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\begin{document}
\title{The Short Ruler on Cones}
\author{\name{Peter Stadler\thanks{CONTACT Peter Stadler. Email: peter.stadler@student.uibk.ac.at}}\affil{Department of Mathematics, University of Innsbruck, Austria}}
\maketitle
\begin{abstract}
On complete Riemannian manifolds, a geodesic linking two different points could be approximated by applying short ruler methods to shorten a given path iteratively.
In this paper we take a closer look at a short ruler method on a cone.
To handle the problems caused by the fact that the cone is not smooth at its apex, we unwind the cone isometrically into the plane, and show that the sequence of shortened curves converges to a geodesic.
% On Riemannian manifolds geodesics are locally shortest paths.
% We assume that we have a short ruler, which allows us to construct short geodesics. We want to approximate a geodesic between two given points that are far away from each other.
% Using the short ruler, we can shorten any curve that connects the two points, iteratively.
% On complete Riemannian manifolds, at least a subsequence of the shortened curves converges to a geodesic between starting point and end point.
% In this paper we will take a closer look at this short ruler method on a cone.
% Cones are not smooth in their apex, and therefore they are not complete Riemannian manifolds.
% This leads to some problems for the short ruler method.
% By unwinding the cone isometrically into the plane, we can handle the problems.
% So we can see that the sequence of shortened curves converges to a geodesic.
\end{abstract}
\begin{keywords} curve shortening, short ruler, cone, geodesics, iteration \end{keywords}
\section{Introduction}
We search geodesics between two points on a manifold with an affine connection $\nabla$, i.e., curves $\gamma$ with given $\gamma\mathopen{}\mathclose\bgroup\left(0\aftergroup\egroup\right)$ and $\gamma\mathopen{}\mathclose\bgroup\left(1\aftergroup\egroup\right)$ that satisfy $\nabla_{\dot\gamma}\dot\gamma = 0$.
For example, a Lie Group $G$ is a manifold, and the restrictions of homomorphisms $h\colon\mathbb R\to G$ to compact intervals are geodesic according to the left or right invariant connection.
The affine connection $\nabla$ does not need to be the Levi-Civita connection. We restrict to that case. Then the manifold is a Riemannian manifold.
For calculating geodesics, we can solve the following system of ordinary differential equations:
\begin{equation*}\ddot\gamma^k + \sum_{i,j=1}^m\Gamma^k_{ij}\dot\gamma^i\dot\gamma^j = 0\text{ for }1\le k\le m,\end{equation*}
where $m\in\mathbb N$ is the dimension of the manifold and $\Gamma^k_{ij}$ are the Christoffel symbols.
One could approximate the solution using different methods of numerical analysis.
For a given starting point, one could also choose a vector in its tangent space. Using the differential equations, one could approximate the geodesic that starts at the given point in the direction of the chosen vector.
If we want to reach another given point in the end, we have to look how close the geodesic gets to that end point with the chosen vector.
Then we modify the vector, and recalculate the geodesic adaptively until we reach the end point by the geodesic (or get close enough to it).
Another method is to modify a (non-geodesic) curve in order to approximate a geodesic between its starting point and end point.
On a Riemannian manifold we can measure lengths, and geodesics are locally shortest connections between their starting point and end point.
Assuming that we have a short ruler, which allows us to construct geodesics whose length does not exceed a given length $L>0$, we can shorten any curve connecting starting point and end point.
Roman Liedl proposed to iterate such a short ruler method in \cite{Liedl87}.
For closed curves a similar method was investigated by George D. Birkhoff in \cite{Birkhoff27} and more generally by Jürgen Jost in \cite{Jost95}. The Pilgerschritt transform is another method that modifies a curve to get a geodesic. Wolfgang Förg-Rob gives a summary with exhaustive links to results for the Pilgerschritt transform in \cite{Foerg12}.
We will investigate two short ruler methods, only.
Using this methods, we get sequences of curves.
This leads to the question, if the sequences converge to geodesics.
Some existing results will be introduced.
In normed vector spaces we can apply both the midpoint method as well as the reduced method to any curve.
Applying one of these methods iteratively, we get a sequence of curves that converges to the straight line between its starting point and its end point
\cite[see][]{Stadler12}.
On complete Riemannian manifolds at least a subsequence converges to a geodesic if we iterate the reduced method for any curve
\cite[see][]{Stadler14}.
In this paper, we will take a closer look at cones.
A cone is a quite simple surface, which can be unwinded into the plane isometrically, similar as cylinders (for which I investigated short ruler methods in \cite{Stadler12}). Unlike cylinders, cones are not manifolds as they are not smooth in their apex.
Without its apex a cone would be a Riemannian manifold, but it would not be complete anymore.
This leads to some problems, e.g., we cannot choose a short ruler small enough, such that we get unique geodesics in all cases (other than on complete Riemannian manifolds).
