# Basic Theory

### Curves in the Plane

Perhaps the most used definition of a plane curve is the following: A *plane curve* $C \subset
\R^2$ is the image of some interval $I \subset \R$ under some continuous parameterisation map $\mathbf{r}:
I \to \R^2$, that is, $C = \mathbf{r}(I)$. Other (non-equivalent) definitions are based on the fact that every point
on a 'curve' has a neighbourhood (on the curve) homeomorphic to some open interval of $\R$. This requires that the
curve has no 'endpoints'; in the language of the first definition of a curve, this means that the interval $I$ has to
be *open*. Also, this prohibits self-intersections, which are clearly allowed according to the first definition.
In what follows, we will chiefly allow these kinds of behaviour, and use the first definition. However, we will
generally demand that parameterisation maps be differentiable and with non-vanishing derivatives.

#### Describing a Plane Curve

The are several ways of describing a plane curve, one of which consists simply of giving a parameterisation map
$\mathbf{r}$. For instance, the *unit circle* $C = \mathbf{r}(\left[0,2\pi
\right[)$ where $$\mathbf{r}(t) = (\cos t, \sin t), \quad\quad \forall t \in \R.$$

Another common way of defining a curve is to give its *equation* in some coordinate system of the plane.
For instance, if $(x,y)$ are the Cartesian coordinates in $\R^2$, then the unit circle $$C = \left\{(x,y)\in\R^2:
x^2 + y^2 = 1\right\},$$ and we say that $x^2 + y^2 = 1$ is the equation of the curve. Often, with some abuse of
terminology, we even use the word 'curve' for the formal expression $x^2 + y^2 = 1$ itself. Each new coordinate system
in the plane brings a new way of describing (and defining) curves. For instance, in polar coordinates $(r, \varphi)$,
defined by $$\begin{align}x &= r \cos \varphi\\y &= r \sin \varphi\end{align}$$ with $r \ge 0$ and $\varphi \in \left[0,
2\pi\right[$, the unit circle has the particularly simple equation $r = 1$.

A (continuous) *function* $f : D_f \to \R$ with domain $D_f$ a real interval induces a plane curve, namely,
its *graph* $$C = \left\{(x,y) \in \R: x \in D_f \wedge y=f(x) \right\}.$$ This curve can clearly also be stated
in terms of its *equation* $y = f(x)$ and in terms of the *parameterisation map* $\mathbf{r}: D_f \to \R^2 :
t \mapsto (t, f(t)).$ Notice that this is a very restricted way of describing curves; for instance, the unit circle
is not the graph of any function.

Finally, any (continuous) *scalar field* $F : D_F \to \R$ defined in some (nice) region $D_F \subset \R^2$
gives rise to a family of curves, namely, the *level curves* $$C_\ell = \left\{(x,y) \in D_F: F(x,y) = \ell
\right\}, \quad\quad \forall \ell \in R_F$$ where $R_F = F(D_F)$ is the range of $F$. For instance, the unit circle is the
level curve $C_1$ of the scalar field $F: \R^2 \to \R: (x,y) \mapsto x^2 + y^2$.

#### Curvature

By definition, the *curvature* of a plane curve is the norm of the second derivative (the acceleration)
of a 'unit-speed' parameterisation map of the curve; hence, the curvature is a *local* property, defined
*pointwise* along the curve. A 'unit-speed' parameterisation map is simply a map the derivative of which (the
velocity) is of unit norm. Given a curve $C = \mathbf{r}(I)$ with *any* parameterisation map $\mathbf{r}$ (unit-speed
or not), it is straightforward to show that the curvature at parameter value $t$ is $$\kappa(t) = \frac{|\mathbf{
\dot r}(t)\times\mathbf{\ddot r}(t)|}{|\mathbf{\dot r}(t)|^3}, \quad\quad \forall t\in I.$$ It can also be shown
that the curvature is the absolute value of the rate-of-change of the angle between the curve's tangent vector and the
positive $x$ axis with respect to arc length. This rate of change, with its sign, is called the *signed curvature*
of the curve.

### Curves in Space

Similarly, a curve in space can either be defined as the image of a (continuous) parameterisation map $\mathbf{r}: I \to \R^3$, or by demanding that every point on it should have a neighbourhood homeomorphic to some open subset of $\R$.

#### Describing a Space Curve

Of course, one can describe a space curve by giving a parameterisation map. One can also give *a pair* of
equations (in the spatial coordinates) that are to hold simultaneously; in this case, the curve is the intersection of two
surfaces. For example, the unit circle in the $xy$ plane is the image $C = \mathbf{r}(\left[0,2\pi\right[)$ where $$\mathbf{r}(t)
= (\cos t, \sin t, 0), \quad\quad\forall t\in\R,$$ but it is also the set $$C = \left\{(x,y,z)\in\R^3: x^2 + y^2 = 1, z = 0
\right\},$$ that is, it is the intersection between the cylinder $x^2 + y^2 = 1$ and the plane $z = 0$. Again, every spatial coordinate system yields a new way of describing space curves. For instance,
in cylindrical coordinates $(r,\varphi,z)$, defined by $$\begin{align}x &= r \cos \varphi\\y &= r \sin \varphi\\z &= z\end{align}$$ and
$r \ge 0, \varphi \in \left[0,2\pi\right[$, the same curve has equations $r = 1, z = 0$ (where each individual equation retains its
previous interpretation, i.e., as a cylinder and a plane, respectively).

