2024年5月26日发(作者：终娴雅)
Gauss Bonnet公式
[1] 测地曲率
测地曲率是曲面上的曲线的曲率向量在切平面上的投影, 如图,Г: r=r(s) 以弧长s为参数, =α×n
k
g
= kβ· = kβ· = r″·(α×n) = (r′, r″,n) 即
k
g
= (r′, r″,n).
由于k
n
是kβ在n上的投影, 所以下面的关系式成立:
k
2
=k
n
2
+k
g
2
[2] 测地曲率的计算
首先计算正交坐标曲线的测地曲率.
设u-线、v-线的弧长分别为s
1
、s
2
, 单位切向量分别为e
1
、e
2
, 则
e
1
= r
u
/
图: 测地曲率
Г
r
0
θ
β
n
r
v
e
2
α
r
u
e
1
E
, e
2
= r
v
/
G
, ds
1
=
E
du, ds
2
=
G
dv.
α= r
u
u′+ r
v
v′ = e
1
E
u′+ e
2
G
v′ = cosθ e
1
+ sinθ e
2
.
cosθ=
E
u′, sinθ=
G
v′.
沿u-线, v=常数. u-线的测地曲率
k
u
= (e
1
,
d(r
u
/E)
de
1
de
du
, n) = (r
u
/
E
,
1
, n) = (r
u
,, n) /E =
du
ds
1
duds
1
33
= (r
u
,
r
uu
/E
, n) /E = (r
u
, r
uu
, n) /
E
= (r
u
× r
uu
)·n/
E
= (r
u
× r
uu
)·(r
u
× r
v
)/
E
4
G
=
= (r
u
2
) (r
uu
·r
v
)/
E
4
G
= -E (E
v
/2)/
E
4
G
= - E
v
/2 E
G
=
k
u
=
v-线的测地曲率
lnE
/2G
. 即
v
lnE
/2G
. 参见P149
v
k
v
= (e
2
,
d(r
v
/G)
de
2
de
dv
, n) = (r
v
/
G
,
2
, n) = (r
v
,, n) /G =
dv
dvds
2
ds
2
= (r
v
,
r
vv
/G
, n) /G = (r
v
, r
vv
, n) /
G
3
= (r
v
× r
vv
)·n/
G
3
= (r
v
× r
vv
)·(r
u
× r
v
)/
G
4
E
=
= -(r
v
2
) (r
vv
·r
u
)/
G
4
E
= G (G
u
/2)/
G
4
E
= G
u
/2G
E
=
lnG
/2E
.
u
k
v
=
lnG
u
/2E
.
说明: 从 r
u
·r
v
=0, 得到 r
uu
·r
v
+r
u
·r
vu
=0, 从而r
uu
·r
v
= - r
u
·r
vu
= -E
v
/2.
同样, 从 r
uv
·r
v
+r
u
·r
vv
=0, 得到r
vv
·r
u
= - r
v
·r
vu
= -G
u
/2.
下面计算一般曲线Г的测地曲率.
k
g
= (r′, r″,n) = (r
u
u′+ r
v
v′, r
uu
u′
2
+2r
uv
u′v′+ r
vv
v′
2
+ r
u
u″+ r
v
v″,n) =
= (r
u
u′, r
uu
u′
2
+2r
uv
u′v′+ r
vv
v′
2
+ r
v
v″,n) + ( r
v
v′, r
uu
u′
2
+2r
uv
u′v′+ r
vv
v′
2
+ r
u
u″,n) =
= (r
u
, r
uu
,n) u′
3
+ 2 u′
2
v′ (r
u
, r
uv
,n) + (r
u
, r
vv
,n) u′v′
2
+ (r
u
u′, r
v
v″,n) +
+ (r
v
, r
uu
,n) v′u′
2
+ 2u′v′
2
(r
v
, r
uv
,n) + (r
v
, r
vv
,n) v′
3
+ ( r
v
v′, r
u
u″,n)
(r
u
, r
v
,n)= -(r
v
, r
u
,n)= (r
u
× r
v
)·n =
EG
(r
u
, r
uu
,n)= (r
u
× r
uu
)·n = (r
u
× r
uu
)·(r
u
× r
v
)/
EG
= r
u
2
·r
uu
r
v
/
EG
= -E·E
v
/2
EG
.