Nevertheless, it is possible to define the reduced method usefully.
The main result in theorem \ref{thm:end} shows that we can apply the reduced method to any curve on a cone iteratively to yield a sequence of curves, which converges to a geodesic.
\section{Review of the Short Ruler Method}
In this section, we look at some previous results on the short ruler methods, which are useful in the proofs of the new results.
\subsection{Normed Vector Spaces}
In the following section $\mathopen{}\mathclose\bgroup\left(V,\mathopen{}\mathclose\bgroup\left\lVert\aftergroup\egroup\right\rVert\aftergroup\egroup\right)$ denotes a normed vector space. We want to shorten an arbitrary curve in it that is parameterized by arc length: $\alpha\colon[0,a]\to V$.
\begin{remark}[Polygonal Line] \label{rem:construct}
From the path $\alpha$, we construct the nodes of a polygonal line by
\begin{equation*}p_k:=\alpha\mathopen{}\mathclose\bgroup\left(kL\aftergroup\egroup\right)\text{ for }k\in\mathbb Z.\end{equation*}
Hereby, we extend $\alpha$ to $\mathbb R$ constantly: $\alpha\mathopen{}\mathclose\bgroup\left(<0\aftergroup\egroup\right)=\alpha\mathopen{}\mathclose\bgroup\left(0\aftergroup\egroup\right)$ and $\alpha\mathopen{}\mathclose\bgroup\left(>a\aftergroup\egroup\right)=\alpha\mathopen{}\mathclose\bgroup\left(a\aftergroup\egroup\right)$.
Hence, we get finitely many different points:
\begin{equation*}p_0,\dots p_n \text{ with }n=\left\lceil \frac aL \right\rceil.\end{equation*}
\end{remark}
\begin{figure}\centering
\includegraphics[width=.54\textwidth, height=4.8cm]{plane-s0.jpg}
\caption{Construction of a polygonal line from the path. (Remark \ref{rem:construct})}
\end{figure}
\begin{definition}[Midpoint Method on Normed Vector Spaces]\label{def:short}
Let $\mathopen{}\mathclose\bgroup\left(V,\mathopen{}\mathclose\bgroup\left\lVert\aftergroup\egroup\right\rVert\aftergroup\egroup\right)$ be a normed vector space with a curve $\alpha$ that is parameterized by arc length.
The midpoint method shortens the curve by substituting the nodes of the polygonal line $p_k:=\alpha\mathopen{}\mathclose\bgroup\left(kL\aftergroup\egroup\right)$ by their midpoints. It is the following linear operator:
\begin{equation*}S\colon V^\mathbb Z\to V^\mathbb Z, \mathopen{}\mathclose\bgroup\left(Sp\aftergroup\egroup\right)_k=\frac{1}{2}p_k+\frac{1}{2}p_{k-1}.\end{equation*}
\end{definition}
\begin{figure}\centering
\subfigure[$1$ step]{\includegraphics[width=.45\textwidth, height=4cm]{plane-s1.jpg}}
\subfigure[$2$ steps]{\includegraphics[width=.45\textwidth, height=4cm]{plane-s2.jpg}}
\caption{The midpoint method. (Definition \ref{def:short})}
\end{figure}
\begin{theorem}\label{thm:result}
The polygonal lines that we get by applying the midpoint method iteratively, converge uniformly to the straight line between start and end point of the path $\alpha$:
\begin{equation*}\forall\varepsilon>0\colon\exists r\in\mathbb N\colon\forall t>r\colon\sup_{k\in\mathbb Z}\mathopen{}\mathclose\bgroup\left\lVert\mathopen{}\mathclose\bgroup\left(S^tp\aftergroup\egroup\right)_k-g_k^t\aftergroup\egroup\right\rVert <\varepsilon.\end{equation*}
\end{theorem}
\begin{figure}\centering
\subfigure[$10$ steps]{\includegraphics[width=.45\textwidth, height=4cm]{plane-s10.jpg}}
\subfigure[$1000$ steps]{\includegraphics[width=.45\textwidth, height=4cm]{plane-s1000.jpg}}
\caption{The midpoint method converges. (Theorem \ref{thm:result})}
\end{figure}
\begin{definition}[Reduced Method on Normed Vector Spaces]\label{def:reduced}
Let $\mathopen{}\mathclose\bgroup\left(V,\mathopen{}\mathclose\bgroup\left\lVert\aftergroup\egroup\right\rVert\aftergroup\egroup\right)$ be a normed vector space with a curve $\alpha$ that is parameterized by arc length. The reduced method shortens the curve by applying two times the midpoint method $S$. Then it skips the first and the last of the new midpoints using the short ruler. The reduced method is the following linear operator for the nodes of the polygonal line $p_k:=\alpha\mathopen{}\mathclose\bgroup\left(kL\aftergroup\egroup\right)$:
