Smooth Surfaces

Regular Surface

We focus our study on differentiable surfaces embedded in 3D space, also known as regular surfaces in $\mathbb{R}^3$.

Definition: A regular 2-surface embedded into $\mathbb{R}^3$ is a subset $\mathcal{M} \subset \mathbb{R}^3$. For each point $p \in \mathcal{M}$ there is an open neighborhood $p \in V_{p} \subset \mathbb{R}^3$ and a map $f: U \to V_p \cap \mathcal{M}$ of an open set $U \subset \mathbb{R}^2$ onto $V_p \cap \mathcal{M}$ such that:

1. The map $f$ is differentiable.
2. $f$ is a homeomorphism.
3. For each $q \in U$, the differential $df_q: \mathbb{R}^2 \to \mathbb{R}^3$ is one-to-one. The map $f$ is called a local parametrization (or charts) of the surface. Just like texture maps on a mesh, $f$ maps a 2D piece onto $\mathcal{M}$ in a continuous way. This property ensures that such maps can be combined together to form a set of parametrizations into an atlas over the whole surface.

Differential

Consider a local parametrization $f(s,t)$ where $s$ and $t$ are the 2D parametrized coordinates. At a point $q \in \mathbb{R}^2$ such that $f(q) \in \mathcal{M}$, the differential $df_q$ is represented by the Jacobian of $f(s,t)$:

$df_q = \begin{pmatrix}\partial_s f_1 && \partial_t f_1 \\ \partial_s f_2 && \partial_t f_2\\ \partial_s f_3 && \partial_t f_3 \\ \end{pmatrix}$

The Jacobian maps directions in the parameter space to tangent vectors in the tangent space $T_{f(q)}\mathcal{M}$ at the point $f(q)$.

Using the differential, we can measure the inner product (i.e. the angle) betwen any two tangent vectors $df(u)$ and $df(v)$ and get the metric $g$ on the surface $\mathcal{M}$:

$g(u, v) = df(u) \cdot df(v)$

Curvature of Surfaces

Curvature quantifies the bendiness and stretchy-ness of surfaces in 3D.

We use curves on surfaces to define curvatures on surfaces. Intuitively, following along a curve on a surface, we can sense the change in acceleration caused by the bumps of the underlying surface. Such change in acceleration along the curve can help us identify the curvature of the surface.

We first define a map $n: M \to S^2$ that maps from a point on the surface to the normal vector at that point. The differential of this map $dn_p$ at point $p$ is a map from the tangent space $T_p(\mathcal{M})$ on our surface to the tangent space of the unit sphere $S^2$. This differential is known as the shape operator. Intuitively, given a vector $v$ in the tangent plane $T_p(\mathcal{M})$, the differential $dn_p$ measures the change in the normal of our surface when moving along direction $v$ on the surface.

More formally, given a vector $v \in T_{p}(\mathcal{M})$, take any curve $\gamma(t): (-\epsilon, \epsilon) \to \mathcal{M}$ such that $\gamma(0) = p$ and $\gamma'(0) = v$. By definition of the differential, $dn_p(v) = (n_p \circ v)' (0)$.

Note that the change in normal $dn_p(v)$ along any direction $v$ is always orthogonal to the normal $n_p$, and thus $dn_p(v) \in T_p(\mathcal{M})$.

Second Fundamental Form

Using the differential $dn_p$ we can define the second fundamental form $\mathrm{II}$:

$\mathrm{II}(v, w) = -v \cdot dn_p(w)$

The second fundamental form takes in two vectors $v,w \in T_{p}(\mathcal{M})$ at the tangent space at point $p$, and outputs a scalar value, which is the dot product between vector $v$ and the change of normal when moving along $w$.

Consider the case when $v=w$, then the second fundamental form is $\mathrm{II} = -v \cdot d n_p (v)$. We show that the second fundamental form is related to the principal curvature of the surface. Let $\gamma: (-\epsilon, \epsilon) \to M$ be a curve parametrized by arc length such that $\gamma(0) = p$ and $\gamma'(0) = v$. Since the tangent and normal of the curve are orthogonal to each other, $\gamma'(s) \cdot n(\gamma(s)) = 0$. Taking the derivative of this equation, $\partial_s \left(\gamma'(s) \cdot n(\gamma(s))\right) = k n(s) \cdot n(\gamma(s)) + \gamma'(s) dn_{\gamma(s)}(\gamma'(s)) = 0$, where $k$ is the curvature of the curve and we use the fact that $\gamma''(s) = k n(s)$. Let $s = 0$ and since $\gamma'(s) = v$, we have $\mathrm{II} = kn(0) \cdot n(p)$. Therefore, the second fundamental form measures the dot product between the normal of the curve and the normal of the surface, multiplied by the curvature of the curve. Intuitively, when moving along the curve, we only measures the curvature of the curve that is in the normal direction of the surface, and discount the tangential portion of the curvature. This makes sense because the tangential portion of the curvature is due to the wiggle-ness of the path $\gamma$, where as the normal portion of the curvature is due to the curvature of the underlying surface. No matter how we change the curve $\gamma$, we cannot change the normal portion of the curvature because the curve cannot leave the surface.

