# The horizontal chord set

Diana Davis  and  Serge Troubetzkoy

###### Abstract.

We study the set of lengths of the horizontal chords of a continuous function.
We give a new proof of Hopf’s characterization of this set, and show that it
implies that no matter which function we choose, at least half of the possible
lengths occur. We prove several results about functions for which _all_ the
possible lengths occur.

##  1\. Story

The mathematical conference center CIRM, on the outskirts of Marseille in
France, was created in 1981 by the French mathematics community and hosts over
4500 mathematicians each year. CIRM is situated on an estate with a thousand-
year history, at the edge of the Calanques National Park.

On a beautiful summer day, two mathematicians walk from the CIRM research
institute down, down, down to the Mediterranean Sea at the beautiful Calanque
de Sugiton. After briefly appreciating the cliffs, the waves, and the people
jumping off of the former into the latter, they then walk back up, up, up via
the same path, the whole trip taking one hour. They wonder: is there a spot on
the route that they passed by two times, exactly 23 minutes apart? How about
$\ell$ hours apart, for any $\ell\in[0,1]$?

In this paper, we show that the answer to the above question is “yes.” Such a
spot exists for every $\ell$ even if the mathematicians wander back and forth
on the trail along the way (left and middle pictures in Figure 1). On the
other hand, if the mathematicians pass by CIRM on their return and walk in the
other direction, and then go back to part of the way to the Calanque before
returning to CIRM (right picture in Figure 1), there is no such guarantee;
however, at least half of the possible times must be achieved. The main tool
in the proof of this last result is Hopf’s characterization of the possible
set of lengths $\ell$ which can occur, and we also give a new proof of Hopf’s
result.

Figure 1. A simple hike, a meandering hike, and a wandering hike. Time is on
the horizontal axis, and distance from CIRM towards the Calanque is on the
vertical axis.

A different application of the issues discussed in this article, to average
pace over time, was given in [1].

##  2\. Definitions

By shifting and scaling, we may assume that length (in time) of the hike is
$1$, and that the common value of $f(0)$ and $f(1)$ is $0$. Let

$\mathcal{C}:=\\{f:[0,1]\to\mathbb{R},f\text{ is continuous such that
}f(0)=f(1)=0\\}.$

The horizontal chord set is the set of lengths horizontally connecting two
points on the graph, i.e.,

$S(f):=\\{\ell\in[0,1]:\text{ there exists }s\in[0,1]\text{ with
}f(s)=f(s+\ell)\\}.$

Hopf has characterized which sets $S\subset\mathbb{R}$ can be a horizontal
chord set for some $f\in\mathcal{C}$ ([4], see [1] for an English version of
Hopf’s proof). We study the size and structure of $S(f)$. We begin with the
following question: for which functions $f\in\mathcal{C}$ does $S(f)=[0,1]$?
We call this the full chord property.

######  Definition.

Suppose $f$ is a continuous function, $s_{1}<s_{2}$, $a>0$,
$f(s_{1})=f(s_{2})=0$ and $f((s_{1},s_{2}))=(0,a]$. Then we call
$f|_{[s_{1},s_{2}]}$ a mountain with endpoints $s_{1}$ and $s_{2}$. Fix
$t\in[s_{1},s_{2}]$ so that $f(t)=a$. (Note that while there may be several
choices for $t$, none of our results depend on the choice.) Then we call
$f|_{[s_{1},t]}$ the ascent, $f|_{[t,s_{2}]}$ the descent, $a$ the height, and
$|s_{2}-s_{1}|$ the width.

A valley is exactly analogous to the above, with $a<0$ and “ascent” and
“descent” exchanged.

A mountain range is a union of contiguous mountains, and possibly also
intervals with $f(x)=0$ (including intervals degenerate to a point), i.e.,
essentially the same definition as above but with $f([s_{1},s_{2}])=[0,a]$.
The height of a mountain range is the height of its tallest mountain, which
exists because a continuous function has a maximum on a compact set. The
ascent, descent, and width are as above. We define a valley range analogously.

Note that mountain ranges and valley ranges can include horizontal intervals
satisfying $f(x)=0$. Such a horizontal piece between a mountain range and a
valley range can be considered as part of either one; the choice does not
affect our results.

When we refer to a mountain or a valley range we will always implicitly assume
it is a maximal mountain or valley range, modulo the choice of assignment of
horizontal pieces at $f(x)=0$ as described above.

##  3\. There and back again

######  Theorem 1.

If $f$ is continuous and $f|_{[s_{1},s_{2}]}$ is a mountain range, then
$[0,s_{2}-s_{1}]\subset S(f)$.

######  Corollary 2.

If $f(x)\geq 0$ for all $x\in[0,1]$ then $f$ has the full chord property.

In particular the walk to Sugiton has the full chord property. Both of these
results also hold for valley ranges.

