§8.1 Gaussian Curvature of a Surface

8.1 Gaussian Curvature of a Surface

We will give a brief introduction to the curvature of a surface for heuristic intuition.

Suppose $𝑈 \subseteq ℝ^{2}$ is a region, and let $(𝑢, 𝑣) \in 𝑈$ . Consider a surface parameterized via

\vec{𝐫} (𝑢, 𝑣) = (𝑥 (𝑢, 𝑣), 𝑦 (𝑢, 𝑣), 𝑧 (𝑢, 𝑣)) \in ℝ^{3},

where $𝑥, 𝑦, 𝑧 \in 𝐶^{2} (𝑈)$ . If ${\vec{𝐫}}_{𝑢}^{'} \times {\vec{𝐫}}_{𝑣}^{'}$ never vanishes for $(𝑢, 𝑣) \in 𝑈$ , then $\vec{𝐫} (𝑈)$ defines a smooth surface $Σ$ . For a fixed $(𝑢, 𝑣) \in 𝑈$ , the vectors ${\vec{𝐫}}_{𝑢}^{'}$ and ${\vec{𝐫}}_{𝑣}^{'}$ form the basis of the tangent space (a plane) of $Σ$ at $𝑃 = \vec{𝐫} (𝑢, 𝑣)$ , denoted by $𝑇_{𝑃} Σ = span ({\vec{𝐫}}_{𝑢}^{'} (𝑃), {\vec{𝐫}}_{𝑣}^{'} (𝑃))$ .

The square of the length of the vector infinitesimal $d \vec{𝐫} = {\vec{𝐫}}_{𝑢}^{'} d 𝑢 + {\vec{𝐫}}_{𝑣}^{'} d 𝑣$ , or

I = d 𝑠^{2} = 𝐸 d 𝑢^{2} + 2 𝐹 d 𝑢 d 𝑣 + 𝐺 d 𝑣^{2}, (8.1 . 1)

is known as the first fundamental form of $Σ$ , where $𝐸 = {\vec{𝐫}}_{𝑢}^{'} \cdot {\vec{𝐫}}_{𝑢}^{'}$ , $𝐹 = {\vec{𝐫}}_{𝑢}^{'} \cdot {\vec{𝐫}}_{𝑣}^{'}$ , and $𝐺 = {\vec{𝐫}}_{𝑣}^{'} \cdot {\vec{𝐫}}_{𝑣}^{'}$ .

Let $𝑄 = \vec{𝐫} (𝑢 + Δ 𝑢, 𝑣 + Δ 𝑣)$ be near $𝑃$ . It follows that $\vec{𝑃 𝑄} = \vec{𝐫} (𝑢 + Δ 𝑢, 𝑣 + Δ 𝑣) - \vec{𝐫} (𝑢, 𝑣)$ . The distance between $𝑄$ and $𝑇_{𝑃} Σ$ is $\vec{𝑃 𝑄} \cdot \hat{𝐧}$ , where $\hat{𝐧} = \frac{{\vec{𝐫}}_{𝑢}^{'} \times {\vec{𝐫}}_{𝑣}^{'}}{‖ {\vec{𝐫}}_{𝑢}^{'} \times {\vec{𝐫}}_{𝑣}^{'} ‖}$ . By application of the multivariate Taylor’s Theorem, we have

\vec{𝑃 𝑄} = {\vec{𝐫}}_{𝑢}^{'} Δ 𝑢 + {\vec{𝐫}}_{𝑣}^{'} Δ 𝑣 + \frac{1}{2} ({\vec{𝐫}}_{𝑢 𝑢}^{″} Δ 𝑢^{2} + 2 {\vec{𝐫}}_{𝑢 𝑣}^{″} Δ 𝑢 Δ 𝑣 + {\vec{𝐫}}_{𝑣 𝑣}^{″} Δ 𝑣^{2}) + 𝒪︀ (Δ 𝑢^{3} + Δ 𝑣^{3}),

and therefore,

\vec{𝑃 𝑄} \cdot \hat{𝐧} = \frac{1}{2} ({\vec{𝐫}}_{𝑢 𝑢}^{″} \cdot \hat{𝐧} Δ 𝑢^{2} + 2 {\vec{𝐫}}_{𝑢 𝑣}^{″} \cdot \hat{𝐧} Δ 𝑢 Δ 𝑣 + {\vec{𝐫}}_{𝑣 𝑣}^{″} \cdot \hat{𝐧} Δ 𝑣^{2}) + 𝒪︀ (Δ 𝑢^{3} + Δ 𝑣^{3}),

where the first two linear terms vanished by properties of the scalar triple product. Omitting the higher order terms and the coefficient, the second fundamental form of $Σ$ is defined as

