Curves And Surfaces
Level: 0 1 2 3 4 5 6 7
MSC Classification: 53 (Differential geometry)

Getting Oriented

The simplest objects of study in differential geometry are curves and surfaces in 3-space. These inherit notions of distance and area from the ambient space.

There are two ways to approach differential geometry. Classical differential geometry looks at the local structure of curves and surfaces: what they look like in the neighborhood of a point. There is also global differential geometry, which looks at properties depending on the entire curve. Beginning with the local theory allows us to keep our statements very precise.

Curves

We will consider curves to be maps of a straight line into $\mathbb{R}^3$. To each point of the line (called the parameter) we associate a point in $\mathbb{R}^3$. We will see that we can pretend the parameter is actually the arc length, and we will also discover how to quantify how ‘curvy’ and ‘twisty’ the curve is. But we begin our discussion by reviewing properties of the vector space $\mathbb{R}^3$.

Vector Products

All vector spaces have exactly two orientations: two bases give the same orientation iff the determinant of the change of basis matrix is positive. In our case, $\mathbb{R}^3$ has a positive orientation (corresponding to the standard basis $\{\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3\} = \{(1,0,0),(0,1,0),(0,0,1)\}$) and a negative orientation.

The inner product of two vectors $\mathbf{u}=(u_1,u_2,u_3)$ and $\mathbf{v}=(v_1,v_2,v_3)$ is

(1)
\begin{align} \mathbf{u}\cdot\mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 = |\mathbf{u}| |\mathbf{v}| \cos \theta, \end{align}

where $\theta$ is the angle between the two vectors. We use the inner product to define the norm of a vector as $|\mathbf{u}|=\sqrt{\mathbf{u}\cdot\mathbf{u}} = \sqrt{u_1^2+u_2^2+u_3^2}$. If $\mathbf{u}$ and $\mathbf{v}$ both depend on a parameter $t$, then we can differentiate the above formula to obtain:

(2)
\begin{align} \tfrac{d}{dt}(\mathbf{u}(t)\cdot\mathbf{v}(t)) = \mathbf{u}'(t)\cdot\mathbf{v}(t) +\mathbf{u}(t)\cdot \mathbf{v}'(t). \end{align}

The vector (cross) product can be defined as the unique vector $\mathbf{u} \times \mathbf{v}$ such that

(3)
\begin{align} (\mathbf{u}\times\mathbf{v})\cdot\mathbf{w} = \det(\mathbf{u},\mathbf{v},\mathbf{w}) = \begin{vmatrix}u_1&u_2&u_3\\v_1&v_2&v_3\\w_1&w_2&w_3\end{vmatrix} \end{align}

It is common to use the shorthand notation with a matrix having $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$:

(4)
\begin{align} \mathbf{u}\times\mathbf{v} = \begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\u_1&u_2&u_3\\v_1&v_2&v_3\end{vmatrix} = \begin{vmatrix}u_2&u_3\\v_2&v_3\end{vmatrix} \mathbf{e}_1 - \begin{vmatrix}u_1&u_3\\v_1&v_3\end{vmatrix} \mathbf{e}_2 + \begin{vmatrix}u_1&u_2\\v_1&v_2\end{vmatrix} \mathbf{e}_3. \end{align}

This second definition allows us to verify the properties (a) $\mathbf{u}\times\mathbf{v} = - \mathbf{v}\times\mathbf{u}$, (b) $\mathbf{u}\times\mathbf{v}=0$ iff $\mathbf{u}$ and $\mathbf{v}$ are linearly dependent, (c) $(\mathbf{u}\times\mathbf{v})\cdot\mathbf{u} = 0$ and $(\mathbf{u}\times\mathbf{v})\cdot\mathbf{v} = 0$. Less obvious properties include the relation

