\part{???}


\chapter{Topological curves}

Consider the following example of a jump.

The quick brown fox jumped over the lazy dog with altitude according to $f(t)=\sin(\pi t)$ with $t \in [0,1]$ indexing the start and end of the jump.

Such a jump is \emph{impressive}.


\begin{definition}
A \emph{topological curve}
\end{definition}

\begin{definition}
A \emph{closed curve}
\end{definition}


\begin{definition}
A \emph{simple closed curve}
\end{definition}

This is also known as a Jordan curve.



\chapter{Differentiable curves}

\begin{definition}
A \emph{differentiable curve}
\end{definition}

\begin{definition}
The \emph{arc-length of a curve}
\end{definition}

\begin{definition}
A \emph{reparametrization of a curve}
\end{definition}

\begin{definition}
A \emph{regular curve}
\end{definition}

\begin{definition}
An \emph{arc-length reparametrization of a differentiable curve}
\end{definition}


\begin{theorem}
A differentiable curve is \emph{arc-length reparametrizable} iff it is regular.
\end{theorem}


\begin{definition}
The \emph{winding number}
\end{definition}

If two curves have identical start and end points, one may ask if curve A can be continuously deformed into curve B and vice versa (like moving a piece of string). Such a deformation is known as a \emph{homotopy}.


\begin{definition}
Two functions are \emph{homotopic} iff 
\end{definition}

\begin{definition}
A \emph{homotopy}
\end{definition}

\chapter{Frenet-Serret frame}


A \emph{group} is an ordered pair $(G, \cdot )$ of a set $G$ and a binary operation $\cdot$ acting on $G$ with the following properties:
\begin{itemize}
	\item $\cdot$ is associative on $G$
	\item $G$ contains an identity element with respect to $\cdot$
	\item Every element of $G$ is invertible with respect to $\cdot$
\end{itemize}


\begin{definition}
The \emph{tangent vector}
\end{definition}

\begin{definition}
The \emph{normal vector}
\end{definition}

\begin{definition}
The \emph{binormal vector}
\end{definition}

\begin{definition}
The \emph{Frenet-Serret frame}
\end{definition}



\begin{definition}
The \emph{curvature}
\end{definition}

\begin{definition}
The \emph{torsion}
\end{definition}

\begin{definition}
The \emph{Frenet-Serret equations}
\end{definition}

\begin{Theorem}\emph{Fundamental uniqueness and existence theorem for differentiable curves}
\end{definition}