I I = 𝐿 d 𝑢^{2} + 2 𝑀 d 𝑢 d 𝑣 + 𝑁 d 𝑣^{2}, (8.1 . 2)

where $𝐿 = {\vec{𝐫}}_{𝑢 𝑢}^{″} \cdot \hat{𝐧}$ , $𝑀 = {\vec{𝐫}}_{𝑢 𝑣}^{″} \cdot \hat{𝐧}$ , and $𝑁 = {\vec{𝐫}}_{𝑣 𝑣}^{″} \cdot \hat{𝐧}$ . Since ${\vec{𝐫}}_{𝑢}^{'} \cdot \hat{𝐧} = 0$ and ${\vec{𝐫}}_{𝑣}^{'} \cdot \hat{𝐧} = 0$ , by differentiation, we have

\begin{aligned} {\vec{𝐫}}_{𝑢 𝑢}^{″} \cdot \hat{𝐧} + {\vec{𝐫}}_{𝑢}^{'} \cdot {\hat{𝐧}}_{𝑢}^{'} & = 0, & {\vec{𝐫}}_{𝑢 𝑣}^{″} \cdot \hat{𝐧} + {\vec{𝐫}}_{𝑢}^{'} \cdot {\hat{𝐧}}_{𝑣}^{'} & = 0, \\ {\vec{𝐫}}_{𝑢 𝑣}^{″} \cdot \hat{𝐧} + {\vec{𝐫}}_{𝑣}^{'} \cdot {\hat{𝐧}}_{𝑢}^{'} & = 0, & {\vec{𝐫}}_{𝑣 𝑣}^{″} \cdot \hat{𝐧} + {\vec{𝐫}}_{𝑣}^{'} \cdot {\hat{𝐧}}_{𝑣}^{'} & = 0 . \end{aligned}

It follows that $𝐿 = - {\vec{𝐫}}_{𝑢}^{'} \cdot {\hat{𝐧}}_{𝑢}^{'}$ , $𝑀 = - {\vec{𝐫}}_{𝑢}^{'} \cdot {\hat{𝐧}}_{𝑣}^{'} = - {\vec{𝐫}}_{𝑣}^{'} \cdot {\hat{𝐧}}_{𝑢}^{'}$ , and $𝑁 = - {\vec{𝐫}}_{𝑣}^{'} \cdot {\hat{𝐧}}_{𝑣}^{'}$ . Because $d \hat{𝐧} = {\hat{𝐧}}_{𝑢}^{'} d 𝑢 + {\hat{𝐧}}_{𝑣}^{'} d 𝑣$ ,

I I = - d \vec{𝐫} \cdot d \hat{𝐧} .

Figure 8.1:

𝑄

has a greater heuristic distance to

𝑇_{𝑃} Σ

for a more curved surface.

The second fundamental form, in a rough sense, measures the curvature of the surface $Σ$ at $𝑃$ (refer to Figure 8.1 $Figure 8.1 Figure 8.1$ ). Both the first and second fundamental forms are geometric invariants; they are independent of the parameterization $\vec{𝐫}$ of $Σ$ . The first fundamental form is also referred to as the intrinsic metric (we will not delve into the metric tensor here) of $Σ$ , and the second fundamental form is an extrinsic property of $Σ$ as it is invariant up to the orientation of the surface (consequent direction of the normal vector).

Let $𝛾 \subset Σ$ be a curve parameterized by arc length, $\vec{𝐫} (𝑠) = \vec{𝐫} (𝑢 (𝑠), 𝑣 (𝑠))$ . Then the unit tangent vector at $𝑃 = \vec{𝐫} (𝑠)$ is

\vec{𝐓} (𝑠) = \frac{d \vec{𝐫}}{d 𝑠} = {\vec{𝐫}}_{𝑢}^{'} \frac{d 𝑢}{d 𝑠} + {\vec{𝐫}}_{𝑣}^{'} \frac{d 𝑣}{d 𝑠} .