(5)
\begin{align} (\mathbf{u}\times\mathbf{v})\cdot(\mathbf{x}\times\mathbf{y}) = \begin{vmatrix}\mathbf{u}\cdot\mathbf{x}&\mathbf{v}\cdot\mathbf{x}\\ \mathbf{u}\cdot\mathbf{y}&\mathbf{v}\cdot\mathbf{y} \end{vmatrix}, \end{align}

and the implicated equation $|\mathbf{u}\times\mathbf{v}| = |\mathbf{u}| |\mathbf{v}| \sqrt{1-\cos^2\theta} = A$. This last provides a geometric interpretation of the cross product: it points in a direction such that $\{\mathbf{u},\mathbf{v},\mathbf{u}\times\mathbf{v}\}$ is a positive basis, and its magnitude is the area $A$ of the parallelogram with sides $\mathbf{u}$ and $\mathbf{v}$. Finally, we have the formula for the non-associative triple vector product:

(6)
\begin{align} (\mathbf{u}\times\mathbf{v})\times\mathbf{w} = (\mathbf{u}\cdot\mathbf{w})\mathbf{v} - (\mathbf{v}\cdot\mathbf{w})\mathbf{u}, \end{align}

and the formula for the derivative of the cross product:

(7)
\begin{align} \tfrac{d}{dt}(\mathbf{u}(t)\times\mathbf{v}(t)) = \ddtfrac{\mathbf{u}}{t}\times \mathbf{v}(t) + \mathbf{u}(t) \times\ddtfrac{\mathbf{v}}{t}. \end{align}

Parametrized Curves

We begin with the definition of a curve:

Parametrized differentiable curve
a differentiable map $\alpha:I\to \mathbb{R}^3$, usually denoted $\alpha(t)=(x(t),y(t),z(t))$. By differentiable (or smooth), we mean the functions $x(t)$, $y(t)$, and $z(t)$ have derivatives of all orders.

The trace of the curve is the image set $\alpha(I)\subset\mathbb{R}^3$. The tangent vector is $\alpha'(t)=(x'(t),y'(t),z'(t))$ and the acceleration vector is $\alpha''(t)=(x''(t),y''(t),z''(t))$. It is certainly possible for the tangent vector may be zero at some points. Also, the curve may not be injective (if it crosses itself, for example), and it is common for two distinct curves to have the same trace.

A curve is regular if its tangent vector is nowhere vanishing: $\alpha'(t)\neq 0$ for all $t$. We can define the arc length from a given point $t_0$ by:

(8)
\begin{align} s(t)=\int_{t_0}^t |\alpha'(t)| dt. \end{align}

If the curve is regular, we can differentiate the arc length to obtain the speed $\frac{ds}{dt}=|\alpha'(t)|$. Curves with unit speed everywhere are said to be parametrized by arc length. We can always reparametrize a curve by arc length; inverting the function $s(t)$ gives us the curve $\alpha(t(s))$, which has unit speed. For simplicity, we now assume our curves are parametrized by arc length.

Local Theory of Curves

In this section we will see how a curve is determined solely by its local curvature and torsion. To define these we introduce a local coordinate system, called the Frenet trihedron. We begin with:

Curvature
the nonnegative function $k(s)=|\alpha''(s)|$. Thus, it is the magnitude of the acceleration.
Torsion
the function $\tau(s)$ defined by $\mathbf{b}'(s)=\tau(s) \mathbf{n}(s)$ for the $\mathbf{n}$ and $\mathbf{b}$ below.

The Frenet Trihedron

If the curvature is everywhere nonzero we have the Frenet trihedron $\{\mathbf{t}(s),\mathbf{n}(s),\mathbf{b}(s)\}$ consisting of

  1. the tangent vector $\mathbf{t}(s)=\alpha'(s)$;
  2. the normal vector $\mathbf{n}(s)$, defined as the unit vector in the direction of acceleration or by the equation $\alpha''(s)=k(s)\mathbf{n}(s)$;
  3. the binormal vector $\mathbf{b}(s)$, defined by $\mathbf{b}(s)=\mathbf{t}(s)\times \mathbf{n}(s)$.