\begin{equation*}R\colon V^\mathbb Z\to V^\mathbb Z, \mathopen{}\mathclose\bgroup\left(Rp\aftergroup\egroup\right)_k=\begin{cases}
\frac{1}{4}p_{k-1}+\frac{1}{2}p_k+\frac{1}{4}p_{k+1},\quad 00\colon \exists r\in \mathbb N \colon \forall t>r\colon \sup_{k\in \mathbb Z }\mathopen{}\mathclose\bgroup\left\lVert p_0+\frac kn\mathopen{}\mathclose\bgroup\left(p_n-p_0\aftergroup\egroup\right)-\mathopen{}\mathclose\bgroup\left(R^tp\aftergroup\egroup\right)_k\aftergroup\egroup\right\rVert<\varepsilon.\end{equation*}
\end{theorem}
\begin{figure}\centering
\subfigure[$10$ steps]{\includegraphics[width=.45\textwidth, height=4cm]{plane-r10.jpg}}
\hfill
\subfigure[$20$ steps]{\includegraphics[width=.45\textwidth, height=4cm]{plane-r20.jpg}}
\caption{The reduced method converges. (Theorem \ref{thm:reduced})}
\end{figure}
\subsection{Riemannian Manifolds}
In the following section $\mathopen{}\mathclose\bgroup\left(M,g\aftergroup\egroup\right)$ denotes a \textit{complete} Riemannian manifold. We want to shorten a curve that is parameterized by arc length: $\alpha\colon[0,a]\to M$.
Similarly as in normed vector spaces, we can construct a piecewise geodesic line along the path if we limit the length of the ruler, which produces short geodesics. We apply then the reduced method to it, and see that all paths stay in the following compact ball around $\alpha\mathopen{}\mathclose\bgroup\left(0\aftergroup\egroup\right)$: \begin{equation*}K:=\overline{B_{\ell\mathopen{}\mathclose\bgroup\left(\alpha\aftergroup\egroup\right)}\mathopen{}\mathclose\bgroup\left(\alpha\mathopen{}\mathclose\bgroup\left(0\aftergroup\egroup\right)\aftergroup\egroup\right)}:=\left\{x\in M \mid d\mathopen{}\mathclose\bgroup\left(x,\alpha\mathopen{}\mathclose\bgroup\left(0\aftergroup\egroup\right)\aftergroup\egroup\right)\le \ell\mathopen{}\mathclose\bgroup\left(\alpha\aftergroup\egroup\right)\right\}\text{ where }\ell(\alpha)\text{ is the length of }\alpha.\end{equation*}
\begin{remark}[Piecewise Geodesic]\label{prop:construction}
We can choose $L$ small enough, so that for all points $x,y\in K$ with distance $d\mathopen{}\mathclose\bgroup\left(x,y\aftergroup\egroup\right)\le L$, there exists an unique geodesic that is parameterized by arc length, connects $x$ and $y$ and has length $d\mathopen{}\mathclose\bgroup\left(x,y\aftergroup\egroup\right)$, i.e., the injectivity radius of the compact subset $K\subseteq M$ is greater than $L>0$.
From the path $\alpha$, we construct the nodes of a piecewise geodesic by
\begin{equation*}p_k:=\alpha\mathopen{}\mathclose\bgroup\left(kL\aftergroup\egroup\right)\text{ for }k\in\mathbb Z.\end{equation*}
Hereby, we extend $\alpha$ to $\mathbb R$ constantly: $\alpha\mathopen{}\mathclose\bgroup\left(<0\aftergroup\egroup\right)=\alpha\mathopen{}\mathclose\bgroup\left(0\aftergroup\egroup\right)$ and $\alpha\mathopen{}\mathclose\bgroup\left(>a\aftergroup\egroup\right)=\alpha\mathopen{}\mathclose\bgroup\left(a\aftergroup\egroup\right)$.
Hence, we get finitely many different points:
\begin{equation*}p_0,\dots p_n \text{ with }n=\left\lceil \frac aL \right\rceil.\end{equation*}
\end{remark}
\begin{definition}[Reduced Method on Complete Riemannian Manifolds]
Let $\mathopen{}\mathclose\bgroup\left(M,g\aftergroup\egroup\right)$ be a complete Riemannian manifold with a curve $\alpha$ that is parameterized by arc length. We choose $L>0$, such that for all $x,y\in K:=\overline{B_{\ell\mathopen{}\mathclose\bgroup\left(\alpha\aftergroup\egroup\right)}\mathopen{}\mathclose\bgroup\left(\alpha\mathopen{}\mathclose\bgroup\left(0\aftergroup\egroup\right)\aftergroup\egroup\right)}$ with distance $d\mathopen{}\mathclose\bgroup\left(x,y\aftergroup\egroup\right)\le L$, there is a unique geodesic with length $d\mathopen{}\mathclose\bgroup\left(x,y\aftergroup\egroup\right)$ connecting the points $x$ and $y$.