Consequently,

{\vec{𝐓}}^{'} (𝑠) = {\vec{𝐫}}_{𝑢 𝑢}^{″} {(\frac{d 𝑢}{d 𝑠})}^{2} + 2 {\vec{𝐫}}_{𝑢 𝑣}^{″} (\frac{d 𝑢}{d 𝑠}) (\frac{d 𝑣}{d 𝑠}) + {\vec{𝐫}}_{𝑣 𝑣}^{″} {(\frac{d 𝑣}{d 𝑠})}^{2} + {\vec{𝐫}}_{𝑢}^{'} \frac{d^{2} 𝑢}{d 𝑠^{2}} + {\vec{𝐫}}_{𝑣}^{'} \frac{d^{2} 𝑣}{d 𝑠^{2}},

where the last two terms are in $𝑇_{𝑃} Σ$ . Because $‖ \vec{𝐓} (𝑠) ‖ = 1$ for all $𝑠$ by the arc-length parameterization, we have

0 = \frac{d {‖ \vec{𝐓} (𝑠) ‖}^{2}}{d 𝑠} = \frac{d \vec{𝐓} (𝑠) \cdot \vec{𝐓} (𝑠)}{d 𝑠} = 2 \vec{𝐓} (𝑠) \cdot {\vec{𝐓}}^{'} (𝑠) .

Hence, $\vec{𝐓} (𝑠)$ and ${\vec{𝐓}}^{'} (𝑠)$ are orthogonal and ${\vec{𝐓}}^{'} (𝑠)$ is a normal to the curve $𝛾$ . Let $\hat{𝐧} = \frac{{\vec{𝐫}}_{𝑢}^{'} \times {\vec{𝐫}}_{𝑣}^{'}}{‖ {\vec{𝐫}}_{𝑢}^{'} \times {\vec{𝐫}}_{𝑣}^{'} ‖}$ be the unit normal to $Σ$ at $𝑃$ . The normal curvature of $𝛾$ at $𝑃$ in $Σ$ is defined as

𝜅_{𝑛} = {\vec{𝐓}}^{'} (𝑠) \cdot \hat{𝐧} = [{\vec{𝐫}}_{𝑢 𝑢}^{″} {(\frac{d 𝑢}{d 𝑠})}^{2} + 2 {\vec{𝐫}}_{𝑢 𝑣}^{″} (\frac{d 𝑢}{d 𝑠}) (\frac{d 𝑣}{d 𝑠}) + {\vec{𝐫}}_{𝑣 𝑣}^{″} {(\frac{d 𝑣}{d 𝑠})}^{2}] \cdot \hat{𝐧} .

The quotient

𝜅_{𝑛} = \frac{I I}{I} = \frac{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}},

varies depending on the curve traversing $Σ$ (and ultimately, depending on the direction induced by $d 𝑢$ and $d 𝑣$ ). On $𝛾$ , the two representations are equivalent since $I = d 𝑠^{2}$ . The maximum and minimum values of $𝜅_{𝑛}$ are known as the principal curvatures $𝜅_{1}$ and $𝜅_{2}$ of $Σ$ at $𝑃$ , achieved along the principal directions of the (unit) tangent vectors at $𝑃$ .

The mean curvature of $Σ$ at $𝑃$ is defined to be $𝐻 = \frac{𝜅_{1} + 𝜅_{2}}{2}$ . Let $𝑟_{1}, 𝑟_{2}$ be the radii of curvature corresponding to $𝜅_{1}$ and $𝜅_{2}$ . The product of the two principal curvatures is known as the Gaussian curvature of $Σ$ at $𝑃$ , denoted by $𝐾 = 𝜅_{1} 𝜅_{2}$ . We will now heuristically derive the explicit formulas for $𝐻$ and $𝐾$ in terms of $𝐸, 𝐹, 𝐺, 𝐿, 𝑀, 𝑁$ .

