# 法曲率与 Weingarten 变换设曲面 S : r ⃗ = r ⃗ ( u , v ) S:\vec r=\vec r(u,v) S : r = r ( u , v ) C : r ⃗ = r ⃗ ( u ( s ) , v ( s ) ) C:\vec r=\vec r(u(s),v(s)) C : r = r ( u ( s ) , v ( s ) ) s s s C C C
t ⃗ = r ⃗ ˙ = r ⃗ u d u d s + r ⃗ v d v d s \vec t=\dot {\vec r}=\vec r_u\frac {\mathrm du}{\mathrm ds}+\vec r_v\frac {\mathrm dv}{\mathrm ds} t = r ˙ = r u d s d u + r v d s d v
则 C C C
d t ⃗ d s = r ⃗ u u ( d u d s ) 2 + 2 r ⃗ u v d u d s d v d s + r ⃗ v v ( d v d s ) 2 + r ⃗ u d 2 u d s 2 + r ⃗ v d 2 v d s 2 \dfrac {\mathrm d\vec t}{\mathrm ds}=\vec r_{uu}(\frac {\mathrm du}{\mathrm ds})^2+2\vec r_{uv}\frac {\mathrm du}{\mathrm ds}\frac {\mathrm dv}{\mathrm ds}+\vec r_{vv}(\frac {\mathrm dv}{\mathrm ds})^2+\vec r_u\frac {\mathrm d^2u}{\mathrm ds^2}+\vec r_v\frac {\mathrm d^2v}{\mathrm ds^2} d s d t = r u u ( d s d u ) 2 + 2 r u v d s d u d s d v + r v v ( d s d v ) 2 + r u d s 2 d 2 u + r v d s 2 d 2 v
Definition 设曲面 S S S C : r ⃗ = r ⃗ ( u ( s ) , v ( s ) ) C:\vec r=\vec r(u(s),v(s)) C : r = r ( u ( s ) , v ( s ) ) s s s r ⃗ ˙ \dot {\vec r} r ˙
κ n = ⟨ d t ⃗ d s , n ⃗ ⟩ = L ( d u d s ) 2 + 2 M d u d s d v d s + N ( d v d s ) 2 \kappa_n=\langle \dfrac {\mathrm d\vec t}{\mathrm ds},\vec n\rangle=L(\frac {\mathrm du}{\mathrm ds})^2+2M\frac {\mathrm du}{\mathrm ds}\frac {\mathrm dv}{\mathrm ds}+N(\frac {\mathrm dv}{\mathrm ds})^2 κ n = ⟨ d s d t , n ⟩ = L ( d s d u ) 2 + 2 M d s d u d s d v + N ( d s d v ) 2
为曲线 C C C P P P 法曲率 。
Remark 法曲率就是曲率向量在单位法向量方向上的投影。
Remark 给定曲面 S S S d u d s \dfrac {\mathrm du}{\mathrm ds} d s d u d v d s \dfrac {\mathrm dv}{\mathrm ds} d s d v r ⃗ u , r ⃗ v \vec r_u,\vec r_v r u , r v κ n = κ n ( t ⃗ ) \kappa_n=\kappa_n(\vec t) κ n = κ n ( t ) C , C ~ C,\tilde C C , C ~ P P P t ⃗ \vec t t P P P
Remark 由于曲面第一基本形式 ( d s ) 2 = E ( d u ) 2 + 2 F d u d v + G ( d v ) 2 (\mathrm d s)^2=E(\mathrm du)^2+2F\mathrm du\mathrm dv+G(\mathrm dv)^2 ( d s ) 2 = E ( d u ) 2 + 2 F d u d v + G ( d v ) 2
