Parallel surface

Question

For a regular surface $\mathbf{x} = \mathbf{x}(u,v)$

Define $\mathbf{y}(u,v) = \mathbf{x}(u,v) + t \mathbf{N} (u,v)$ where $\mathbf{N}$ is the unit normal of $\mathbf{x}$

How could I show the following? $\mathbf{y}_u \times \mathbf{y}_v = (1-2Ht+Kt^2)\mathbf{x}_u \times \mathbf{x}_v$

where $H$ is the mean curvature of $\mathbf{x}$ and $K$ is its Gaussian curvature

Answer 1

Fundamental Forms $$ \begin{pmatrix} E & F \\ F & G \end{pmatrix}= \begin{pmatrix} \mathbf{x}_{u} \\ \mathbf{x}_{v} \end{pmatrix} \begin{pmatrix} \mathbf{x}_{u} & \mathbf{x}_{v} \end{pmatrix} $$

$$ \begin{pmatrix} e & f \\ f & g \end{pmatrix}= -\begin{pmatrix} \mathbf{N}_{u} \\ \mathbf{N}_{v} \end{pmatrix} \begin{pmatrix} \mathbf{x}_{u} & \mathbf{x}_{v} \end{pmatrix} $$

Let $\begin{pmatrix} \mathbf{N}_{u} \\ \mathbf{N}_{v} \end{pmatrix}= \mathbb{A} \begin{pmatrix} \mathbf{x}_{u} \\ \mathbf{x}_{v} \end{pmatrix}$ where $\mathbb{A}= \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$.

Now \begin{align*} \begin{pmatrix} \mathbf{N}_{u} \\ \mathbf{N}_{v} \end{pmatrix} \begin{pmatrix} \mathbf{x}_{u} & \mathbf{x}_{v} \end{pmatrix} &= \mathbb{A} \begin{pmatrix} \mathbf{x}_{u} \\ \mathbf{x}_{v} \end{pmatrix} \begin{pmatrix} \mathbf{x}_{u} & \mathbf{x}_{v} \end{pmatrix} \\ -\begin{pmatrix} e & f \\ f & g \end{pmatrix} &= \mathbb{A} \begin{pmatrix} E & F \\ F & G \end{pmatrix} \\ \mathbb{A} &= -\begin{pmatrix} e & f \\ f & g \end{pmatrix} \begin{pmatrix} E & F \\ F & G \end{pmatrix}^{-1} \end{align*}

The principal curvatures $k_{1}, k_{2}$ are the eigenvalues of $-\mathbb{A}$. That is

$$H=\frac{k_{1}+k_{2}}{2}=-\frac{1}{2} \operatorname{tr} \mathbb{A}$$ $$K=k_{1} k_{2}=(-1)^{2} \det \mathbb{A}$$

\begin{align*} \mathbf{y}_{u} &= \mathbf{x}_{u}+t\mathbf{N}_{u} \\ &= \mathbf{x}_{u}+t(a_{11} \mathbf{x}_{u}+a_{12} \mathbf{x}_{v}) \\ \mathbf{y}_{v} &= \mathbf{x}_{v}+t\mathbf{N}_{v} \\ &= \mathbf{x}_{v}+t(a_{21} \mathbf{x}_{u}+a_{22} \mathbf{x}_{v}) \\ \mathbf{y}_{u} \times \mathbf{y}_{v} &= [1+t(a_{11}+a_{22})+t^{2}(a_{11}a_{22}-a_{12}a_{21})] \mathbf{x}_{u} \times \mathbf{x}_{v} \\ &= (1-2Ht+Kt^{2}) \mathbf{x}_{u} \times \mathbf{x}_{v} \end{align*}