Suppose $𝑝 \in Σ$ . Adopt the matrix notation of $𝐈$ , $𝐈 𝐈$ as in

𝐈 = (\begin{matrix} 𝐸 & 𝐹 \\ 𝐹 & 𝐺 \end{matrix}), 𝐈 𝐈 = (\begin{matrix} 𝐿 & 𝑀 \\ 𝑀 & 𝑁 \end{matrix}),

to reduce to the optimization problem of

𝜅_{𝑛} = \frac{{\vec{𝐯}}^{⊤} 𝐈 𝐈 \vec{𝐯}}{{\vec{𝐯}}^{⊤} 𝐈 \vec{𝐯}}, \vec{𝐯} \in 𝑇_{𝑝} Σ .

We may restrict $\vec{𝐯} = (𝑣_{1}, 𝑣_{2})$ so that the denominator is always $1$ , aiming to optimize the numerator. By the method of Lagrange multipliers, we write

ℒ︀ (\vec{𝐯}, 𝜆) = {\vec{𝐯}}^{⊤} 𝐈 𝐈 \vec{𝐯} - 𝜆 ({\vec{𝐯}}^{⊤} 𝐈 \vec{𝐯} - 1) .

The equation $𝛁 ℒ︀ = 𝟎$ for $𝛁 = (\frac{𝜕}{𝜕 𝑣_{1}}, \frac{𝜕}{𝜕 𝑣_{2}}, \frac{𝜕}{𝜕 𝜆})$ can then be decomposed into (where $\vec{𝐯} = (𝑣_{1}, 𝑣_{2})$ ):

\begin{matrix} 2 𝐿 𝑣_{1} + 2 𝑀 𝑣_{2} - 𝜆 (2 𝐸 𝑣_{1} + 2 𝐹 𝑣_{2}) = 0, \\ 2 𝑀 𝑣_{1} + 2 𝑁 𝑣_{2} - 𝜆 (2 𝐹 𝑣_{1} + 2 𝐺 𝑣_{2}) = 0, \\ ({\vec{𝐯}}^{⊤} 𝐈 \vec{𝐯} = 1) . \end{matrix}

The first two equations can be written as

(\begin{matrix} 𝐿 - 𝜆 𝐸 & 𝑀 - 𝜆 𝐹 \\ 𝑀 - 𝜆 𝐹 & 𝑁 - 𝜆 𝐺 \end{matrix}) \vec{𝐯} = 𝟎 . (8.1 . 3)

Let the matrix on the left be denoted by $𝐌$ . In order for non-trivial $(𝐯 \neq 𝟎)$ to exist, we must have $det 𝐌 = 0$ . That is,

\begin{aligned} (𝐿 - 𝜆 𝐸) (𝑁 - 𝜆 𝐺) - {(𝑀 - 𝜆 𝐹)}^{2} \\ \begin{matrix}  \end{matrix} = 𝜆^{2} (𝐸 𝐺 - 𝐹^{2}) + 𝜆 (2 𝑀 𝐹 - 𝐸 𝑁 - 𝐺 𝐿) + 𝐿 𝑁 - 𝑀^{2} = 0 . \end{aligned}

This is a quadratic giving two solutions for $𝜆$ . From

𝛁 ({\vec{𝐯}}^{⊤} 𝐈 𝐈 \vec{𝐯}) = 𝜆 𝛁 ({\vec{𝐯}}^{⊤} 𝐈 \vec{𝐯})

it is apparent that the roots $𝜆_{1}, 𝜆_{2} \in ℝ$ . Moreover, from (8.1.3) $(8.1.3) (8.1.3)$ we have

𝐈 𝐈 \vec{𝐯} = 𝜆 𝐈 \vec{𝐯} ⟹ 𝜆 = \frac{{\vec{𝐯}}^{⊤} 𝐈 𝐈 \vec{𝐯}}{{\vec{𝐯}}^{⊤} 𝐈 \vec{𝐯}} .

Hence, the two roots $𝜆_{1}, 𝜆_{2}$ are precisely the principal curvatures. Vieta’s formulas give that

𝐾 = 𝜆_{1} 𝜆_{2} = \frac{𝐿 𝑁 - 𝑀^{2}}{𝐸 𝐺 - 𝐹^{2}}, 𝐻 = \frac{𝐸 𝑁 + 𝐺 𝐿 - 2 𝑀 𝐹}{2 𝐸 𝐺 - 2 𝐹^{2}} .