(r
u
, r
uv
,n)= (r
u
× r
uv
)·n = (r
u
× r
uv
)·(r
u
× r
v
)/
EG
= r
u
2
·r
uv
r
v
/
EG
= E·G
u
/2
EG
.
(r
u
, r
vv
,n)= (r
u
× r
vv
)·n = (r
u
× r
vv
)·(r
u
× r
v
)/
EG
= r
u
2
·r
vv
r
v
/
EG
= E·G
v
/2
EG
.
(r
v
, r
uu
,n) = (r
v
× r
uu
)·n = (r
v
× r
uu
)·(r
u
× r
v
)/
EG
= -r
v
2
·r
uu
r
u
/
EG
= -G·E
u
/2
EG
.
(r
v
, r
uv
,n) = (r
v
× r
uv
)·n = (r
v
× r
uv
)·(r
u
× r
v
)/
EG
= -r
v
2
·r
uv
r
u
/
EG
= -G·E
v
/2
EG
.
(r
v
, r
vv
,n)= (r
v
× r
vv
)·n = (r
v
× r
vv
)·(r
u
× r
v
)/
EG
= -r
v
2
·r
vv
r
u
/
EG
= G·G
u
/2
EG
.
u′=cosθ/
E
, v′=sinθ/
G
.
k
g
= u′v″
EG
-v′u″
EG
+
- u′
3
E·E
v
/2
EG
+ v′
3
G·G
u
/2
EG
+ u′
2
v′E·G
u
/
EG
+ u′v′
2
E·G
v
/2
EG
+
- v′u′
2
G·E
u
/2
EG
- u′v′
2
G·E
v
/
EG
参见P148
从 cosθ=
E
u′, sinθ=
G
v′ 得 u″=-θ′sinθ
E
1
+ cosθ(
E
1
)′, v″=θ′cosθ
G
1
+sinθ(
G
1
)′,
u′v″-v′u″ = cosθ
E
1
(θ′cosθ
G
1
+sinθ(
G
1
)′) - sinθ
G
1
(-θ′sinθ
E
1
+ cosθ(
E
1
)′) =
=θ′/
EG
+sinθcosθ(
E
1
(
G
1
)′ -
G
1
(
E
1
)′ ) .
故
2024年5月26日发(作者：终娴雅)
Gauss Bonnet公式
[1] 测地曲率
测地曲率是曲面上的曲线的曲率向量在切平面上的投影, 如图,Г: r=r(s) 以弧长s为参数, =α×n
k
g
= kβ· = kβ· = r″·(α×n) = (r′, r″,n) 即
k
g
= (r′, r″,n).
由于k
n
是kβ在n上的投影, 所以下面的关系式成立:
k
2
=k
n
2
+k
g
2
[2] 测地曲率的计算
首先计算正交坐标曲线的测地曲率.
设u-线、v-线的弧长分别为s
1
、s
2
, 单位切向量分别为e
1
、e
2
, 则
e
1
= r
u
/
图: 测地曲率
Г
r
0
θ
β
n
r
v
e
2
α
r
u
e
1
E
, e
2
= r
v
/
G
, ds
1
=
E
du, ds
2
=
G
dv.
α= r
u
u′+ r
v
v′ = e
1
E
u′+ e
2
G
v′ = cosθ e
1
+ sinθ e
2
.
cosθ=
E
u′, sinθ=
G
v′.
沿u-线, v=常数. u-线的测地曲率
k
u
= (e
1
,
d(r
u
/E)
de
1
de
du
, n) = (r
u
/
E
,
1
, n) = (r
u
,, n) /E =
du
ds
1
duds
1
33
= (r
u
,
r
uu
/E
, n) /E = (r
u
, r
uu
, n) /
E
= (r
u
× r
uu
)·n/
E
= (r
u
× r
uu
)·(r
u
× r
v
)/
E
4
G
=
= (r
u
2
) (r
uu
·r
v
)/
E
4
G
= -E (E
v
/2)/
E
4
G
= - E
v
/2 E
G
=
k
u
=
v-线的测地曲率
lnE
/2G
. 即
v
lnE
/2G
. 参见P149
v
k
v
= (e
2
,
d(r
v
/G)
de
2
de
dv
, n) = (r
v
/
G
,
2
, n) = (r
v
,, n) /G =
dv
dvds
2
ds
2
= (r
v
,
r
vv
/G
, n) /G = (r
v
, r
vv
, n) /
G
3
= (r
v
× r
vv
)·n/
G
3
= (r
v
× r
vv
)·(r
u
× r
v
)/
G
4
E
=
= -(r
v
2
) (r
vv
·r
u
)/
G
4
E
= G (G
u
/2)/
G
4
E
= G
u
/2G
E
=
lnG
/2E
.
u
k
v
=
lnG
u
/2E
.
