We remind that for the parametric surface r ( u , v ) r(u,v) r ( u , v )
I = d s 2 = E d u 2 + 2 F d u d v + G d v 2 , I=ds^2=Edu^2+2Fdudv+Gdv^2, I = d s 2 = E d u 2 + 2 F d u d v + G d v 2 , E = ( r u , r u ) , F = ( r u , r v ) , E=(r_u,r_u), F=(r_u,r_v), E = ( r u , r u ) , F = ( r u , r v ) , G = ( r v , r v ) G=(r_v,r_v) G = ( r v , r v )
The second fundamental form is: I I = L d u 2 + 2 M d u d v + N d v 2 II=Ldu^2+2Mdudv+Ndv^2 II = L d u 2 + 2 M d u d v + N d v 2 L = − ( r u , N u ) , M = − ( r v , N u ) , N = − ( r v , N v ) L=-(r_u,\text{N}_u),M=-(r_v,\text{N}_u),N=-(r_v,\text{N}_v) L = − ( r u , N u ) , M = − ( r v , N u ) , N = − ( r v , N v )
The normal vector is: N = r u × r v = ∣ i j k ( r u ) 1 ( r u ) 2 ( r u ) 3 ( r v ) 1 ( r v ) 2 ( r v ) 3 ∣ \text{N}=r_u\times r_v=\begin{vmatrix}
i & j& k \\
(r_u)_1 & (r_u)_2& (r_u)_3\\
(r_v)_1 & (r_v)_2& (r_v)_3
\end{vmatrix} N = r u × r v = ∣ ∣ i ( r u ) 1 ( r v ) 1 j ( r u ) 2 ( r v ) 2 k ( r u ) 3 ( r v ) 3 ∣ ∣ ( ) j ()_j ( ) j
The Weingarten matrix is the matrix of the shape operator of surface s with respect to variables u and v (see https://resources.wolframcloud.com/FunctionRepository/resources/WeingartenMatrix).
It is given by ( M F − L G E G − F 2 L F − M E E G − F 2 N F − M G E G − F 2 M F − N E E G − F 2 ) \begin{pmatrix}
\frac{MF-LG}{EG-F^2} & \frac{LF-ME}{EG-F^2} \\
\frac{NF-MG}{EG-F^2} & \frac{MF-NE}{EG-F^2}
\end{pmatrix} ( EG − F 2 MF − L G EG − F 2 NF − MG EG − F 2 L F − ME EG − F 2 MF − NE )
a). r θ = ( − s i n θ c o s φ , − s i n θ s i n φ , c o s θ ) , r_{\theta}=(-sin\,\theta\,cos\varphi,-sin\,\theta\,sin\varphi,cos\,\theta), r θ = ( − s in θ cos φ , − s in θ s in φ , cos θ ) ,
r φ = ( − c o s θ s i n φ , c o s θ c o s φ , 0 ) , r_{\varphi}=(-cos\,\theta\,sin\varphi,cos\,\theta\,cos\varphi,0), r φ = ( − cos θ s in φ , cos θ cos φ , 0 ) ,
E = ( r θ , r θ ) = s i n 2 θ c o s 2 φ + s i n 2 θ s i n 2 φ + c o s 2 θ = 1 E=(r_{\theta},r_{\theta})=sin^2\theta\,cos^2\,\varphi+sin^2\theta\,sin^2\,\varphi+cos^2\theta=1 E = ( r θ , r θ ) = s i n 2 θ co s 2 φ + s i n 2 θ s i n 2 φ + co s 2 θ = 1
F = ( r θ , r φ ) = s i n θ c o s θ c o s φ s i n φ − s i n θ c o s θ c o s φ s i n φ = 0 , F=(r_{\theta},r_{\varphi})=sin\,\theta\,\,cos\,\theta\,cos\varphi\, sin\varphi-sin\,\theta\,\,cos\,\theta\,cos\varphi\, sin\varphi=0, F = ( r θ , r φ ) = s in θ cos θ cos φ s in φ − s in θ cos θ cos φ s in φ = 0 ,
G = ( r φ , r φ ) = c o s 2 θ s i n 2 φ + c o s 2 θ c o s 2 φ = c o s 2 θ . G=(r_{\varphi},r_{\varphi})=cos^2\,\theta\,sin^2\varphi+cos^2\,\theta\,cos^2\varphi=cos^2\,\theta. G = ( r φ , r φ ) = co s 2 θ s i n 2 φ + co s 2 θ co s 2 φ = co s 2 θ .