Now, assume a parameterization of $Σ$ by $\vec{𝐫} (𝑢, 𝑣)$ (thrice continuously differentiable) such that

I (𝑢, 𝑣) = 𝜌^{2} d 𝑢^{2} + 𝜌^{2} d 𝑣^{2} = 𝜌^{2} (d 𝑢^{2} + d 𝑣^{2})

(which we will later formalize as a conformal metric). Then there is an alternate representation of the Gaussian curvature in terms of $𝜌$ .

By definition, $𝐸 \equiv 𝐺 \equiv 𝜌^{2}$ while $𝐹 \equiv 0$ . Moreover,

\begin{aligned} 𝐿 𝑁 & = ({\vec{𝐫}}_{𝑢 𝑢}^{″} \cdot \frac{{\vec{𝐫}}_{𝑢}^{'} \times {\vec{𝐫}}_{𝑣}^{'}}{‖ {\vec{𝐫}}_{𝑢}^{'} \times {\vec{𝐫}}_{𝑣}^{'} ‖}) ({\vec{𝐫}}_{𝑣 𝑣}^{″} \cdot \frac{{\vec{𝐫}}_{𝑢}^{'} \times {\vec{𝐫}}_{𝑣}^{'}}{‖ {\vec{𝐫}}_{𝑢}^{'} \times {\vec{𝐫}}_{𝑣}^{'} ‖}) \\ = \frac{det (\begin{matrix} {\vec{𝐫}}_{𝑢 𝑢}^{″} & {\vec{𝐫}}_{𝑢}^{'} & {\vec{𝐫}}_{𝑣}^{'} \end{matrix}) det (\begin{matrix} {\vec{𝐫}}_{𝑣 𝑣}^{″} & {\vec{𝐫}}_{𝑢}^{'} & {\vec{𝐫}}_{𝑣}^{'} \end{matrix})}{{‖ {\vec{𝐫}}_{𝑢}^{'} ‖}^{2} {‖ {\vec{𝐫}}_{𝑣}^{'} ‖}^{2} - {({\vec{𝐫}}_{𝑢}^{'} \cdot {\vec{𝐫}}_{𝑣}^{'})}^{2}} \\ = \frac{det (\begin{matrix} {\vec{𝐫}}_{𝑢 𝑢}^{″} & {\vec{𝐫}}_{𝑢}^{'} & {\vec{𝐫}}_{𝑣}^{'} \end{matrix}) det (\begin{matrix} {\vec{𝐫}}_{𝑣 𝑣}^{″} & {\vec{𝐫}}_{𝑢}^{'} & {\vec{𝐫}}_{𝑣}^{'} \end{matrix})}{𝐸 𝐺 - 𝐹^{2}} \\ = \frac{1}{𝜌^{4}} det (\begin{matrix} {\vec{𝐫}}_{𝑣 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑢 𝑢}^{″} & {\vec{𝐫}}_{𝑣 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑢}^{'} & {\vec{𝐫}}_{𝑣 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑣}^{'} \\ {\vec{𝐫}}_{𝑢}^{'} \cdot {\vec{𝐫}}_{𝑢 𝑢}^{″} & {\vec{𝐫}}_{𝑢}^{'} \cdot {\vec{𝐫}}_{𝑢}^{'} & {\vec{𝐫}}_{𝑢}^{'} \cdot {\vec{𝐫}}_{𝑣}^{'} \\ {\vec{𝐫}}_{𝑣}^{'} \cdot {\vec{𝐫}}_{𝑢 𝑢}^{″} & {\vec{𝐫}}_{𝑣}^{'} \cdot {\vec{𝐫}}_{𝑢}^{'} & {\vec{𝐫}}_{𝑣}^{'} \cdot {\vec{𝐫}}_{𝑣}^{'} \end{matrix}) . \end{aligned}