It is clear by definition that the Frenet trihedron is an orthonormal, positive basis. The lines in the directions of $\mathbf{t}(s)$, $\mathbf{n}(s)$, and $\mathbf{b}(s)$ are called the tangent line, the principal normal, and the binormal. The planes spanned by the vectors are the osculating plane $\mathsf{Span}\{\mathbf{t},\mathbf{n}\}$, the rectifying plane $\mathsf{Span}\{\mathbf{t},\mathbf{b}\}$, and the normal plane $\mathsf{Span}\{\mathbf{n},\mathbf{b}\}$.

The Fundamental Theorem of the Local Theory of Curves

The derivatives of $\mathbf{t}$, $\mathbf{n}$, and $\mathbf{b}$ allow us to interpret the curvature and torsion geometrically. We compute:

(9)
\begin{eqnarray} \mathbf{t}'&=k\mathbf{n}\\ \mathbf{b}'&=\tau\mathbf{n}\\ \mathbf{n}' &= -k\mathbf{t}-\tau\mathbf{b}. \end{eqnarray}

Thus, the curvature measures how fast the curve pulls away from the tangent line, while the torsion measures how fast it pulls away from the osculating plane. The curvature is 0 iff the curve is a straight line, while the torsion is 0 iff the curve is contained in a plane. Another geometric interpretation is given by the radius of curvature $R(s)=\frac{1}{k(s)}$, since a circle of this radius will have curvature $k(s)$.

One global property of a curve is its orientation, since there are two possible directions for each tangent vector. Switching the orientation of the curve leaves $k(s)$, $\tau(s)$, and $\mathbf{n}(s)$ invariant, while $\mathbf{t}(s)$ goes to $-\mathbf{t}(s)$ and $\mathbf{b}(s)$ to $-\mathbf{b}(s)$. For a plane curve one may give $k(s)$ a sign by requiring that $\{\mathbf{t},\mathbf{n}\}$ have the same orientation as the standard basis $\{\mathbf{e}_1,\mathbf{e}_2\}$; in this case the sign of $k(s)$ switches with the orientation of the curve.

Theorem (Fundamental Theorem of the Local Theory of Curves)
One can find a regular curve $\alpha(s)$ which has arbitrarily defined curvature $k(s)>0$ and torsion $\tau(s)$. This curve is unique up to rigid motion.

The Local Canonical Form

By introducing a coordinate system based on the Frenet trihedron, meaning we apply a rigid motion to transform the trihedron to the standard basis of $\mathbb{R}^3$, we can describe the behavior of a curve in the neighborhood of an arbitrary point. The first few terms of the Taylor expansion give

(10)
\begin{eqnarray} x(s) =& s - \tfrac{1}{6}k^2s^3 + \cdots\\ y(s) =& \half ks^2 - \tfrac{1}{6}k's^3 + \cdots\\ z(s) =& -\tfrac{1}{6}k\tau s^3 + \cdots \end{eqnarray}

These formulae imply the following geometric properties:

  1. the sign of $-\tau$ is the sign of $z'(s)$, so the torsion is positive if the curve pulls ‘down’ from the osculating plane, and negative if it pulls ‘up’;
  2. $y(s)\geq 0$ and $y(s)=0$ only when $s=0$ in some neighborhood of $s$, so that the curve is entirely on one side of the rectifying plane;
  3. the osculating plane is the limit of the planes spanned by the tangent line and the point $\alpha{s+h}$ as $h\to 0$.

Global properties of planar curves

We now move onto global properties of curves. We will take $\alpha: [0,l] \to \mathbb{R}^2$ to be a regular planar curve of length $l$ parametrized by arc length. The curve is closed if its endpoints match, so $\alpha(0)=\alpha(l)$ and all derivatives agree. It is simple if there are no other self-intersections. The Jordan Curve Theorem says that every simple closed curve bounds a region of the plane, called the interior. We will orient the curve positively, meaning the interior would be on the left when walking around the curve in the direction of orientation.

