20-1 Geodesic, geodesic circle, Darboux frame, Riemann coordinates 419
From these mapping equations, we derive cha-cha-cha (“change from one chart to another chart”)
tan L = −
tan V
cos U
versus tan U = −
tan B
cos L
,
(20.7)
tan B =
tan U
1 + tan
2 V / cos 2 U
=
sin U
√
cos 2 U + tan
2 V
versus
tan V =
tan L
1 + tan
2 B/ cos 2 L
=
sin L
√
cos 2 L + tan
2 B
,
(20.8)
in particular, the diffeomorphism
⎡
⎣
dU
dV
⎤
⎦ =
=
−
sin L tan B
cos 2 L+tan 2 B
−
cos L
cos 2 B cos 2 L+sin 2 B
cos L
(cos 2 L+tan 2 B) 1/2
− sin L tan B
(cos 2 L+tan 2 B) 1/2
⎡
⎣
dL
dB
⎤
⎦ ,
(20.9)
⎡
⎣
dL
dB
⎤
⎦ =
=
⎡
⎢
⎣
− tan V sin U
cos 2 U +tan 2 V
− cos U
cos 2 V (cos 2 U +tan 2 V )
cos U
(cos 2 U +tan 2 V ) 1/2
− sin U tan V
(cos 2 U +tan 2 V ) 1/2
⎤
⎥
⎦
⎡
⎣
dU
dV
⎤
⎦ .
(20.10)
Box 20.1 and Box 20.2 summarize the surface geometry of E
2
A 1 ,A 2
⊂ {R
3 , δ IJ }, in particular, the
matrices F, G, H, and J of type Frobenius, Gauß, Hesse, and Jacobi as well as the curvature matrix
K := −HG
−1 , especially the surface fundamental forms {I, II, III}.
Let C : [0, ∞] → {E
2
A 1 ,A 2
, G KL } be a smooth curve which is parameterized by arc length. Denote by
{D 1 , D 2 , D 3 } its Darboux frame defined by (20.11). Its derivational equations are given by (20.12),
introducing the antisymmetric connection matrix Ω (κ g , κ n , τ g ) containing geodetic curvature κ g ,
normal curvature κ n , and geodetic torsion τ g . In terms of the first fundamental form, in particular
{G KL }, the second fundamental form, in particular {H KL }, the Riemann connection, in particular the
Christoffel symbols {
M
KL }, the curvature measures of the curve C as a submanifold of {E
2
A 1 ,A 2
, G KL }
can be represented by Corollary 20.1.
D 1 :=
dX{U
k (S)}
dS
=
∂X
∂U K U
K = G K U
K ,
D 2 := ∗(D 3 ∧ D 1 ) = D 3 × D 1 ,
D 3 := G 3 ,
(20.11)
D
1 = +κ g D 2 + κ n D 3 ,
D
2 = −κ g D 1 + τ g D 3 ,
D
3 = −κ n D 1 − τ g D 2 ,
⎡
⎣
D
1
D
2
D
3
⎤
⎦ =
⎡
⎣
0
κ g κ n
−κ g
0
τ g
−κ n −τ g 0
⎤
⎦
⎡
⎣
D 1
D 2
D 3
⎤
⎦ , D
= ΩD , Ω =
⎡
⎣
0
κ g κ n
−κ g
0
τ g
−κ n −τ g 0
⎤
⎦ .
(20.12)
From these mapping equations, we derive cha-cha-cha (“change from one chart to another chart”)
tan L = −
tan V
cos U
versus tan U = −
tan B
cos L
,
(20.7)
tan B =
tan U
1 + tan
2 V / cos 2 U
=
sin U
√
cos 2 U + tan
2 V
versus
tan V =
tan L
1 + tan
2 B/ cos 2 L
=
sin L
√
cos 2 L + tan
2 B
,
(20.8)
in particular, the diffeomorphism
⎡
⎣
dU
dV
⎤
⎦ =
=
−
sin L tan B
cos 2 L+tan 2 B
−
cos L
cos 2 B cos 2 L+sin 2 B
cos L
(cos 2 L+tan 2 B) 1/2
− sin L tan B
(cos 2 L+tan 2 B) 1/2
⎡
⎣
dL
dB
⎤
⎦ ,
(20.9)
⎡
⎣
dL
dB
⎤
⎦ =
=
⎡
⎢
⎣
− tan V sin U
cos 2 U +tan 2 V
− cos U
cos 2 V (cos 2 U +tan 2 V )
cos U
(cos 2 U +tan 2 V ) 1/2
− sin U tan V
(cos 2 U +tan 2 V ) 1/2
⎤
⎥
⎦
⎡
⎣
dU
dV
⎤
⎦ .
(20.10)
Box 20.1 and Box 20.2 summarize the surface geometry of E
2
A 1 ,A 2
⊂ {R
3 , δ IJ }, in particular, the
matrices F, G, H, and J of type Frobenius, Gauß, Hesse, and Jacobi as well as the curvature matrix
K := −HG
−1 , especially the surface fundamental forms {I, II, III}.
Let C : [0, ∞] → {E
2
A 1 ,A 2
, G KL } be a smooth curve which is parameterized by arc length. Denote by
{D 1 , D 2 , D 3 } its Darboux frame defined by (20.11). Its derivational equations are given by (20.12),
introducing the antisymmetric connection matrix Ω (κ g , κ n , τ g ) containing geodetic curvature κ g ,
normal curvature κ n , and geodetic torsion τ g . In terms of the first fundamental form, in particular
{G KL }, the second fundamental form, in particular {H KL }, the Riemann connection, in particular the
Christoffel symbols {
M
KL }, the curvature measures of the curve C as a submanifold of {E
2
A 1 ,A 2
, G KL }
can be represented by Corollary 20.1.
D 1 :=
dX{U
k (S)}
dS
=
∂X
∂U K U
K = G K U
K ,
D 2 := ∗(D 3 ∧ D 1 ) = D 3 × D 1 ,
D 3 := G 3 ,
(20.11)
D
1 = +κ g D 2 + κ n D 3 ,
D
2 = −κ g D 1 + τ g D 3 ,
D
3 = −κ n D 1 − τ g D 2 ,
⎡
⎣
D
1
D
2
D
3
⎤
⎦ =
⎡
⎣
0
κ g κ n
−κ g
0
τ g
−κ n −τ g 0
⎤
⎦
⎡
⎣
D 1
D 2
D 3
⎤
⎦ , D
= ΩD , Ω =
⎡
⎣
0
κ g κ n
−κ g
0
τ g
−κ n −τ g 0
⎤
⎦ .
(20.12)