Similarly,

\begin{aligned} 𝑀^{2} & = \frac{1}{𝜌^{4}} det (\begin{matrix} {\vec{𝐫}}_{𝑢 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑢 𝑣}^{″} & 𝜌 𝜌_{𝑣}^{'} & 𝜌 𝜌_{𝑢}^{'} \\ 𝜌 𝜌_{𝑣}^{'} & 𝜌^{2} & 0 \\ 𝜌 𝜌_{𝑢}^{'} & 0 & 𝜌^{2} \end{matrix}) \\ = \frac{1}{𝜌^{4}} [{\vec{𝐫}}_{𝑢 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑢 𝑣}^{″} 𝜌^{4} - 𝜌^{4} {(𝜌_{𝑣}^{'})}^{2} - 𝜌^{4} {(𝜌_{𝑢}^{'})}^{2}] \\ = {\vec{𝐫}}_{𝑢 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑢 𝑣}^{″} - {(𝜌_{𝑣}^{'})}^{2} - {(𝜌_{𝑢}^{'})}^{2} . \end{aligned}

By differentiation of the equations

{\vec{𝐫}}_{𝑢}^{'} \cdot {\vec{𝐫}}_{𝑣}^{'} \equiv 𝐹 \equiv 0, {\vec{𝐫}}_{𝑢}^{'} \cdot {\vec{𝐫}}_{𝑢}^{'} \equiv 𝐸 \equiv 𝐺 \equiv {\vec{𝐫}}_{𝑣}^{'} \cdot {\vec{𝐫}}_{𝑣}^{'} \equiv 𝜌^{2},

we have

{\vec{𝐫}}_{𝑢 𝑢}^{″} \cdot {\vec{𝐫}}_{𝑣}^{'} + {\vec{𝐫}}_{𝑢}^{'} \cdot {\vec{𝐫}}_{𝑢 𝑣}^{″} \equiv 0, {\vec{𝐫}}_{𝑢 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑣}^{'} + {\vec{𝐫}}_{𝑢}^{'} \cdot {\vec{𝐫}}_{𝑣 𝑣}^{″} \equiv 0, (8.1 . 4)

and

2 {\vec{𝐫}}_{𝑢 𝑢}^{″} \cdot {\vec{𝐫}}_{𝑢}^{'} \equiv 2 𝜌 𝜌_{𝑢}^{'} \equiv 2 {\vec{𝐫}}_{𝑢 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑣}^{'}, 2 {\vec{𝐫}}_{𝑢 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑢}^{'} \equiv 2 𝜌 𝜌_{𝑣}^{'} \equiv 2 {\vec{𝐫}}_{𝑣 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑣}^{'} . (8.1 . 5)

Substituting (8.1.5) $(8.1.5) (8.1.5)$ into (8.1.4) $(8.1.4) (8.1.4)$ then gives

{\vec{𝐫}}_{𝑢 𝑢}^{″} \cdot {\vec{𝐫}}_{𝑣}^{'} = - 𝜌 𝜌_{𝑣}^{'}, ({\vec{𝐫}}_{𝑣 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑢}^{'} = - 𝜌 𝜌_{𝑢}^{'}) .

Differentiating these give

\begin{matrix} {\vec{𝐫}}_{𝑢 𝑢 𝑣}^{‴} \cdot {\vec{𝐫}}_{𝑣}^{'} + {\vec{𝐫}}_{𝑢 𝑢}^{″} \cdot {\vec{𝐫}}_{𝑣 𝑣} = - {(𝜌_{𝑣}^{'})}^{2} - 𝜌 𝜌_{𝑣 𝑣}^{″}, \\ ({\vec{𝐫}}_{𝑣 𝑣 𝑢}^{‴} \cdot {\vec{𝐫}}_{𝑢}^{'} + {\vec{𝐫}}_{𝑣 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑢 𝑢} = - {(𝜌_{𝑢}^{'})}^{2} - 𝜌 𝜌_{𝑢 𝑢}^{″}) . \end{matrix}

Differentiating the inner two expressions of (8.1.5) $(8.1.5) (8.1.5)$ , we have

\begin{matrix} {\vec{𝐫}}_{𝑢 𝑢 𝑣}^{‴} \cdot {\vec{𝐫}}_{𝑣}^{'} + {\vec{𝐫}}_{𝑢 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑢 𝑣} = {(𝜌_{𝑢}^{'})}^{2} + 𝜌 𝜌_{𝑢 𝑢}^{″}, \\ ({\vec{𝐫}}_{𝑣 𝑣 𝑢}^{‴} \cdot {\vec{𝐫}}_{𝑢}^{'} + {\vec{𝐫}}_{𝑢 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑢 𝑣} = {(𝜌_{𝑣}^{'})}^{2} + 𝜌 𝜌_{𝑣 𝑣}^{″}) . \end{matrix}

