Gaussian and Mean Curvature of a Graph Surface¶

Let a surface in Euclidean three-space be the graph

$$ X(x,y)=(x,y,f(x,y)). $$

This notebook derives both fundamental forms and then computes the Gaussian curvature $K$ and mean curvature $H$. Egison keeps the derivatives of the arbitrary function $f$ symbolic.

Tangent plane¶

The coordinate tangent vectors are

$$ X_x=(1,0,f_x),\qquad X_y=(0,1,f_y). $$

Their cross product is $(-f_x,-f_y,1)$, so its norm is $W=\sqrt{1+f_x^2+f_y^2}$.

declare symbol x, y: MathValue

def f : MathValue := function (x, y)
def X : Vector MathValue := [| x, y, f x y |]

def vx : Vector MathValue := [| 1, 0, ∂/∂ (f x y) x |]
def vy : Vector MathValue := [| 0, 1, ∂/∂ (f x y) y |]

vx

$\begin{pmatrix} 1 \\ 0 \\ \frac{\partial f}{\partial 1}\\ \end{pmatrix}$

vy

$\begin{pmatrix} 0 \\ 1 \\ \frac{\partial f}{\partial 2}\\ \end{pmatrix}$

Oriented unit normal¶

We choose the upward-pointing normal

$$ n=\frac{X_x\times X_y}{\lVert X_x\times X_y\rVert}. $$

Reversing this orientation changes the sign of $H$ but not of $K$.

def normalNumerator : Vector MathValue := crossProduct vx vy
def W : MathValue := sqrt (V.* normalNumerator normalNumerator)
def normal : Vector MathValue := normalNumerator / W

normalNumerator

$\begin{pmatrix} -\frac{\partial f}{\partial 1} \\ -\frac{\partial f}{\partial 2} \\ 1\\ \end{pmatrix}$

normal

$\begin{pmatrix} -\sqrt{\frac{\partial f}{\partial 2}^{2} + \frac{\partial f}{\partial 1}^{2} + 1}^{-1} \frac{\partial f}{\partial 1} \\ -\sqrt{\frac{\partial f}{\partial 2}^{2} + \frac{\partial f}{\partial 1}^{2} + 1}^{-1} \frac{\partial f}{\partial 2} \\ \sqrt{\frac{\partial f}{\partial 2}^{2} + \frac{\partial f}{\partial 1}^{2} + 1}^{-1}\\ \end{pmatrix}$

First and second fundamental forms¶

With

$$ I=E\,dx^2+2F\,dx\,dy+G\,dy^2, \qquad II=L\,dx^2+2M\,dx\,dy+N\,dy^2, $$

the coefficients are dot products of tangent vectors and derivatives of tangent vectors with the chosen normal.

def E : MathValue := V.* vx vx
def F : MathValue := V.* vx vy
def G : MathValue := V.* vy vy

def L : MathValue := V.* (∂/∂ vx x) normal
def M : MathValue := V.* (∂/∂ vx y) normal
def N : MathValue := V.* (∂/∂ vy y) normal

(E, F, G)

$(\frac{\partial f}{\partial 1}^{2} + 1, \frac{\partial f}{\partial 1} \frac{\partial f}{\partial 2}, \frac{\partial f}{\partial 2}^{2} + 1)$

(L, M, N)

$(\sqrt{\frac{\partial f}{\partial 2}^{2} + \frac{\partial f}{\partial 1}^{2} + 1}^{-1} \frac{\partial^2 f}{\partial 1^2}, \sqrt{\frac{\partial f}{\partial 2}^{2} + \frac{\partial f}{\partial 1}^{2} + 1}^{-1} \frac{\partial^2 f}{\partial 1 \partial 2}, \sqrt{\frac{\partial f}{\partial 2}^{2} + \frac{\partial f}{\partial 1}^{2} + 1}^{-1} \frac{\partial^2 f}{\partial 2^2})$

Curvature¶

The determinant and trace of the shape operator give

$$ K=\frac{LN-M^2}{EG-F^2},\qquad H=\frac{EN-2FM+GL}{2(EG-F^2)}. $$

These formulas apply wherever the graph chart is regular. For a graph, $EG-F^2=1+f_x^2+f_y^2$ is always positive.

def K : MathValue := (L * N - M^2) / (E * G - F^2)
def H : MathValue := (E * N - 2 * F * M + G * L) / (2 * (E * G - F^2))

$\frac{-\frac{\partial^2 f}{\partial 1 \partial 2}^{2} + \frac{\partial^2 f}{\partial 1^2} \frac{\partial^2 f}{\partial 2^2}}{\frac{\partial f}{\partial 2}^{4} + \frac{\partial f}{\partial 1}^{4} + 2 \frac{\partial f}{\partial 1}^{2} \frac{\partial f}{\partial 2}^{2} + 2 \frac{\partial f}{\partial 2}^{2} + 2 \frac{\partial f}{\partial 1}^{2} + 1}$

$\frac{\sqrt{\frac{\partial f}{\partial 2}^{2} + \frac{\partial f}{\partial 1}^{2} + 1}^{-1} \frac{\partial^2 f}{\partial 1^2} \frac{\partial f}{\partial 2}^{2} + \sqrt{\frac{\partial f}{\partial 2}^{2} + \frac{\partial f}{\partial 1}^{2} + 1}^{-1} \frac{\partial f}{\partial 1}^{2} \frac{\partial^2 f}{\partial 2^2} - 2 \sqrt{\frac{\partial f}{\partial 2}^{2} + \frac{\partial f}{\partial 1}^{2} + 1}^{-1} \frac{\partial f}{\partial 1} \frac{\partial^2 f}{\partial 1 \partial 2} \frac{\partial f}{\partial 2} + \sqrt{\frac{\partial f}{\partial 2}^{2} + \frac{\partial f}{\partial 1}^{2} + 1}^{-1} \frac{\partial^2 f}{\partial 2^2} + \sqrt{\frac{\partial f}{\partial 2}^{2} + \frac{\partial f}{\partial 1}^{2} + 1}^{-1} \frac{\partial^2 f}{\partial 1^2}}{2 \frac{\partial f}{\partial 2}^{2} + 2 \frac{\partial f}{\partial 1}^{2} + 2}$

Interpretation¶

$K$ is intrinsic: it can be recovered entirely from distances measured on the surface. Positive, zero, and negative values correspond locally to elliptic, parabolic, and hyperbolic behavior. $H$ is extrinsic and depends on the embedding and orientation; the equation $H=0$ is the minimal-surface equation for a graph.