The Isoperimetric Inequality

Our first global property states that no other curve bounds more area for its length than the circle:

Theorem (Isoperimetric Inequality)
Let $C$ be a (regular) simple closed curve with length $l$, and $A$ the area of the region bounded by $C$. Then, $l^2-4\pi A \geq 0$, and equality holds iff $C$ is a circle.

This theorem is also true in general when $\alpha$ is piecewise $C^1$, that is guaranteed to have at least one derivative at all but a finite number of points.

The Four-Vertex Theorem

For the next theorem, we define the tangent indicatrix $t:I\to R^2$ of a planar simple closed curve $\alpha(s)=(x(s),y(s))$ by $t(s)=(x'(s),y'(s))$. Then, $\frac{dt}{ds}=\alpha''(s)=k(s)n(s)$ for the signed curvature $k(s)$. One can define the angle $\theta(s)$ between the $x$-axis and $t(s)$ by either the formula $\theta(s)=\arctan\frac{y'(s)}{x(s)}$ or by the integral $\theta(s)=\int_0^s k(s) ds$; they are equivalent up to multiples of $2\pi$ since $\frac{dt}{ds}=\theta'n$. This allows us to define the rotation index $I$ of the curve by $I=\frac{1}{2\pi}\int_0^l k(s)ds = \theta(l)-\theta(0)$. The Theorem of Turning Tangents states that the rotation index of a simple closed curve is $\pm 1$, depending only on orientation.

A convex curve lies to one side of any of its tangent lines. A vertex of a curve is a point $s\in[a,b]$ where $k'(s)=0$, i.e., local maxima/minima of the curvature. We can now state:

Theorem (Four-Vertex Theorem)
A convex simple closed curve has at least four vertices.

The theorem is also true for non-convex curves, although the proof is harder. It is also true that a planar closed curve is convex iff it is simple and the curvature never changes sign. This implies the curvature of a convex closed curve is either constant or has at least two maxima, two minima. A converse to this also holds: any $k(s)>0$ which is either constant or has at least two maxima and two minima is the curvature of some simple closed curve.

The Cauchy-Crofton Formula

Our third theorem gives a way to ‘measure’ how many lines meet a given curve. This measure on the set $\mathcal{L}$ of straight lines is given by identifying a line with the formula $\rho=x\cos\theta+y\sin\theta$ for constant $(\rho,\theta)$, with appropriate equivalence relations. The measure of a subset $\mathcal{S}\subset\mathcal{L}$ is then defined to be $\int\int_{\mathcal{S}} d\rho d\theta,$ which is the only measure (up to a constant) invariant under rigid motions.

Theorem (Cauchy-Crofton Formula)
The measure of the set of straight lines, counted with multiplicity, meeting a curve of length $l$ is $2l$.

This theorem gives a way to define the ‘length’ of a non-rectifiable curve, one whose arc length is undefined. It can also give a way to estimate curve length by counting how many times an appropriately dense set of lines meets the curve.

Surfaces

Regular Surfaces

In contrast to the definition of a curve as a map of an interval into $\mathbb{R}^3$, a surface is defined as (the image of) a collection of maps which are ‘glued together’:

Regular Surface
a subset $S\subset\mathbb{R}^3$ such that each point $p\in S$ has a neighborhood $V$ in $\mathbb{R}^3$ with a map $\mathbf{x}:U\to V\cap S$ for open $U\subset\mathbb{R}^2$ which is a differentiable homeomorphism with injective differential $d\mathbf{x}_q:\mathbb{R}^2\to\mathbb{R}^3$. The map $\mathbf{x}$ is called a parametrization or a system of local coordinates.