#### Curvature

By definition, the *curvature* of a space curve is the norm of the second derivative (the acceleration)
of a unit-speed parameterisation map of the curve; hence, the curvature is a *local* property, defined
*pointwise* along the curve. Given a curve $C = \mathbf{r}(I)$ with *any* parameterisation map $\mathbf{r}$ (unit-speed
or not), it is straightforward to show that the curvature at parameter value $t$ is $$\kappa(t) = \frac{|\mathbf{
\dot r}(t)\times\mathbf{\ddot r}(t)|}{|\mathbf{\dot r}(t)|^3}, \quad\quad \forall t\in I.$$

#### Tangent, Normal, and Binormal

Given a curve $C = \mathbf{r}(I)$ with $\mathbf{r}$ unit-speed, the unit *tangent* vector at $t \in I$ is $$\mathbf{\hat t}(t) =
\mathbf{\dot r}(t).$$ Although $\mathbf{\hat t}(t)$ depends upon the actual parameterisation map
$\mathbf{r}$ (namely, upon the *orientation* of the curve), the set $\left\{\mathbf{\hat t}(t), -\mathbf{\hat t}(t)\right\}$
is independent of parameterisation function at corresponding points on the curve. The standard unit *normal* vector is $$\mathbf{\hat
n}(t) = \frac{1}{|\mathbf{\ddot r}(t)|}\mathbf{\ddot r}(t) = \frac{1}{\kappa(t)}\mathbf{\ddot r}(t).$$ This vector is independent of
parameterisation, and it is obviously only well-defined at points of non-vanishing curvature. Since $\mathbf{r}$
is unit-speed, $\mathbf{\hat t}(t) \perp \mathbf{\hat n}(t)$ for every $t\in I$, and so by adding a third vector $$\mathbf{\hat b}(t)
=\mathbf{\hat t}(t)\times\mathbf{\hat n}(t),$$ the so-called *binormal* vector (which is also clearly seen to be unit), we have
in fact a natural basis $\mathbf{\hat t}(t), \mathbf{\hat n}(t), \mathbf{\hat b}(t)$ of $\R^3$ at each $t \in I$ for which $\kappa(t) \ne 0$.

#### Torsion

Intuitively, the *torsion* of a space curve is a measure of the amount by which the curve 'fails' to lie within a single
plane, locally. Specifically, the torsion $\tau$ is defined by $\mathbf{\hat b}\prime = −\tau \mathbf{\hat n}$ where $\mathbf{\hat n}$
and $\mathbf{\hat b}$ are the standard unit normal and standard unit binormal of some unit-speed parameterisation of the curve. It
can be shown that, given any curve $C = \mathbf{r}(I)$, with $\mathbf{r}$ not necessarily unit-speed, the torsion is $$\tau(t) = \frac
{(\mathbf{\dot r}(t)\times\mathbf{\ddot r}(t))\cdot \mathbf{\dddot r}(t)}{|\mathbf{\dot r}(t)\times\mathbf{\ddot r}(t)|^2}, \quad\quad
\forall t\in I: \kappa(t) \ne 0.$$

### Surfaces

Not surprisingly, there are pricipally two common ways of defining the concept of a (two-dimensional) surface (in $\R^3$). Either, one may
define a *surface* as the image of some (nice) parameter-plane region $U \subset \R^2$ under a (continuous) parameterisation function $\mathbf{r}:
U \to \R^3$, or one may define a *surface* as a subset $S \subset \R^3$ such that every point on the 'surface' has a neighbourhood
(on the surface) homeomorphic to some open disk in $\R^2$. Essentially the same remarks can be made here as we did in the case
of curves, namely, the latter approach cannot allow surfaces with 'edges' (so $U$ has to be open) and nor can it allow self-intersections.
We will mainly use the first approach.
We will, however, make the standard requirements that the parameterisation maps $\mathbf{r}$ be continuously differentiable, and that
the derivatives $\mathbf{r}_u(u,v)$ and $\mathbf{r}_v(u,v)$ be linearly independent at every point $(u,v)\in U$.