$m\mathopen{}\mathclose\bgroup\left(x,y\aftergroup\egroup\right)$ denotes the midpoint on this geodesic.
The reduced method shortens the curve $\alpha$ by applying the following map to the nodes of the piecewise geodesic $p_k:=\alpha\mathopen{}\mathclose\bgroup\left(kL\aftergroup\egroup\right)$:
\begin{equation*} R\colon p \longmapsto \biggl( p_0, \Bigl(m\bigl(m\mathopen{}\mathclose\bgroup\left(p_{k-1},p_k\aftergroup\egroup\right), m\mathopen{}\mathclose\bgroup\left(p_k,p_{k+1}\aftergroup\egroup\right)\bigr)\Bigr)_{k=1}^{n-1}, q_n\biggr).\end{equation*}
\end{definition}
\begin{theorem}
On a complete Riemannian manifold $\mathopen{}\mathclose\bgroup\left(M,g\aftergroup\egroup\right)$, we can choose for every curve $\alpha$ that is parameterized by arc length a length of a short ruler $L>0$, which allows to define a piecewise geodesic $p_k:=\alpha\mathopen{}\mathclose\bgroup\left(kL\aftergroup\egroup\right)$.
There is a subsequence of the iteration of the reduced method $R^tp$, which converges to a geodesic $g$.
If no other geodesic, which connects start point and end point of the curve, has the same length than $g$, then the sequence produced by the reduced method converges to the geodesic $g$.
\end{theorem}
\section{Cone}
In this section $\alpha$ denotes an arbitrary path which is parameterized by arc length in a cone $C$. For a short ruler with length $L>0$ the supporting points of the piecewise geodesic for $\alpha$ are $p_k:=\alpha\mathopen{}\mathclose\bgroup\left(kL\aftergroup\egroup\right)$ as before.
\begin{definition}[Cone]
We define a (right circular) cone as:
\begin{equation*} C:=\mathopen{}\mathclose\bgroup\left\lbrace\mathopen{}\mathclose\bgroup\left(x,y,z\aftergroup\egroup\right)\in\mathbb R^3\mathrel{}\middle|\mathrel{} z=-\sqrt{f^2-1}\sqrt{x^2+y^2}\aftergroup\egroup\right\rbrace \text{ with a constant } f>1.\end{equation*}
If $f$ is larger, then the cone is more pointed.
\end{definition}
A cone is not a manifold, because it is not smooth in the apex $0$. Without the apex the cone becomes a Riemannian manifold, but is not complete anymore. Therefore, we cannot use the results for complete Riemannian manifolds directly.
\begin{figure}\centering
\subfigure[Cone]{\includegraphics[width=.2\textwidth]{cone.jpg}}\hfil
\subfigure[$1$ time]{\includegraphics[width=.15\textwidth]{param.jpg}}\hfil
\subfigure[$2$ times]{\includegraphics[width=.3\textwidth]{param2x.jpg}}\hfil
\subfigure[$3$ times]{\includegraphics[width=.3\textwidth]{param3x.jpg}}\hfil
\caption{Cone unwinded. (Proposition \ref{prop:unwind})}
\end{figure}
\begin{proposition}\label{prop:unwind}
The cone is covered by
\begin{equation*}F\colon [0,\infty[{}\times\mathbb R\to C\colon \mathopen{}\mathclose\bgroup\left(r,\varphi\aftergroup\egroup\right) \longmapsto \begin{pmatrix}r\cos\mathopen{}\mathclose\bgroup\left(f\varphi\aftergroup\egroup\right)\\ r\sin\mathopen{}\mathclose\bgroup\left(f\varphi\aftergroup\egroup\right)\\ -r\sqrt{f^2-1}\end{pmatrix}.
\end{equation*}
So we can unwind a curve $\gamma\colon t\mapsto F\mathopen{}\mathclose\bgroup\left(r\mathopen{}\mathclose\bgroup\left(t\aftergroup\egroup\right), \varphi\mathopen{}\mathclose\bgroup\left(t\aftergroup\egroup\right)\aftergroup\egroup\right)$ on the cone by considering $\mathopen{}\mathclose\bgroup\left(r,\varphi\aftergroup\egroup\right)$ as polar coordinates of $\mathbb R^2$:
\begin{equation*}\begin{pmatrix}r\mathopen{}\mathclose\bgroup\left(t\aftergroup\egroup\right)\cos\mathopen{}\mathclose\bgroup\left(\varphi\mathopen{}\mathclose\bgroup\left(t\aftergroup\egroup\right)\aftergroup\egroup\right) \\ r\mathopen{}\mathclose\bgroup\left(t\aftergroup\egroup\right)\sin\mathopen{}\mathclose\bgroup\left(\varphi\mathopen{}\mathclose\bgroup\left(t\aftergroup\egroup\right)\aftergroup\egroup\right)\end{pmatrix}.\end{equation*}
This is a straight line in the plane iff. $\gamma$ is a geodesic on the cone.
