Toroidal Coordinates

Author

Eric W. Weisstein
April 15, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/CoordinateGeometry/ToroidalCoordinates.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Definition

x[u_,v_,ϕ_]:=a Sinh[v]Cos[ϕ]/(Cosh[v]-Cos[u])
y[u_,v_,ϕ_]:=a Sinh[v]Sin[ϕ]/(Cosh[v]-Cos[u])
z[u_,v_,ϕ_]:=a Sin[u]/(Cosh[v]-Cos[u])

conds=a>0&&v>0&&0<=u<2π&&0≤ϕ<2π;

Identities

Surfaces of constant v = toroids

(a^2)+((x(u,v,ϕ))^2)+((y(u,v,ϕ))^2)+((z(u,v,ϕ))^2)==2 a (Sqrt[((x(u,v,ϕ))^2)+((y(u,v,ϕ))^2)]) coth(v)

Simplify[Subtract@@{x[u,v,ϕ]^2+y[u,v,ϕ]^2+z[u,v,ϕ]^2+a^2,2a Sqrt[x[u,v,ϕ]^2+y[u,v,ϕ]^2]Coth[v]},conds]

Surfaces of constant u = spherical bowls

(((z(u,v,ϕ)-a cot(u)))^2)+((x(u,v,ϕ))^2)+((y(u,v,ϕ))^2)=((a^2)/((sin^2)(u)))

FullSimplify[Subtract@@{x[u,v,ϕ]^2+y[u,v,ϕ]^2+(z[u,v,ϕ]-a Cot[u])^2,a^2/Sin[u]^2},conds]

2 a z(u,v,ϕ) cot(u)=-(a^2)+((x(u,v,ϕ))^2)+((y(u,v,ϕ))^2)+((z(u,v,ϕ))^2)

FullSimplify[Subtract@@{2a z[u,v,ϕ] Cot[u],
x[u,v,ϕ]^2+y[u,v,ϕ]^2+z[u,v,ϕ]^2-a^2},conds]

Scale Factors

(h=Simplify[Sqrt[Plus@@Function[q,D[q,#]^2]/@{x[u,v,ϕ],y[u,v,ϕ],z[u,v,ϕ]}]&/@{u,v,ϕ},conds])//TraditionalForm

hs=Times@@Rest[#]/First[#]&/@Partition[h,3,1,1]

Laplacian

Simplify[{(1/(Times@@h)),hs},conds]

Built-In Properties

<<Calculus`VectorAnalysis`
coord=Toroidal;

Usage

?Toroidal

ScaleFactors[coord]

CoordinatesToCartesian[Coordinates[coord],coord]

CoordinatesFromCartesian[{x,y,z},coord]

Div[Through[{F1,F2,F3}@@Coordinates[coord]],coord]//TraditionalForm

Curl[Through[{F1,F2,F3}@@Coordinates[coord]],coord]//TraditionalForm

Grad[f@@Coordinates[coord],coord]//TraditionalForm

Laplacian[f@@Coordinates[coord],coord]//TraditionalForm

Coordinates[Toroidal[u,v,ϕ]]

{h1,h2,h3}=ScaleFactors[Toroidal[u,v,ϕ]]

{h1 h2 h3,h2 h3/h1,h1 h3/h2,h1 h2/h3}

FullSimplify[Laplacian[F[u,v,ϕ],Toroidal[u,v,ϕ]]]

f[u,v,ϕ]:=Sqrt[Cosh[u]-Cos[v]]U[u]V[v]Φ[ϕ];

FullSimplify[Laplacian[f[u,v,ϕ],Toroidal[u,v,ϕ]]]

Expand[%68/((((-Cos[v]+Cosh[u]))^(5/2)) ((Csch[u])^2) )/(U[u]V[v]Φ[ϕ])]