#### Describing a Surface

Of course, a particular surface can be defined by a suitable parameterisation map $\mathbf{r}: U \to \R^3$. For instance, the
*unit sphere* is the image $\mathbf{r}(U)$ where $$\mathbf{r}(\theta,\varphi) = \basis\begin{pmatrix}
\sin\theta\cos\varphi\\\sin\theta\sin\varphi\\\cos\theta\end{pmatrix}, \quad\quad\forall (\theta, \varphi)\in U$$ and $U := \left[0, \pi\right]\times
\left[0,2\pi\right[$. Since a surface in $\R^3$, like a curve in $\R^2$, is an object of codimension $1$, it can also be specified
by an equation. For instance, the unit sphere is $$S = \left\{(x,y,z)\in\R^3: x^2 + y^2 + z^2 = 1\right\},$$ and we say that $x^2 + y^2 +z^2 = 1$
is the equation of the surface. Often, with some abuse of terminology, we even use the word 'surface' for the formal expression $x^2 + y^2 + z^2 = 1$
itself. Each new coordinate system in space brings a new way of describing (and defining) surfaces. For instance, in spherical coordinates $(r,
\theta, \varphi)$, defined by $$\begin{align}x &= r \sin \theta \cos \varphi\\y &= r \sin \theta \sin \varphi\\z &= r \cos \theta\end{align}$$ with $r \ge 0$,
$\theta \in \left[0,2\pi\right]$, and $\varphi \in \left[0,2\pi\right[$, the unit sphere has the particularly simple equation $r = 1$.

Given a (continuous) function $f : D_f \to \R$ with (nice) domain $D_f \subset \R^2$, its *graph* is the surface $$S = \left\{(x,y,z)
\in \R^3: (x,y) \in D_f \wedge z=f(x,y)\right\}.$$ Clearly, this surface can also be given by its eqation $z = f(x,y)$ and by the parameterisation
map $\mathbf{r}: D_f \to \R^3: (x,y) \mapsto (x, y, f(z))$. Analogous to the case of curves in the plane, this is a very restricted way of
describing a surface; for example, the unit sphere is not the graph of any function.

Finally, given any (continous) scalar field $F: D_F \to \R$ defined in some (nice) domain $D_F \subset \R^3$, the *level surfaces*
belong to the family of surfaces $$S_\ell = \left\{(x,y,z)\in D_F: F(x,y,z) = \ell\right\}, \quad\quad\forall \ell\in R_F$$ where $R_F=F(D_F)$
is the range of $F$. For example, the unit sphere is the level surface $S_1$ of the scalar field $\mathbf{r}: \R^3 \to \R: (x,y,z) \mapsto
x^2+y^2+z^2$.

#### Properties of Surfaces

Let $S = \mathbf{r}(U)$ be a surface. At every coordinate $(u,v)\in U$, $\mathbf{r}_u(u,v)$ and $\mathbf{r}_v(u,v)$ forms a basis of the
*tangent space* of $S$ at $\mathbf{r}(u,v)$. The plane $$\Pi_{(u,v)} = \left\{(x,y,z)\in\R^3: (x,y,z) = \mathbf{r}(u,v) + s\mathbf{r}_u(u,v)
+t\mathbf{r}_v(u,v)\right\}$$ is called the *tangent plane* of $S$ at $\mathbf{r}(u,v)$, and is independent of the choice
of parameterisation function $\mathbf{r}$. Below, a tangent plane on the unit sphere is shown.

The vector field $$\mathbf{\hat N}(u,v) = \frac{1}{|\mathbf{r}_u(u,v) \times \mathbf{r}_v(u,v)|} \mathbf{r}_u(u,v) \times \mathbf{r}_v(u,v),
\quad\quad\forall(u,v)\in U$$ is called the *standard unit normal vector field* of the surface, and, up to the sign of the vectors,
it is independent of the particular choice of parameterisation map as long as the surface is orientable. Below, the normal vector field
on the unit sphere is shown.

*area element*of the surface is $$dA = |\mathbf{r}_u(u,v) \times \mathbf{r}_v(u,v)|~dudv.$$ This is the 'area scale' between the parameter plane and the surface. Specifically, if $D \subset U$ is a (nice) subset of the parameter plane, then the area of the corresponding part on the surface is $$\iint_D dA.$$ In particular, for $D = U$, this yields the total area $A$ of the surface, i.e., $$A = \iint_U dA.$$

#### Curves on Surfaces. Parameter Curves

Consider a surface $S = \mathbf{r}(U)$, and a curve $\gamma = \mathbf{p}(I) \subset U$ in the parameter-plane region $U$ of $S$. Then the image $C = \mathbf{r}(\gamma) = (\mathbf{r}\circ\mathbf{p})(I) \subset S$ is a space curve on the surface $S$. For example, one can draw the butterfly curve on a cylinder:

In fact, in all the pictures of surfaces given above, we have not plotted the surfaces themselves, but rather the
so-called *parameter curves* of the surfaces, that is, the images of rectangular grids in the parameter planes. Below
a rectangular grid $G \subset U = \left[-\pi,\pi\right[\times\left[-1,1\right]$ is shown, together with its image $\mathbf{r}(G)$
where $\mathbf{r}(u,v) = 5(\cos u,\sin u,v)$ parameterizes a cylinder.