\end{proposition}
\begin{figure}\centering
\subfigure[$f=2\upi$]{\includegraphics[width=.2\textwidth]{cone1side.jpg}}\hfil
\subfigure[from top]{\includegraphics[width=.27\textwidth]{cone1.jpg}}\hfil
\subfigure[unwinded]{\includegraphics[width=.25\textwidth]{param1.jpg}}\hfil
\\\hfil
\subfigure[$f=\upi$]{\includegraphics[width=.24\textwidth]{cone2side.jpg}}\hfil
\subfigure[from top]{\includegraphics[width=.27\textwidth]{cone2.jpg}}\hfil
\subfigure[unwinded]{\includegraphics[width=.2\textwidth]{param2.jpg}}\hfil
\\\hfil
\subfigure[$f=\frac{2\upi}5$]{\includegraphics[width=.32\textwidth]{cone5side.jpg}}\hfil
\subfigure[from top]{\includegraphics[width=.27\textwidth]{cone5.jpg}}\hfil
\subfigure[unwinded]{\includegraphics[width=.25\textwidth]{param5.jpg}}\hfil
\caption{Cones with geodesics.}
\end{figure}
% All geodesics between two points differ in the winding number around the apex.
For every compact subset of a complete Riemannian manifold, the injectivity radius was positive: $L>0$. This is false in a neighborhood of the apex of a cone, as the following example shows.
\begin{example}\label{ex:antipodal}
For all $L>0$, there are antipodal points $p$ and $q$ with $d\mathopen{}\mathclose\bgroup\left(p,q\aftergroup\egroup\right)\le L$ and \emph{two} (different) geodesics, which connect $p$ and $q$ and both have the length $d\mathopen{}\mathclose\bgroup\left(p,q\aftergroup\egroup\right)$.
\end{example}
\begin{proof} The map $F\mathopen{}\mathclose\bgroup\left(r,\cdot\aftergroup\egroup\right)$ is periodic with period $\frac{2\upi}f$. Therefore, the following points are antipodal (for all $r$):
\begin{equation*}p:=F\mathopen{}\mathclose\bgroup\left(r,0\aftergroup\egroup\right)\text{ and }q:=F\mathopen{}\mathclose\bgroup\left(r,\varphi\aftergroup\egroup\right)=F\mathopen{}\mathclose\bgroup\left(r,-\varphi\aftergroup\egroup\right)\text{ with }\varphi:=\frac{\upi}f.\end{equation*}
In the plane there are two different straight segments connecting $\begin{pmatrix} r\cos\mathopen{}\mathclose\bgroup\left(0\aftergroup\egroup\right)\\ r\sin\mathopen{}\mathclose\bgroup\left(0\aftergroup\egroup\right)\end{pmatrix}$ with the following points each:
\begin{equation*}\begin{pmatrix} r\cos\mathopen{}\mathclose\bgroup\left(\varphi\aftergroup\egroup\right)\\ r\sin\mathopen{}\mathclose\bgroup\left(\varphi\aftergroup\egroup\right)\end{pmatrix}\text{ and }\begin{pmatrix} r\cos\mathopen{}\mathclose\bgroup\left(-\varphi\aftergroup\egroup\right)\\ r\sin\mathopen{}\mathclose\bgroup\left(-\varphi\aftergroup\egroup\right)\end{pmatrix}.\end{equation*}
The corresponding geodesics on the cone are different, too. Both geodesics connect the points $p$ and $q$ and have the length $d\mathopen{}\mathclose\bgroup\left(p,q\aftergroup\egroup\right)$. If we choose $r$ small enough the points are close to the apex with $d\mathopen{}\mathclose\bgroup\left(p,q\aftergroup\egroup\right)\le L$, e.g., $r:=\frac L\upi$.
\end{proof}
This example shows that we cannot define the reduced method as before, because sometimes no short ruler gives us \emph{unique} midpoints for sure. Therefore, we choose them arbitrarily (but fixed) to define the reduced method on the cone.