It follows that

{\vec{𝐫}}_{𝑢 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑢 𝑣} - {\vec{𝐫}}_{𝑢 𝑢}^{″} \cdot {\vec{𝐫}}_{𝑣 𝑣}^{″} = {(𝜌_{𝑢}^{'})}^{2} + {(𝜌_{𝑣}^{'})}^{2} + 𝜌 \nabla^{2} 𝜌,

where $\nabla^{2}$ here is $𝜕^{2} / 𝜕 𝑢^{2} + 𝜕^{2} / 𝜕 𝑣^{2}$ . Then

\begin{aligned} 𝐿 𝑁 & = \frac{1}{𝜌^{4}} det (\begin{matrix} {\vec{𝐫}}_{𝑣 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑢 𝑢}^{″} & - 𝜌 𝜌_{𝑢}^{'} & 𝜌 𝜌_{𝑣}^{'} \\ 𝜌 𝜌_{𝑢}^{'} & 𝜌^{2} & 0 \\ - 𝜌 𝜌_{𝑣}^{'} & 0 & 𝜌^{2} \end{matrix}) \\ = \frac{1}{𝜌^{4}} [{\vec{𝐫}}_{𝑣 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑢 𝑢}^{″} 𝜌^{4} + 𝜌^{4} {(𝜌_{𝑢}^{'})}^{2} + 𝜌^{4} {(𝜌_{𝑣}^{'})}^{2}] \\ = {\vec{𝐫}}_{𝑣 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑢 𝑢}^{″} + {(𝜌_{𝑢}^{'})}^{2} + {(𝜌_{𝑣}^{'})}^{2}, \end{aligned}

and

\begin{aligned} 𝑀^{2} & = \frac{1}{𝜌^{4}} det (\begin{matrix} {\vec{𝐫}}_{𝑢 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑢 𝑣}^{″} & 𝜌 𝜌_{𝑣}^{'} & 𝜌 𝜌_{𝑢}^{'} \\ 𝜌 𝜌_{𝑣}^{'} & 𝜌^{2} & 0 \\ 𝜌 𝜌_{𝑢}^{'} & 0 & 𝜌^{2} \end{matrix}) \\ = \frac{1}{𝜌^{4}} [{\vec{𝐫}}_{𝑢 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑢 𝑣}^{″} 𝜌^{4} - 𝜌^{4} {(𝜌_{𝑣}^{'})}^{2} - 𝜌^{4} {(𝜌_{𝑢}^{'})}^{2}] \\ = {\vec{𝐫}}_{𝑢 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑢 𝑣}^{″} - {(𝜌_{𝑣}^{'})}^{2} - {(𝜌_{𝑢}^{'})}^{2} . \end{aligned}

Combining the two expressions, we have

\begin{aligned} 𝐾 & = \frac{𝐿 𝑁 - 𝑀^{2}}{𝐸 𝐺 - 𝐹^{2}} = \frac{{\vec{𝐫}}_{𝑣 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑢 𝑢}^{″} + 2 {(𝜌_{𝑢}^{'})}^{2} + 2 {(𝜌_{𝑣}^{'})}^{2} - {\vec{𝐫}}_{𝑢 𝑣}^{″} \cdot {\vec{𝐫}}_{𝑢 𝑣}^{″}}{𝜌^{4}} \\ = \frac{{(𝜌_{𝑢}^{'})}^{2} + {(𝜌_{𝑣}^{'})}^{2} - 𝜌 \nabla^{2} 𝜌}{𝜌^{4}} = - \frac{1}{𝜌^{2}} \nabla^{2} (log 𝜌) . (8.1 . 6) \end{aligned}

