing

Integrate[Boole[Evaluate[eqn/.{x->r Cos[θ],y->r Sin[θ]}]<0]r,{r,0,1},{θ,0,2π},{z,-1,1}]//Timing

Cylindrical

FullSimplify[((((x^2)))^(1/3))+((((y^2)))^(1/3))+((((z^2)))^(1/3))<1/.{x->r Cos[θ],y->r Sin[θ]},r>0&&z∈Reals&&0<θ<2π]

NIntegrate[Boole[((Abs[z])^(2/3))+((((r^2) ((Cos[θ])^2)))^(1/3))+((((r^2) ((Sin[θ])^2)))^(1/3))<1]r,{θ,0,2π},{z,-1,1},{r,0,1}]//Timing

Integrate[Boole[((Abs[z])^(2/3))+((((r^2) ((Cos[θ])^2)))^(1/3))+((((r^2) ((Sin[θ])^2)))^(1/3))<1]r,{θ,0,2π},{z,-1,1},{r,0,1}]//Timing

Spherical Algebraic

NIntegrate[Boole[Evaluate[eqn/.{x->r Cos[θ]Sin[ϕ],y->r Sin[θ]Sin[ϕ],z->r Cos[ϕ]}]<0](r^2)Sin[ϕ],{r,0,1},{θ,0,2π},{ϕ,0,π}]//Timing

Integrate[Boole[Evaluate[eqn/.{x->r Cos[θ]Sin[ϕ],y->r Sin[θ]Sin[ϕ],z->r Cos[ϕ]}]<0](r^2)Sin[ϕ],{r,0,1},{θ,0,2π},{ϕ,0,π}]//Timing

Spherical

FullSimplify[((((x^2)))^(1/3))+((((y^2)))^(1/3))+((((z^2)))^(1/3))<1/.{x->r Cos[θ]Sin[ϕ],y->r Sin[θ]Sin[ϕ],z->r Cos[ϕ]},r>0&&0<θ<2π&&0<ϕ<π]

NIntegrate[Boole[((((r^2) ((Cos[ϕ])^2)))^(1/3))+((((r^2) ((Cos[θ])^2) ((Sin[ϕ])^2)))^(1/3))+((((r^2) ((Sin[θ])^2) ((Sin[ϕ])^2)))^(1/3))<1](r^2)Sin[ϕ],{θ,0,2π},{ϕ,0,π},{r,0,1}]//Timing

Integrate[Boole[((((r^2) ((Cos[ϕ])^2)))^(1/3))+((((r^2) ((Cos[θ])^2) ((Sin[ϕ])^2)))^(1/3))+((((r^2) ((Sin[θ])^2) ((Sin[ϕ])^2)))^(1/3))<1](r^2)Sin[ϕ],{θ,0,2π},{ϕ,0,π},{r,0,1}]//Timing

Inertia Tensor

can't get because we can't even get V

With[{ρ=(M/((4/3)π (a^3)))},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,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0]
]//TraditionalForm

Metric Properties

Metric Tensor

MetricTensor[HyperbolicOctahedron[u,v],{u,v}]//FullSimplify

Christoffel Symbol of the Second Kind

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

Vector Properties

eqn=a {Cos[u] Cos[v],Sin[u] Cos[v],Sin[v]}^3;
rules={Abs[x_]^(n_/;Mod[n,2]==0):>x^n};
assum=a>0&&-π/2<u<π/2&&-π<v<π;

Vector Length

Assuming[assum,FullSimplify[Norm[eqn]/.rules]]

Normal Vector

Assuming[assum,FullSimplify[NormalVector[eqn,{u,v}]/.rules]]