For this definition, the length of the ruler does not matter, it changes nothing if we restrict to $d\mathopen{}\mathclose\bgroup\left(x,y\aftergroup\egroup\right)\le L$ for some $L>0$.
\begin{figure}\centering
\subfigure[from top]{\includegraphics[width=.5\textwidth]{cone-midpoint.jpg}}\hfil
\subfigure[unwinded]{\includegraphics[width=.38\textwidth]{param-midpoint.jpg}}\hfil
\caption{Two shortest geodesics for antipodal points with midpoints. (Example \ref{ex:antipodal})}
\end{figure}
\begin{definition}[Reduced Method on Cones] \label{def:Rcone} For a cone $C$ with two points $x,y\in C$, we have two cases:
\begin{enumerate}
\item There are more than one shortest geodesic connecting $x$ and $y$: We choose one of them arbitrarily, and select the midpoint $m\mathopen{}\mathclose\bgroup\left(x,y\aftergroup\egroup\right)$ on this geodesic, e.g., we choose the one with the positive winding number around the apex.
\footnote
{This case arises only if the the points are antipodal. Then, there are exactly two shortest geodesics connecting them; these have winding numbers $\pm\frac 12$ respectively.}
\item The points $x$ and $y$ are not antipodal, i.e. the shortest geodesic connecting them is uniquely determined: Then choose the midpoint $m\mathopen{}\mathclose\bgroup\left(x,y\aftergroup\egroup\right)$ on this geodesic.
\end{enumerate}
Let $\alpha$ be a curve on the cone $C$ that is parameterized by arc length. We choose $L>0$ arbitrarily and $p_k:=\alpha\mathopen{}\mathclose\bgroup\left(kL\aftergroup\egroup\right)$. As we have unique midpoints, we can define the reduced method for shortening the curve $\alpha$ on the whole cone $C$ just as before:
\begin{equation*} R\colon C^{n+1}\to C^{n+1}, p\longmapsto \Biggl(p_0, \biggl(m\Bigl(m\bigl(p_{k-1},p_k\bigr), m\bigl(p_k, p_{k+1}\bigr)\Bigr)\biggr)_{k=1}^{n-1}, p_n\Biggr).\end{equation*}
\end{definition}
\begin{figure}\centering
\subfigure[beginning]{\includegraphics[width=.3\textwidth]{cone-R0.jpg}}\hfil
\subfigure[$1$ step]{\includegraphics[width=.3\textwidth]{cone-R1.jpg}}\hfil
\subfigure[$2$ steps]{\includegraphics[width=.3\textwidth]{cone-R2.jpg}}\hfil
\\\hfil
\subfigure{\includegraphics[width=.27\textwidth]{param-R0.jpg}}\hfil
\subfigure{\includegraphics[width=.27\textwidth]{param-R1.jpg}}\hfil
\subfigure{\includegraphics[width=.27\textwidth]{param-R2.jpg}}\hfil
\caption{The reduced method on the cone. (Definition \ref{def:Rcone})}
\end{figure}
\begin{lemma}\label{lem:subsequence}
The iterated reduced method has a convergent subsequence: $R^{s_t}p\to g$.
\end{lemma}
\begin{proof} The maximum of the distances from the apex $0$ to the points in $p$ exists (finitely many):
\begin{equation*} M:=\max_{0\le k\le n}d\mathopen{}\mathclose\bgroup\left(p_k, 0\aftergroup\egroup\right).\end{equation*}
The subset $K:=C\cap\overline{B_M\mathopen{}\mathclose\bgroup\left(0\aftergroup\egroup\right)}$ is compact, and for all $x,y\in K$ the midpoint $m\mathopen{}\mathclose\bgroup\left(x,y\aftergroup\egroup\right)$ is in $K$, too. Since all $p_k$ are in $K$, also all $Rp_k$ and inductively all $R^tp_k$ are in $K$.
So, $R^tp$ is a sequence in the compact set $K^{n+1}$. Therefore, $R^tp$ has a convergent subsequence.
\end{proof}
\begin{figure}\centering
\subfigure[$10$ steps]{\includegraphics[width=.3\textwidth]{cone-R10+g.jpg}}\hfil
\subfigure[$11$ steps]{\includegraphics[width=.3\textwidth]{cone-R11.jpg}}\hfil
\subfigure[$12$ steps]{\includegraphics[width=.3\textwidth]{cone-R12.jpg}}\hfil
\\\hfil
\subfigure{\includegraphics[width=.27\textwidth]{param-R10+g.jpg}}\hfil
\subfigure{\includegraphics[width=.27\textwidth]{param-R11.jpg}}\hfil
\subfigure{\includegraphics[width=.27\textwidth]{param-R12.jpg}}\hfil
\caption{Sometimes the reduced method skips a geodesic and the apex. (Remark \ref{rem:Rcone})}\label{fig:skip}
\end{figure}
\begin{remark}\label{rem:Rcone}
In some cases if the reduced method is iterated, the polygonal geodesic skips the apex at one step. For example in figure \ref{fig:skip} it skips the apex and the nearby geodesic, and is converging to the shortest geodesic then.