To understand the motivation for which $d 𝑠^{2}$ is said to be conformal, consider two curves in the $𝑢 𝑣$ -plane, parameterized by $𝜸_{1} (𝑡) = (𝑢_{1} (𝑡), 𝑣_{1} (𝑡))$ and $𝜸_{2} (𝑡) = (𝑢_{2} (𝑡), 𝑣_{2} (𝑡))$ such that $𝜸_{1} (0) = 𝜸_{2} (0) = 𝐰_{0} = (𝑢_{0}, 𝑣_{0})$ . Their images via $𝐫$ are $𝛂_{1} (𝑡) = 𝐫 \circ 𝜸_{1} (𝑡)$ and $𝛂_{2} (𝑡) = 𝐫 \circ 𝜸_{2} (𝑡)$ so that they intersect at some point $𝑃 \in Σ$ . Let $𝜸_{1}^{'} (0) = 𝐯_{1} = 𝑎 𝐞_{𝑢} + 𝑏 𝐞_{𝑣}$ and $𝜸_{2}^{'} (0) = 𝐯_{2} = 𝑐 𝐞_{𝑢} + 𝑑 𝐞_{𝑣}$ be two tangent vectors. Then the corresponding vectors in $𝑇_{𝑃} Σ$ are

d 𝐫_{𝐰_{0}} (𝐯_{1}) = 𝑎 d 𝐫_{𝐰_{0}} (𝐞_{𝑢}) + 𝑏 d 𝐫_{𝐰_{0}} (𝐞_{𝑣})

and

d 𝐫_{𝐰_{0}} (𝐯_{2}) = 𝑐 d 𝐫_{𝐰_{0}} (𝐞_{𝑢}) + 𝑑 d 𝐫_{𝐰_{0}} (𝐞_{𝑣}) .

Since¹ $𝐞_{𝑢} = \frac{𝜕}{𝜕 𝑢}$ , $𝐞_{𝑣} = \frac{𝜕}{𝜕 𝑣}$ and $d 𝐫_{𝐰_{0}} (\frac{𝜕}{𝜕 𝑢}) = {\frac{𝜕 𝐫}{𝜕 𝑢} |}_{𝐰_{0}}$ , $d 𝐫_{𝐰_{0}} (\frac{𝜕}{𝜕 𝑣}) = {\frac{𝜕 𝐫}{𝜕 𝑣} |}_{𝐰_{0}}$ . Then

d 𝐫_{𝐰_{0}} (𝐯_{1}) = 𝑎 𝐫_{𝑢}^{'} (𝐰_{0}) + 𝑏 𝐫_{𝑣}^{'} (𝐰_{0}), d 𝐫_{𝐰_{0}} (𝐯_{2}) = 𝑐 𝐫_{𝑢}^{'} (𝐰_{0}) + 𝑑 𝐫_{𝑣}^{'} (𝐰_{0}) .

The angle $𝜃_{𝑢 𝑣}$ between $𝐯_{1}$ and $𝐯_{2}$ on the $𝑢 𝑣$ -plane satisfies

cos 𝜃_{𝑢 𝑣} = \frac{𝐯_{1} \cdot 𝐯_{2}}{‖ 𝐯_{1} ‖ ‖ 𝐯_{2} ‖} = \frac{𝑎 𝑐 + 𝑏 𝑑}{\sqrt{𝑎^{2} + 𝑏^{2}} \sqrt{𝑐^{2} + 𝑑^{2}}},

while the angle $𝜃_{Σ}$ between the two tangent vectors in $𝑇_{𝑃} Σ$ satisfies

\begin{aligned} cos 𝜃_{Σ} & = {\frac{(𝑎 𝐫_{𝑢}^{'} + 𝑏 𝐫_{𝑣}^{'}) \cdot (𝑐 𝐫_{𝑢}^{'} + 𝑑 𝐫_{𝑣}^{'})}{‖ 𝑎 𝐫_{𝑢}^{'} + 𝑏 𝐫_{𝑣}^{'} ‖ ‖ 𝑐 𝐫_{𝑢}^{'} + 𝑑 𝐫_{𝑣}^{'} ‖} |}_{𝐰_{0}} \\ = {\frac{𝜌^{2} (𝑎 𝑐 + 𝑏 𝑑)}{\sqrt{𝑎^{2} 𝜌^{2} + 𝑏^{2} 𝜌^{2}} \sqrt{𝑐^{2} 𝜌^{2} + 𝑑^{2} 𝜌^{2}}} |}_{𝐰_{0}} = cos 𝜃_{𝑢 𝑣} . \end{aligned}