The homeomorphism condition guarantees that the surface has no self-intersections, while the last condition indicates that in the differential

(12)
\begin{align} d\mathbf{x}_q = \begin{pmatrix} x_u & x_v \\ y_u & y_v \\ z_u & z_v \end{pmatrix}, \end{align}

at least one of the three minors/Jacobian determinants such as $\delfrac{(x,y)}{(u,v)} = \bigl(\begin{smallmatrix}x_u & x_v \\ y_u & y_v\end{smallmatrix}\bigr)$ is nonzero. This guarantees the existence of a well-defined tangent plane normal to the vector $\mathbf{x}_u \times \mathbf{x}_v$.

To directly verify that some surface is regular requires covering the surface by open sets and giving the parametrizations explicitly. The unit sphere $S^2$, for example, can be covered by (1) six open hemispheres with parametrizations projecting the hemispheres to coordinate planes, (2) two open neighborhoods corresponding to parametrizations $\mathbf{x}(\theta,\phi)=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ for suitable ranges of $\theta$ and $\phi$.

Proving a Surface is Regular

There are other methods to more easily verify that a surface is regular, including:

Proposition
A surface which is the graph of a regular function $f:U\to\mathbb{R}$ is regular, that is the set of points $(x,y,f(x,y))$ for $(x,y)\subset U$. Conversely, any point on a regular surface has a neighborhood which is the graph of a differentiable function in two of the variables $\{x,y,z\}$.

This proposition gives a much shorter proof that the sphere is a regular function, being the inverse image of $f(x,y,z)=x^2+y^2+z^2=1$. Ellipses, hyperboloids, and the torus can also be proven regular with this method.

Differentiable Functions on Surfaces

If we are to define a function on a surface, we must ensure that the definition is independent of the parametrization. This is generally true because the change of coordinates is a diffeomorphism:

Proposition
For a point $p\in S$ on a regular surface with local parametrizations $\mathbf{x}:U\subset\mathbb{R}^2\to S$ and $\mathbf{y}:V\subset\mathbb{R}^2\to S$, the change of coordinates $h=\mathbf{x}^{-1}\circ\mathbf{y}$ on the overlapping section $W=x(U)\cap y(V)$ is a diffeomorphism from $\mathbf{y}^{-1}(W)$ to $\mathbf{x}^{-1}(W)$, \ie differentiable with differentiable inverse.

This theorem shows that the following definitions are well-defined:

Differentiable Function $f:V\subset S\to \mathbb{R}$
a function with parametrization $\mathbf{x}:U\subset\mathbb{R}^2\to S$ such that the composite $f\circ\mathbf{x}:U\subset\mathbb{R}^2\to\mathbb{R}$ is differentiable at $\mathbf{x}^{-1}(p)$.
Differentiable Map $\phi:V \subset S_1 \to S_2$
a map between surfaces such that the composition $\mathbf{x}_2^{-1}\circ\phi\circ\mathbf{x}_1$ is differentiable at $\mathbf{x}^{-1}(p)$ for some parametrizations $\mathbf{x}_1,\mathbf{x}_2$. Expressed in local coordinates, this means $\phi(u,v)=(\phi_1(u,v),\phi_2(u,v))$ has continuous partials of all orders. If $f:S_1\to S_2$ is a homeomorphism with differentiable inverse, then it is a diffeomorphism, and the surfaces $S_1$ and $S_2$ are diffeomorphic.

Examples of differentiable functions on surfaces include height functions $h(p)=p\cdot v$ for some unit vector $v$ and distance functions $d(p)=|p-p_0|^2$.

Diffeomorphism is the appropriate notion of equivalence between surfaces; such surfaces are indistinguishable from the perspective of differential geometry. For example, an open set $U$ and its image $\mathbf{x}(U)$ are diffeomorphic, as is a surface and its mirror image.

Going Further

The Road Ahead

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