But it is not always converging to the shortest geodesic (compare figure \ref{fig:conv}). Although in both examples the limit $g:=\lim R^tp$ is a fixed point of the reduced method, this is not always the case, since $R$ is not continuous (it can jump over the apex in some cases).
\end{remark}
\begin{figure}\centering
\subfigure[converges to the shortest geodesic.]{\includegraphics[width=.45\textwidth]{cone-R20+g.jpg}}\hfil
\subfigure[converges to a longer geodesic.]{\includegraphics[width=.45\textwidth]{cone-27-R15+g.jpg}}\hfil
\\\hfil
\subfigure{\includegraphics[width=.39\textwidth]{param-R20+g.jpg}}\hfil
\subfigure{\includegraphics[width=.39\textwidth]{param-27-R15+g.jpg}}\hfil
\caption{The reduced method converges. (Theorem \ref{thm:end})}\label{fig:conv}
\end{figure}
\begin{theorem}[Main Result]\label{thm:end}
For every curve $\alpha$ that is parameterized by arc length on a cone $C$, we can choose the short ruler $L>0$ arbitrarily and define a piecewise geodesic $p_k:=\alpha\mathopen{}\mathclose\bgroup\left(kL\aftergroup\egroup\right)$.
If the reduced method from definition \ref{def:Rcone} is applied iteratively to the piecewise
geodesic
$p$, it converges to a geodesic $g$:
\begin{equation*} R^tp\to g.\end{equation*}
\end{theorem}
\begin{proof} According to the lemma \ref{lem:subsequence} the iterated reduced method $R^tp$ has a convergent subsequence:
$R^{s_t}p\to g$.
We want to show that $g$ is a fixed point of the reduced method: $Rg=g$. But that is not possible in every case, since $R$ is not continuous (remark \ref{rem:Rcone}). Instead, we will define another (continuous) midpoint method $S$ to show that $g$ is geodesic.
First we look at the case that starting point $p_0$ and end point $p_n$ are not the apex $0$. Then, we can assume WLOG that the whole curve $\alpha$ is not running through the apex, i.e. all $p_k\ne0$; else, this is true after some iteration steps $R^rp$ because the apex is never a midpoint: $m\mathopen{}\mathclose\bgroup\left(x,y\aftergroup\egroup\right)\ne0$. Therefore, we never reach the apex by iterating the reduced method $R^t p_k \ne0$, too.
The apex is not a limit point of $R^tp_k$ neither: If it gets close enough to the apex after some steps, the apex can be skipped using the short ruler $L>0$. The reduced method moves the piecewise geodesic away from the apex in the subsequent steps. So, all piecewise geodesics defined by the points and by the limit points of the iterated reduced method stay outside of some ball around the apex:\footnote{The image operator ($\im$) means the whole piecewise geodesic not only its supporting points.}
\begin{equation}\label{eq:ball} \exists\varepsilon>0\: \forall t\ge0\colon \im\mathopen{}\mathclose\bgroup\left(R^tp\aftergroup\egroup\right), \im\mathopen{}\mathclose\bgroup\left(g\aftergroup\egroup\right)\in C\setminus B_\varepsilon\mathopen{}\mathclose\bgroup\left(0\aftergroup\egroup\right).\end{equation}
On $C\setminus B_\varepsilon\mathopen{}\mathclose\bgroup\left(0\aftergroup\egroup\right)$ we choose a short ruler $\tilde L<\frac{\upi\varepsilon}f$. Then, the shortest geodesics between the points $x,y\in C\setminus B_\varepsilon\mathopen{}\mathclose\bgroup\left(0\aftergroup\egroup\right)$ with distance $d\mathopen{}\mathclose\bgroup\left(x,y\aftergroup\egroup\right)\le \tilde L$ are unique. To define the midpoint method $S$ for a curve $\beta$ in $C\setminus B_\varepsilon\mathopen{}\mathclose\bgroup\left(0\aftergroup\egroup\right)$, we calculate:
\begin{equation*} q_k:=\beta\mathopen{}\mathclose\bgroup\left(k\tilde L\aftergroup\egroup\right) \text{ and } S\beta:=Sq:=\bigl(q_0, m\mathopen{}\mathclose\bgroup\left(q_0, q_1\aftergroup\egroup\right), \dots m\mathopen{}\mathclose\bgroup\left(q_{m-1}, q_m\aftergroup\egroup\right), q_m\bigr).\end{equation*}
This midpoint method is continuous for $\beta$ in $C\setminus B_\varepsilon\mathopen{}\mathclose\bgroup\left(0\aftergroup\egroup\right)$ because it cannot skip the apex with the short ruler $\tilde L$.
Since we have (\ref{eq:ball}) the piecewise geodesics with supporting points $g$ respectively $R^tp$ are in $C\setminus B_\varepsilon\mathopen{}\mathclose\bgroup\left(0\aftergroup\egroup\right)$ and we can apply $S$ to them. Then, we write $Sg$ respectively $S\mathopen{}\mathclose\bgroup\left(R^tp\aftergroup\egroup\right)$.