κ n = I I I = L ( d u ) 2 + 2 M d u d v + N ( d v ) 2 E ( d u ) 2 + 2 F d u d v + G ( d v ) 2 = L λ 2 + 2 M λ + N E λ 2 + 2 F λ + G , λ = d u d v \kappa_n=\dfrac {II}{I}=\dfrac {L(\mathrm du)^2+2M\mathrm du\mathrm dv+N(\mathrm dv)^2}{E(\mathrm du)^2+2F\mathrm du\mathrm dv+G(\mathrm dv)^2}=\dfrac {L\lambda^2+2M\lambda +N}{E\lambda^2+2F\lambda +G},\quad \lambda =\dfrac {\mathrm du}{\mathrm dv} κ n = I I I = E ( d u ) 2 + 2 F d u d v + G ( d v ) 2 L ( d u ) 2 + 2 M d u d v + N ( d v ) 2 = E λ 2 + 2 F λ + G L λ 2 + 2 M λ + N , λ = d v d u
所以 κ n \kappa_n κ n λ \lambda λ
Remark 设 v ⃗ ∈ T P ( S ) \vec v\in T_P(S) v ∈ T P ( S ) { v ⃗ , n ⃗ ( P ) } \{\vec v,\vec n(P)\} { v , n ( P ) } 法截面 ，与曲面 S S S C C C 法截线 ，法曲率 κ n \kappa_n κ n
Remark 法曲率 κ n ( v ⃗ ) \kappa_n(\vec v) κ n ( v ) v ⃗ \vec v v
Proposition 法曲率只依赖切方向。
证明：将法曲率推广到非单位切向量上。
(1) 设 t ⃗ = r ⃗ u d u d s + r ⃗ v d v d s \vec t=\vec r_u\dfrac {\mathrm du}{\mathrm ds}+\vec r_v\dfrac {\mathrm dv}{\mathrm ds} t = r u d s d u + r v d s d v t ⃗ \vec t t
κ n ( t ⃗ ) = L ( d u d s ) 2 + 2 M d u d s d v d s + N ( d v d s ) 2 \kappa_n(\vec t)=L(\frac {\mathrm du}{\mathrm ds})^2+2M\frac {\mathrm du}{\mathrm ds}\frac {\mathrm dv}{\mathrm ds}+N(\frac {\mathrm dv}{\mathrm ds})^2 κ n ( t ) = L ( d s d u ) 2 + 2 M d s d u d s d v + N ( d s d v ) 2
(2) 一般地，设 w ⃗ = ξ r ⃗ u + η r ⃗ v ∈ T P ( S ) \vec w=\xi \vec r_u+\eta \vec r_v\in T_P(S) w = ξ r u + η r v ∈ T P ( S ) v ⃗ = w ⃗ ∣ w ⃗ ∣ \vec v=\dfrac {\vec w}{|\vec w|} v = ∣ w ∣ w v ⃗ \vec v v
κ n ( v ⃗ ) = L ( ξ ∣ w ⃗ ∣ ) 2 + 2 M ξ ∣ w ⃗ ∣ η ∣ w ⃗ ∣ + N ( η ∣ w ⃗ ∣ ) 2 = L ξ 2 + 2 M ξ η + N η 2 E ξ 2 + 2 F ξ η + G η 2 : = κ n ( w ⃗ ) \kappa_n(\vec v)=L(\dfrac \xi {|\vec w|})^2+2M\dfrac \xi {|\vec w|}\dfrac \eta {|\vec w|}+N(\dfrac \eta {|\vec w|})^2=\dfrac {L\xi^2+2M\xi\eta +N\eta^2}{E\xi^2+2F\xi\eta +G\eta^2}:=\kappa_n(\vec w) κ n ( v ) = L ( ∣ w ∣ ξ ) 2 + 2 M ∣ w ∣ ξ ∣ w ∣ η + N ( ∣ w ∣ η ) 2 = E ξ 2 + 2 F ξ η + G η 2 L ξ 2 + 2 M ξ η + N η 2 : = κ n ( w )
这只依赖于 ξ η \dfrac \xi \eta η ξ
Proposition 法曲率在同向（反向）参数变换下不变（变号）。
证明：因为 κ n = I I I \kappa_n=\dfrac {II}{I} κ n = I I I I , I I I,II I , I I I I II I I
Proposition 法曲率在曲面刚体运动（反向刚体运动）下不变（变号）。
证明：因为 κ n = I I I \kappa_n=\dfrac {II}{I} κ n = I I I I , I I I,II I , I I I I II I I
Example 平面的法曲率为 κ n = 0 \kappa_n=0 κ n = 0
Example 球面 ( R cos φ cos θ , R cos φ sin θ , R sin φ ) (R\cos\varphi\cos\theta,R\cos\varphi\sin\theta,R\sin\varphi) ( R cos φ cos θ , R cos φ sin θ , R sin φ ) κ n = 1 R \kappa_n=\frac 1R κ n = R 1