I = d θ + c o s 2 θ d φ I=d\theta+cos^2\theta\,d\varphi I = d θ + co s 2 θ d φ
N = ∣ i j k − s i n θ c o s φ − s i n θ s i n φ c o s θ − c o s θ s i n φ c o s θ c o s φ 0 ∣ = i ∣ − s i n θ s i n φ c o s θ c o s θ c o s φ 0 ∣ − j ∣ − s i n θ c o s φ c o s θ − c o s θ s i n φ 0 ∣ + k ∣ − s i n θ c o s φ − s i n θ s i n φ − c o s θ s i n φ c o s θ c o s φ ∣ = \text{N}=\begin{vmatrix}
i & j &k \\
-sin\,\theta\,cos\varphi&-sin\,\theta\,sin\varphi&cos\,\theta\\
-cos\,\theta\,sin\varphi&cos\,\theta\,cos\varphi&0
\end{vmatrix}=i\begin{vmatrix}
-sin\,\theta\,sin\varphi&cos\,\theta\\
cos\,\theta\,cos\varphi&0
\end{vmatrix}-j\begin{vmatrix}
-sin\,\theta\,cos\varphi&cos\,\theta\\
-cos\,\theta\,sin\varphi&0
\end{vmatrix}+k\begin{vmatrix}
-sin\,\theta\,cos\varphi&-sin\,\theta\,sin\varphi\\
-cos\,\theta\,sin\varphi&cos\,\theta\,cos\varphi
\end{vmatrix}= N = ∣ ∣ i − s in θ cos φ − cos θ s in φ j − s in θ s in φ cos θ cos φ k cos θ 0 ∣ ∣ = i ∣ ∣ − s in θ s in φ cos θ cos φ cos θ 0 ∣ ∣ − j ∣ ∣ − s in θ cos φ − cos θ s in φ cos θ 0 ∣ ∣ + k ∣ ∣ − s in θ cos φ − cos θ s in φ − s in θ s in φ cos θ cos φ ∣ ∣ =
= − i c o s 2 θ c o s φ − j c o s θ s i n φ − k s i n θ c o s θ = ( − c o s 2 θ c o s φ , − c o s θ s i n φ , − s i n θ c o s θ ) , =-i\,cos^2\theta\,cos\,\varphi-j\,cos\,\theta\,sin\,\varphi-ksin\,\theta\,cos\,\theta=(-\,cos^2\theta\,cos\,\varphi,-\,cos\,\theta\,sin\,\varphi,-sin\,\theta\,cos\,\theta), = − i co s 2 θ cos φ − j cos θ s in φ − k s in θ cos θ = ( − co s 2 θ cos φ , − cos θ s in φ , − s in θ cos θ ) ,
The second fundamental form is:
I I = L d u 2 + 2 M d u d v + N d v 2 , II=Ldu^2+2Mdudv+Ndv^2, II = L d u 2 + 2 M d u d v + N d v 2 ,
where L = − ( r u , N u ) , M = − ( r v , N u ) , N = − ( r v , N v ) L=-(r_u,\text{N}_u),M=-(r_v,\text{N}_u),N=-(r_v,\text{N}_v) L = − ( r u , N u ) , M = − ( r v , N u ) , N = − ( r v , N v )
The Weingarten matrix is: ( M F − L G E G − F 2 L F − M E E G − F 2 N F − M G E G − F 2 M F − N E E G − F 2 ) \begin{pmatrix}
\frac{MF-LG}{EG-F^2} & \frac{LF-ME}{EG-F^2} \\
\frac{NF-MG}{EG-F^2} & \frac{MF-NE}{EG-F^2}
\end{pmatrix} ( EG − F 2 MF − L G EG − F 2 NF − MG EG − F 2 L F − ME EG − F 2 MF − NE )
b). r u = ( − s i n h u c o s h 2 u c o s v , − s i n h u c o s h 2 u s i n v , 1 c o s h 2 u ) , r_{u}=(-\frac{sinh\,u}{cosh^2\,u}cos\,v,-\frac{sinh\,u}{cosh^2\,u}sin\,v,\frac{1}{cosh^2u}), r u = ( − cos h 2 u s inh u cos v , − cos h 2 u s inh u s in v , cos h 2 u 1 ) ,
r v = ( − s e c h u s i n v , s e c h u c o s v , 0 ) , r_{v}=(-sech\,u\,sin\,v,sech\,u\,cos\,v,0), r v = ( − sec h u s in v , sec h u cos v , 0 ) ,
