Differential Geometry (Notes from Prof. David Forsyth's lectures at UIUC): PART 1

Curves

Parametrization

Curve is an embedding of a line into higher dimensional space

\begin{align}
&\vec{x}: t \rightarrow \mathbb{R}^n \\
&\vec{x}(t) = (x(t), y(t), z(t), \cdots)
\end{align}

Special Case: Arc Length Parametrization

Usually denoted by $s$. In arc length parametrization, a small step $ds$ in the parameter space amounts to moving the same distance along the curve. For any other parametrization, $t$, a step $dt$ in the parameter space leads to covering a distance 

\begin{align}
\|d\vec{x}\| = \left(\sqrt{\left(\frac{dx_1}{dt}\right)^2 + \left(\frac{dx_2}{dt}\right)^2 + \cdots} \right) \cdot dt
\end{align}

By comparison for arc length parametrization

\begin{align}
\sqrt{\left(\frac{dx_1}{ds}\right)^2 + \left(\frac{dx_2}{ds}\right)^2 + \cdots} &= 1 \\
\implies \|\frac{d\vec{x}}{ds}\| &= 1
\end{align}

Tangent to the curve at a point

Tangent to a curve at $t_0$ is defined by 

\begin{align}
\frac{d\vec{x}}{dt}|_{t=t_0}
\end{align}

This can be thought of as a vector in the direction of a ray originating at $\vec{x}(t_0)$ which is in 2 point contact with the curve in the limit of the 2 points being arbitrarily close together.

A unit tangent is usually denoted by $\vec{T}$ and is given by

\begin{align}
\vec{T} = \frac{d\vec{x}/dt}{\|d\vec{x}/dt\|}
\end{align}

For arc length parametrization this reduces to

\begin{align}
\vec{T} = \frac{d\vec{x}/ds}{\|d\vec{x}/ds\|} = \frac{d\vec{x}}{ds}
\end{align}

Curvature at a point on the curve

For a curve parametrized by $s$, rate of change of tangent is along the normal direction and its magnitude is  governed by curvature, $\kappa(s)$. This relation is given by

\begin{align}
\frac{d\vec{T}}{ds} = \kappa(s)\vec{N}(s)
\end{align}

where $\vec{N}(s)$ is a unit normal.

Binormal and Torsion

For plane curves $\vec{T}$ and $\vec{N}$ define a moving coordinate system. For space curves there is an entire family of directions orthogonal to $\vec{T}$ and hence a binormal $\vec{B}$ is also defined. At a point, the space curve is locally planar and $\vec{N}$ is chosen to lie in this plane such that $\vec{N}.\vec{T}=0$ and then $\vec{B} = \vec{T} \times \vec{N}$. For such a coordinate system, for an infinitesimal change in $s$, $\vec{B}$ swings in the direction of the $\vec{N}$ and the magnitude of the rate of this change is governed by torsion, $\tau(s)$. The following equation summarizes the relation between the rate of change of each of these unit vectors

\begin{align}
\frac{d}{ds} \left[
\begin{array}{c}
T \\
N\\
B\\
\end{array}
\right] =
\left[

\begin{array}{ccc}
0 & \kappa & 0\\
-\kappa & 0 & \tau\\
0 & -\tau & 0\\
\end{array}
\right]
\left[
\begin{array}{c}
T \\
N\\
B\\
\end{array}
\right]
\end{align}

Surfaces

A surface is a mapping  $\vec{x} : \mathbb{R}^2 \rightarrow \mathbb{R}^3$.

Gauss Map

Its a mapping from a point on the surface to a point on the sphere which has the same normal. Since the normal at a point $\vec{z}$ on the unit sphere is given by the $\vec{z}$ itself, Gauss map just tells us the normal at every point $\vec{x}$ on the surface $\vec{N}(\vec{x})$.

Gaussian Curvature

Defined as 

\begin{align}
\kappa = \underset{radius\rightarrow 0} \lim \frac{\text{Area on Gauss Map}}{\text{Area on Surface}}
\end{align}

Based on sign of $\kappa$ surfaces can be classified as:

1. Hyperbolic: $\kappa < 0$
2. Parabolic: $\kappa = 0$
3. Elliptic: $\kappa > 0$

Note:

• Bending does not change $\kappa$. Area need to be added or subtracted in order to change it.
• At each point on the surface there are 2 orthogonal directions in which directional curvature is extremal. These are known as principal curvatures, $\kappa_1, \kappa_2$
• Given the principal curvatures a surface can locally be written as \begin{align}
  \vec{x}(s,t) = \left(s,t,\frac{1}{2}(\kappa_1s^2 + \kappa_2s^2)+O(3)\right)
  \end{align}
• Gaussian curvature $= \kappa_1 \cdot \kappa_2$
• Mean curvature $= \frac{\kappa_1 + \kappa_2}{2}$