We can assume WLOG that $\tilde L\le L$. The reduced method with short ruler $L$ shortens a curve not less than the midpoint method with short ruler $\tilde L$. For the piecewise geodesic $R^tp$ we get:
\begin{equation*} \ell\mathopen{}\mathclose\bgroup\left(R^{t+1}p\aftergroup\egroup\right)= \ell\Bigl(R\mathopen{}\mathclose\bgroup\left(R^tp\aftergroup\egroup\right)\Bigr)\le \ell\Bigl(S\mathopen{}\mathclose\bgroup\left(R^tp\aftergroup\egroup\right)\Bigr)\le \ell\mathopen{}\mathclose\bgroup\left(R^tp\aftergroup\egroup\right).\end{equation*}
Especially, the sequence of lengths we get by iterating the reduced method is monotonically decreasing in $\mathbb R$. Therefore, it converges, and so the sequence of lengths of $S\mathopen{}\mathclose\bgroup\left(R^tp\aftergroup\egroup\right)$ converges, too:
\begin{equation*} \lim \ell\mathopen{}\mathclose\bgroup\left(R^{t+1}p\aftergroup\egroup\right) = \lim\ell\Bigl(R\mathopen{}\mathclose\bgroup\left(R^tp\aftergroup\egroup\right)\Bigr) = \lim \ell\Bigl(S\mathopen{}\mathclose\bgroup\left(R^tp\aftergroup\egroup\right)\Bigr) = \lim \ell\mathopen{}\mathclose\bgroup\left(R^tp\aftergroup\egroup\right) = \lim \ell\mathopen{}\mathclose\bgroup\left(R^{s_t}p\aftergroup\egroup\right).\end{equation*}
Since $\ell$ and $S$ are continuous, it holds:
\begin{equation*} \ell\mathopen{}\mathclose\bgroup\left(Sg\aftergroup\egroup\right) = \lim \ell\Bigl(S\mathopen{}\mathclose\bgroup\left(R^tp\aftergroup\egroup\right)\Bigr) = \lim \ell\mathopen{}\mathclose\bgroup\left(R^{s_t}p\aftergroup\egroup\right) = \ell\mathopen{}\mathclose\bgroup\left(g\aftergroup\egroup\right).\end{equation*}
The midpoint method $S$ does not shorten the piecewise geodesic $g$. That is only possible, if the segments between successive midpoints were geodesic before $S$ was applied. So, $g$ must be geodesic.
In the case that the starting point is the apex $p_0=0$, we modify the last step, where we apply the midpoint method: We choose the point $q_t$ on the intersection of the geodesic between $R^tp_0=0$ and $R^tp_1$ with the border of the ball $B_\varepsilon\mathopen{}\mathclose\bgroup\left(0\aftergroup\egroup\right)$. Then, we apply $S$ to the piecewise geodesic with supporting points $\mathopen{}\mathclose\bgroup\left(q_t, R^tp_1, \dots R^tp_n\aftergroup\egroup\right)$, and connect it with $0$ again:
\begin{equation*} \Bigl(0, S\mathopen{}\mathclose\bgroup\left(q_t, R^tp_1, \dots R^tp_n\aftergroup\egroup\right)\Bigr) .\end{equation*}
The limit $g_\varepsilon:=\lim q_t$ lies on the geodesic between $g_0$ and $g_1$.
Similar as above we get that the supporting points $\mathopen{}\mathclose\bgroup\left(g_\varepsilon , g_1, \dots g_n\aftergroup\egroup\right)$ define a geodesic. This geodesic overlaps with the geodesic defined by $\mathopen{}\mathclose\bgroup\left(g_0, g_\varepsilon , g_1\aftergroup\egroup\right)$, and so $g$ is geodesic.
The case that the end point $p_n$ is the apex, can be handled analogously.
It remains to show that the whole sequence $R^tp$ not only one subsequence converges to $g$. If we unwind the geodesic $g$ into the plane, we get a straight line. And as $R^{s_t}p$ converges to $g$, there exists a point $s\in\mathbb N$, such that $R^sp$ is close enough to $g$ that it can be unwinded into the plane (completely into one layer). From this point on ($t\ge s$), we can look at the reduced method $R^{t}p$ in the plane, where it converges (see theorem \ref{thm:result}).
\end{proof}
\section{Summary}
We saw in theorem \ref{thm:end} that on cones the sequence of curves converges if shortened by the reduced method (definition \ref{def:Rcone}) iteratively. The limit is geodesic but neither has to be the shortest connection (other than in the plane) nor must it have the same winding number around the center (other than on cylinders
\cite[cf.][]{Stadler12}).
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\end{document}