E = ( r u , r u ) = s i n h 2 u c o s h 4 u c o s 2 v + 1 c o s h 4 u = c o s 2 v c o s h 2 u E=(r_u,r_u)=\frac{sinh^2\,u}{cosh^4\,u}cos^2\,v+\frac{1}{cosh^4\,u}=\frac{cos^2v}{cosh^2u} E = ( r u , r u ) = cos h 4 u s in h 2 u co s 2 v + cos h 4 u 1 = cos h 2 u co s 2 v
F = 0 , G = ( r v , r v ) = 1 c o s h 2 u ; F=0, G=(r_v,r_v)=\frac{1}{cosh^2u}; F = 0 , G = ( r v , r v ) = cos h 2 u 1 ;
I = c o s 2 v c o s h 2 u d u + 1 c o s h 2 u d v I=\frac{cos^2v}{cosh^2u}du+\frac{1}{cosh^2u}dv I = cos h 2 u co s 2 v d u + cos h 2 u 1 d v
N = ∣ i j k − s i n h u c o s h 2 u c o s v − s i n h u c o s h 2 u s i n v 1 c o s h 2 u − s e c h u s i n v s e c h u c o s v 0 ∣ = i ∣ − s i n h u c o s h 2 u s i n v 1 c o s h 2 u s e c h u c o s v 0 ∣ − j ∣ − s i n h u c o s h 2 u c o s v 1 c o s h 2 u − s e c h u s i n v 0 ∣ + k ∣ − s i n h u c o s h 2 u c o s v − s i n h u c o s h 2 u s i n v − s e c h u s i n v s e c h u c o s v ∣ = \text{N}=\begin{vmatrix}
i & j &k \\
-\frac{sinh\,u}{cosh^2\,u}cos\,v&-\frac{sinh\,u}{cosh^2\,u}sin\,v&\frac{1}{cosh^2u}\\
-sech\,u\,sin\,v&sech\,u\,cos\,v&0
\end{vmatrix}=i\begin{vmatrix}
-\frac{sinh\,u}{cosh^2\,u}sin\,v&\frac{1}{cosh^2u}\\
sech\,u\,cos\,v&0
\end{vmatrix}-j\begin{vmatrix}
-\frac{sinh\,u}{cosh^2\,u}cos\,v&\frac{1}{cosh^2u}\\
-sech\,u\,sin\,v&0
\end{vmatrix}+k\begin{vmatrix}
-\frac{sinh\,u}{cosh^2\,u}cos\,v&-\frac{sinh\,u}{cosh^2\,u}sin\,v\\
-sech\,u\,sin\,v&sech\,u\,cos\,v
\end{vmatrix}= N = ∣ ∣ i − cos h 2 u s inh u cos v − sec h u s in v j − cos h 2 u s inh u s in v sec h u cos v k cos h 2 u 1 0 ∣ ∣ = i ∣ ∣ − cos h 2 u s inh u s in v sec h u cos v cos h 2 u 1 0 ∣ ∣ − j ∣ ∣ − cos h 2 u s inh u cos v − sec h u s in v cos h 2 u 1 0 ∣ ∣ + k ∣ ∣ − cos h 2 u s inh u cos v − sec h u s in v − cos h 2 u s inh u s in v sec h u cos v ∣ ∣ =
N = i c o s v c o s h 3 u + j s i n v c o s h 3 u − k s i n h u c o s h 3 u \text{N}=i\frac{cos\,v}{cosh^3u}+j\frac{sin\,v}{cosh^3u}-k\frac{sinh\,u}{cosh^3\,u} N = i cos h 3 u cos v + j cos h 3 u s in v − k cos h 3 u s inh u
The second fundamental form is:
I I = L d u 2 + 2 M d u d v + N d v 2 , II=Ldu^2+2Mdudv+Ndv^2, II = L d u 2 + 2 M d u d v + N d v 2 ,
where L = − ( r u , N u ) , M = − ( r v , N u ) , N = − ( r v , N v ) . L=-(r_u,\text{N}_u),M=-(r_v,\text{N}_u),N=-(r_v,\text{N}_v). L = − ( r u , N u ) , M = − ( r v , N u ) , N = − ( r v , N v ) .
The Weingarten Matrix is: ( M F − L G E G − F 2 L F − M E E G − F 2 N F − M G E G − F 2 M F − N E E G − F 2 ) \begin{pmatrix}
\frac{MF-LG}{EG-F^2} & \frac{LF-ME}{EG-F^2} \\
\frac{NF-MG}{EG-F^2} & \frac{MF-NE}{EG-F^2}
\end{pmatrix} ( EG − F 2 MF − L G EG − F 2 NF − MG EG − F 2 L F − ME EG − F 2 MF − NE )
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