Example 圆柱面 ( a cos u a , a sin u a , v ) (a\cos \frac ua,a\sin \frac ua,v) ( a cos a u , a sin a u , v ) κ n = − ( d u ) 2 a ( ( d u ) 2 + ( d v ) 2 ) \kappa_n=-\frac {(\mathrm du)^2}{a((\mathrm du)^2+(\mathrm dv)^2)} κ n = − a ( ( d u ) 2 + ( d v ) 2 ) ( d u ) 2
Example 二次曲面 z = 1 2 ( a x 2 + b y 2 ) z=\frac 12(ax^2+by^2) z = 2 1 ( a x 2 + b y 2 )
κ n = 1 1 + a 2 x 2 + b 2 y 2 a ( d x ) 2 + b ( d y ) 2 ( 1 + a 2 x 2 ) ( d x ) 2 + 2 a b x y d x d y + ( 1 + b 2 y 2 ) ( d y ) 2 \kappa_n=\dfrac 1{\sqrt {1+a^2x^2+b^2y^2}}\dfrac {a(\mathrm dx)^2+b(\mathrm dy)^2}{(1+a^2x^2)(\mathrm dx)^2+2abxy\mathrm dx\mathrm dy+(1+b^2y^2)(\mathrm dy)^2} κ n = 1 + a 2 x 2 + b 2 y 2 1 ( 1 + a 2 x 2 ) ( d x ) 2 + 2 a b x y d x d y + ( 1 + b 2 y 2 ) ( d y ) 2 a ( d x ) 2 + b ( d y ) 2
特别地，取曲面上 P 0 P_0 P 0 v ⃗ = λ r ⃗ x + μ r ⃗ y \vec v=\lambda \vec r_x+\mu \vec r_y v = λ r x + μ r y
κ n ( P 0 , v ⃗ ) = a λ 2 + b μ 2 1 + a 2 x 0 2 + b 2 y 0 2 \kappa_n(P_0,\vec v)=\dfrac {a\lambda^2+b\mu^2}{\sqrt {1+a^2x_0^2+b^2y_0^2}} κ n ( P 0 , v ) = 1 + a 2 x 0 2 + b 2 y 0 2 a λ 2 + b μ 2
(1) 当 a b > 0 ab>0 a b > 0 椭圆抛物面 ，κ n ( v ⃗ ) \kappa_n(\vec v) κ n ( v ) a b < 0 ab<0 a b < 0 双曲抛物面 ，κ n ( v ⃗ ) = 0 \kappa_n(\vec v)=0 κ n ( v ) = 0 a b = 0 ab=0 a b = 0 抛物柱面 ，κ n ( v ⃗ ) = 0 \kappa_n(\vec v)=0 κ n ( v ) = 0 z = 1 2 ( a x 2 + b y 2 ) z=\frac 12(ax^2+by^2) z = 2 1 ( a x 2 + b y 2 )
Definition 曲面 S S S v ⃗ = λ r ⃗ u + μ r ⃗ v \vec v=\lambda \vec r_u+\mu \vec r_v v = λ r u + μ r v
κ n = L λ 2 + 2 M λ μ + N μ 2 E λ 2 + 2 F λ μ + G μ 2 \kappa_n=\dfrac {L\lambda^2+2M\lambda \mu +N\mu^2}{E\lambda^2+2F\lambda \mu +G\mu^2} κ n = E λ 2 + 2 F λ μ + G μ 2 L λ 2 + 2 M λ μ + N μ 2
如果 κ n ( v ⃗ ) = 0 \kappa_n(\vec v)=0 κ n ( v ) = 0 v ⃗ \vec v v S S S 渐近方向 ；如果 S S S C C C C C C S S S 渐近曲线 。设 P P P S S S L N − M 2 > 0 LN-M^2>0 L N − M 2 > 0 P P P 椭圆点 ；L N − M 2 < 0 LN-M^2<0 L N − M 2 < 0 P P P 双曲点 ；L N − M 2 = 0 LN-M^2=0 L N − M 2 = 0 P P P 抛物点 。
Definition 1. 映射
d n ⃗ : T P ( S ) → T P ( S ) , w ⃗ ↦ d n ⃗ ( w ⃗ ) d\vec n:T_P(S)\to T_P(S),\quad \vec w\mapsto d\vec n(\vec w) d n : T P ( S ) → T P ( S ) , w ↦ d n ( w )
为曲面的 Weingarten 变换 .