Tangents to the surface at a point

There is a vector subspace of tangents at a point with a basis defined by

\begin{align}
\{ \frac{\partial \vec{x}}{\partial s}, \frac{\partial \vec{x}}{\partial t} \} = \{x_s, x_t\}
\end{align}

The First Fundamental Form

It is an operator defined at every point on the surface that can be used to measure dot product between tangents at that point. Any tangent vector $\vec{u}$ can be written in terms of its basis

\[
\vec{u} = a\vec{x}_s + b\vec{x}_t
\]

Similarly we can write any other tangent vector $\vec{u}$ as 

\[
\vec{v} = c\vec{x}_s + d\vec{x}_t
\]

Then their dot product is given by 

\[
I(\vec{u}, \vec{v}) =
\left[
\begin{array}{cc}
a & b \\
\end{array}
\right]
\left[
\begin{array}{cc}
\vec{x}_s \cdot \vec{x}_s & \vec{x}_s \cdot \vec{x}_t \\
\vec{x}_s \cdot \vec{x}_s & \vec{x}_t \cdot \vec{x}_t \\
\end{array}
\right]
\left[
\begin{array}{c}
c \\
d \\
\end{array}
\right]
\]

The Second Fundamental Form

If we think of $\vec{N}(\vec{x})$ as a map from a point on the surface to a point on the sphere (the Gauss Map), then it would have a directional derivative $d\vec{N}(\vec{u})$  which is a map from the tangent plane to the surface to the tangent plane to the sphere. Some properties:

1. $\vec{N}_s\cdot \vec{N} = 0$ and $\vec{N}_t\cdot \vec{N} = 0$ (follows by differentiating $\vec{N} \cdot \vec{N} = 1$)
2. $d\vec{N}(\vec{u}) = a\vec{N}_s + b\vec{N}_t$
3. It follows from above 2 properties that $d\vec{N}(u)$ lies in the tangent plane to the surface

Now the second fundamental form is defined as 

\[
II(\vec{u}, \vec{v}) = -I(d\vec{N}(\vec{u}),\vec{v})
\]

which turns out to be


\[
II(\vec{u}, \vec{v}) =
\left[
\begin{array}{cc}
a & b \\
\end{array}
\right]
\left[
\begin{array}{cc}
\vec{N} \cdot \vec{x}_{ss}& \vec{N} \cdot \vec{x}_{st} \\
\vec{N} \cdot \vec{x}_{st} & \vec{N} \cdot \vec{x}_{tt} \\
\end{array}
\right]
\left[
\begin{array}{c}
c \\
d \\
\end{array}
\right]
\]

Key results

1. Gaussian Curvature, $\kappa = det(-I^{-1}II)$ 
2. Mean curvature, $H = trace(-I^{-1}II)$
3. These are local geometric properties of the surface which are invariant under 
• Rigid Motion (because $I$ and $II$ do not change)
• Reparametrization $\vec{x}(s(u,v), t(u,v))$ (non-trivially compute $I$ and $II$ with respect to $(u,v)$ and use the above equations)

Isometry

A transformation $\psi$ that maps one surface $S_1$ to another $S_2$ is called an isometry iff lengths along the surface (and hence the angles) are preserved. 

Intrinsic Properties

Geometric properties invariant under an isometry are called Intrinsic properties. 

Lemma:

Given an isometry $\psi: S_1 \rightarrow S_2$, there is a parametrization of $S_2$ such that $I$ does not change. 

Corollary:

Any property that can be expressed in terms of elements of $I$ (or their derivatives) is intrinsic. 

Theorem Egregium:

Gaussian curvature is intrinsic.

Corollary:

Isometric surfaces have same gaussian curvature at corresponding points.

Contour generator

A contour generator on S is a curve where rays from a chosen point $P$ are tangent to the surface. Related but different:

1. Outline: Projection of a contour generator (including invisible regions) into an image plane
2. Silhouette: Visible exterior points of the outline

Contour generator on an implicit surface as a locus

For an implicit surface $\phi(\vec{x}) = 0$, contour generator w.r.t external point $\vec{p}$,  is the locus of points which satisfy the following conditions:

1. $\phi(\vec{x}_1) = 0$
2. $\nabla\phi^T \vec{p} - \nabla \phi^T \vec{x} = 0$ (follows from the fact that tangent plane at a point $\vec{x}_0$ is given by $\nabla\phi^T \vec{x} - \nabla \phi^T \vec{x}_0 = 0$ and the point $\vec{p}$ lies on this plane for every point on the C.G)