If we parametrize the local tangent space $T_p(\mathcal{M})$ using unit vectors $e_1$ and $e_2$, then the second fundamental form can be expressed as a 2x2 matrix where the $i,j$-th entry is $\mathrm{II}(e_i, e_j)$. It is well-known that the second fundamental form is symmetric $\mathrm{II}(v,w) = \mathrm{II}(w,v)$, and thus the matrix is also symmetric. We can find the eigenvectors $e_{\text{min}}, e_{\text{max}}$ of the matrix, which corresponds to the directions of minimum curvature $\kappa_{\text{min}}$ and maximum curvature $\kappa_{\text{max}}$. These eigenvectors give a nice parametrization of the tangent plane $T_p(\mathcal{M})$, and so we set the unit vectors to be the eigenvectors $e_1 = e_{\text{max}}$ and $e_2 = e_{\text{min}}$. This way, for any $v \in T_{p}(\mathcal{M})$, we can first write $v$ in terms of the eigen-basis, $v = e_1 \cos\theta + e_2 \sin\theta$ and then compute the curvature $\kappa$ along the direction $v$ as $\kappa = \kappa_{\text{max}} \cos^2\theta + \kappa_{\text{min}} \sin^2\theta$.

Using the two principal curvatures of the surface, we can define the Gaussian curvature $G$ and the mean curvature $H$:

• Gaussian Curvature $K = \kappa_{\text{min}} \cdot \kappa_{\text{max}}$. The Gaussian curvature tells us whether a surface shapes like a bowl ($K > 0$ when the principal curvatures curve in the same direction) or a saddle ($K < 0$ when the principal curvatures curve in the opposite direction) or an extrusion ($K = 0$ when at least one of the principal curvatures is zero).
• Mean Curvature $H = \frac{1}{2} \left(\kappa_{\text{min}} + \kappa_{\text{max}}\right)$. Mean curvature can also be calculated by integrating curvatures in all directions around the point $p$.

Mean Curvature Flow

Consider a time-varying manifold $\mathcal{M}_t$ and a local parametrization $f_t(u,v): \mathbb{R}^2 \to \mathbb{R}^3$. The mean curvature of the manifold is closely related to the time derivative of the area. Assuming that the time variation is zero outside of our local parametrization, the area (of our interest) is defined as

$A_t = \int\int||\partial_u f_t \times \partial_v f_t||_2 \ du \ dv$

Let $t_u = \partial_u f_t$ and $t_v = \partial_v f_t$, the time derivative of the area is

$\partial_t A_t = \int\int n \cdot \left(\partial_t t_u \times t_v + t_u \times \partial_t t_v \right) \ du \ dv$

Using the property that $a \cdot (b \times c) = b \cdot (c \times a)$, we have

$\partial_t A_t = \int\int \partial_t t_u \cdot(t_v \times n) + \partial_t t_v \cdot (t_u \times n) \ du \ dv$

Let $w_t = \partial_t f_t$, then $\partial_t t_u = \partial_u w_t$ and $\partial_t t_v = \partial_v w_t$. Then using integration by parts, we can shift the derivative on $w_t$,

$\partial_t A_t = -\int\int w_t \cdot \partial_u (t_v \times n) + w_t \cdot \partial_v (t_u \times n) \ du \ dv = -\int\int w_t \cdot (t_v \times \partial_u n) - w_t \cdot (t_u \times \partial_v n) \ du \ dv$

Note that both vectors $t_v \times \partial_u n$ and $t_u \times \partial_v n$ are parallel to the unit normal $n$ at vertex $p$, and thus

$\partial_t A_t = - \int\int (w_t \cdot n) \left( (t_v \times \partial_u n - t_u \times \partial_v n ) \cdot n \right) \ du \ dv$

Using the property $a \cdot (b \times c) = b \cdot (c \times a)$ again, and note that $t_v \times n = t_u$ assuming that $t_u$ and $t_v$ are orthonormal. Then

$\partial_t A_t = -\int\int (w_t \cdot n) (t_u \cdot \partial_u n + t_v \cdot \partial_v n) \ du \ dv$

Note that $\mathrm{II}(u, u) = -t_u \cdot \partial_u n$ and $\mathrm{II}(v, v) = - t_v \cdot \partial_v n$,

$\partial_t A_t = \int\int (w_t \cdot n) 2H \ du \ dv$

Therefore, the first variation of surface area is the mean curvature normal, $H n$. Using this idea, we can minimize the surface area of a surface via mean curvature flow, i.e. solving the partial differential equation $\partial_t f = - H n$.