说明: 从 r
u
·r
v
=0, 得到 r
uu
·r
v
+r
u
·r
vu
=0, 从而r
uu
·r
v
= - r
u
·r
vu
= -E
v
/2.
同样, 从 r
uv
·r
v
+r
u
·r
vv
=0, 得到r
vv
·r
u
= - r
v
·r
vu
= -G
u
/2.
下面计算一般曲线Г的测地曲率.
k
g
= (r′, r″,n) = (r
u
u′+ r
v
v′, r
uu
u′
2
+2r
uv
u′v′+ r
vv
v′
2
+ r
u
u″+ r
v
v″,n) =
= (r
u
u′, r
uu
u′
2
+2r
uv
u′v′+ r
vv
v′
2
+ r
v
v″,n) + ( r
v
v′, r
uu
u′
2
+2r
uv
u′v′+ r
vv
v′
2
+ r
u
u″,n) =
= (r
u
, r
uu
,n) u′
3
+ 2 u′
2
v′ (r
u
, r
uv
,n) + (r
u
, r
vv
,n) u′v′
2
+ (r
u
u′, r
v
v″,n) +
+ (r
v
, r
uu
,n) v′u′
2
+ 2u′v′
2
(r
v
, r
uv
,n) + (r
v
, r
vv
,n) v′
3
+ ( r
v
v′, r
u
u″,n)
(r
u
, r
v
,n)= -(r
v
, r
u
,n)= (r
u
× r
v
)·n =
EG
(r
u
, r
uu
,n)= (r
u
× r
uu
)·n = (r
u
× r
uu
)·(r
u
× r
v
)/
EG
= r
u
2
·r
uu
r
v
/
EG
= -E·E
v
/2
EG
.
(r
u
, r
uv
,n)= (r
u
× r
uv
)·n = (r
u
× r
uv
)·(r
u
× r
v
)/
EG
= r
u
2
·r
uv
r
v
/
EG
= E·G
u
/2
EG
.
(r
u
, r
vv
,n)= (r
u
× r
vv
)·n = (r
u
× r
vv
)·(r
u
× r
v
)/
EG
= r
u
2
·r
vv
r
v
/
EG
= E·G
v
/2
EG
.
(r
v
, r
uu
,n) = (r
v
× r
uu
)·n = (r
v
× r
uu
)·(r
u
× r
v
)/
EG
= -r
v
2
·r
uu
r
u
/
EG
= -G·E
u
/2
EG
.
(r
v
, r
uv
,n) = (r
v
× r
uv
)·n = (r
v
× r
uv
)·(r
u
× r
v
)/
EG
= -r
v
2
·r
uv
r
u
/
EG
= -G·E
v
/2
EG
.
(r
v
, r
vv
,n)= (r
v
× r
vv
)·n = (r
v
× r
vv
)·(r
u
× r
v
)/
EG
= -r
v
2
·r
vv
r
u
/
EG
= G·G
u
/2
EG
.
u′=cosθ/
E
, v′=sinθ/
G
.
k
g
= u′v″
EG
-v′u″
EG
+
- u′
3
E·E
v
/2
EG
+ v′
3
G·G
u
/2
EG
+ u′
2
v′E·G
u
/
EG
+ u′v′
2
E·G
v
/2
EG
+
- v′u′
2
G·E
u
/2
EG
- u′v′
2
G·E
v
/
EG
参见P148
从 cosθ=
E
u′, sinθ=
G
v′ 得 u″=-θ′sinθ
E
1
+ cosθ(
E
1
)′, v″=θ′cosθ
G
1
+sinθ(
G
1
)′,
u′v″-v′u″ = cosθ
E
1
(θ′cosθ
G
1
+sinθ(
G
1
)′) - sinθ
G
1
(-θ′sinθ
E
1
+ cosθ(
E
1
)′) =
=θ′/
EG
+sinθcosθ(
E
1
(
G
1
)′ -
G
1
(
E
1
)′ ) .
故