Coordinate system ( q 1 , q 2 , φ ) , φ ∈ [ 0,2 π )
In Cartesian coordinates ( x , y , z )
h 1
h 2
ϖ
h 1 h 2 ϖ
h 2 h 1 ϖ
Spherical ( r , θ , φ ) r ≥ 0, θ ∈ [ 0, π ]
{ x = r sin ( θ ) cos ( φ ) y = r sin ( θ ) sin ( φ ) z = r cos ( θ )
1
1 r
r s i n ( θ )
1 s i n ( θ )
1 r 2 s i n ( θ )
Modified Spherical ( r , ζ , φ ) r ≥ 0, ζ ∈ [ − 1,1 ]
{ x = r 1 − ζ 2 cos ( φ ) y = r 1 − ζ 2 sin ( φ ) z = r ζ
1 − ζ 2 r
r 1 − ζ 2
1 1 − ζ 2
1 r 2
Prolate Spheroid ( η , θ , φ )
η ≥ 0, θ ∈ [ 0, π ] α > 0
{ x = α sinh ( η ) sin ( θ ) cos ( φ ) y = α sinh ( η ) sin ( θ ) sin ( φ ) z = α cosh ( η ) cos ( θ )
1 a sinh 2 ( η ) + sin 2 ( θ )
a s i n h ( η ) s i n ( θ )
1 a s i n h ( η ) s i n ( θ )
Modified Prolate Spheroid ( τ , ζ , φ )
τ ≥ 1, ζ ∈ [ − 1,1 ] α > 0
{ x = α τ 2 − 1 1 − ζ 2 cos ( φ ) y = α τ 2 − 1 1 − ζ 2 sin ( φ ) z = α τ ζ
τ 2 − 1 a τ 2 − ζ 2
1 − ζ 2 a τ 2 − ζ 2
a τ 2 − 1 1 − ζ 2
1 a ( 1 − ζ 2 )
1 a ( τ 2 − 1 )
Oblate Spheroid ( η , θ , φ )
{ x = α cosh ( η ) sin ( θ ) cos ( φ ) y = α cosh ( η ) sin ( θ ) sin ( φ ) z = α sinh ( η ) cos ( θ )
1 a c o s h 2 ( η ) − sin 2 ( θ )
a c o s h ( η ) s i n ( θ )
1 a c o s h ( η ) s i n ( θ )
Modified Oblate Spheroid ( λ , ζ , φ )
λ ∈ ℝ , ζ ∈ [ − 1,1 ] α > 0
{ x = α λ 2 + 1 1 − ζ 2 cos ( φ ) y = α λ 2 + 1 1 − ζ 2 sin ( φ ) z = α λ ζ
λ 2 + 1 a λ 2 + ζ 2
1 − ζ 2 a λ 2 + ζ 2
a λ 2 + 1 1 − ζ 2
1 a ( 1 + λ 2 )
Parabolic ( μ , ν , φ ) μ , ν ≥ 0
{ x = μ ν cos ( φ ) y = μ ν sin ( φ ) z = μ 2 − ν 2 2
1 μ 2 + ν 2