###### Proof.

Fix a mountain range $f|_{[s_{1},s_{2}]}$, and consider the shifted mountain
range $f(x-\ell)$ (see Figure 2). For any shift $\ell\in[0,s_{2}-s_{1}]$,
exactly one endpoint of the mountain range is inside the other copy. Thus it
follows from the Intermediate Value Theorem that the two mountain ranges
intersect, so $f(x)=f(x-\ell)$ and there is a horizontal chord of length
$\ell$. ∎

A closely related problem is the following: two mountain climbers begin at sea
level at opposite ends of a mountain range, can they find routes along which
to travel, always maintaining equal altitudes, until they eventually meet? A
positive solution implies the full chord property. It holds for piecewise
monotone mountain ranges [8, 3] and for mountain ranges without plateaus [5],
however this stronger property does not hold for certain mountains with
plateaus [5].

Figure 2. A mountain range and its shift always intersect.

##  4\. Structure of the chord set

Next we show that the chord set might not be the entire interval $[0,1]$:

######  Theorem 3.

If $f$ has a mountain at one endpoint and a valley at the other endpoint, then
$f$ does not have the full chord property.

###### Proof.

Let $w_{1},w_{2}$ be the width of the mountain and the valley, respectively.
Then any chord with horizontal displacement in the range
$[1-\min\\{w_{1},w_{2}\\},1)$ must have one endpoint in the mountain and one
endpoint in the valley. Thus the function values at these two endpoints have
different signs, so the chord between them is not horizontal. ∎

Most of this paper consists of exploring closed intervals of chord lengths.
However, the following figure shows that $S(f)$ does not necessarily consist
of unions of closed intervals; there is a possibility that it can include
isolated points.

By making lots of bumps, we can make the set of such additional chord lengths
into a set with accumulation points.

Figure 3. (left) A function with $S(f)=[0,1/2]\cup\\{5/8,3/4,7/8,1\\}$; the
chord with length $5/8$ is marked in bold, and cannot be slid up nor down;
(right) a function where $S(f)$ has an accumulation point at 1.

##  5\. Hopf’s theorem

We begin this section by the following simple result about continuous periodic
functions.

######  Proposition 4.

Let $F:\mathbb{R}\to\mathbb{R}$ be a continuous periodic function with period
$T$. Then for each $a\in\mathbb{R}$ the graph of $F(x-a)$ intersects the graph
of $F(x)$ at least once in each interval of length $T$.

###### Proof.

If $F$ is constant then there is nothing to prove, so suppose that $F$ is non-
constant. Fix $a\in\mathbb{R}$, if $a=0$ there is also nothing to prove, so
assume $a\neq 0$.

Since $[0,T]$ is compact $F|_{[0,T]}$ attains a miminum and a maximum, and
since $F$ is periodic these are global extrema as well. Let $(x_{m},y_{m})$ be
a minimum of $F$ and $(x_{M},y_{M})$ a maximum of $F$. If the result is not
true then either $F(x-a)>F(x)$ for all $x\in\mathbb{R}$ or the opposite
inequality holds. In the first case we get a contradiction by choosing
$x=x_{M}$ and in the second case by choosing $x=x_{m}$.

If $F(x-a)=F(x)$ then $F(x-a+kT)=F(x)$ for all $k\in\mathbb{Z}$ by
periodicity. ∎

This proposition immediately yields a new (and simpler) proof of one direction
of Hopf’s classical theorem.

######  Theorem 5.

Let $f:[0,1]\to\mathbb{R}$ be a continuous function with $f(0)=f(1)=0$, and
let $S(f)$ denote its horizontal chord length set. Then the set
$S^{c}:=\mathbb{R}^{+}\setminus S(f)$ is open and additive; i.e., if $a,b\in
S^{c}$ then $a+b\in S^{c}$.

###### Proof.

Suppose $\ell_{i}\in S(f)$ and $\ell_{i}\to\ell$. Then there are points
$x_{i}\in[0,1]$ such that $x_{i}-\ell_{i}\in[0,1]$ and
$f(x_{i})=f(x_{i}-\ell_{i})$. By passing to a subsequence we can assume that
$x_{i}$ converges to a point $x$ and thus $x_{i}-\ell_{i}\to x-\ell$ and by
continuity of $f$ we have $f(x)=f(x-\ell)$. Thus $S(f)$ is closed and so
$S^{c}$ is open.

For the proof we also consider the signed chord length set

$\bar{S}(f):=\\{\ell\in\mathbb{R}:\text{ there exists }s\in[0,1]\text{ such
that }f(s-\ell)=f(s)\\}.$

Notice that $\bar{S}(f)=S(f)\cup-S(f)$. Suppose $a+b\in S(f)$, so we have
$x_{1}\in[0,1]$ and $x_{2}=x_{1}-(a+b)\in[0,1]$ with $f(x_{1})=f(x_{2})$. Let
$F:\mathbb{R}\to\mathbb{R}$ be such that
$F|_{[x_{1},x_{2}]}=f|_{[x_{1},x_{2}]}$ and $F$ is periodic with period $a+b$.
Apply Proposition 4 to $F$ to find a point $x_{0}$ such that
$F(x_{0}-a)=F(x_{0})$.

By the remark we can choose $x_{0}\in[x_{1},x_{2}]$. If
$x_{0}-a\in[x_{1},x_{2}]$ then we conclude $f(x_{0}-a)=f(x_{0})$, i.e., $a\in
S(f)$. On the other hand if $x_{0}-a<x_{1}$ then
$x_{0}-a+(a+b)=x_{0}+b\in[x_{1},x_{2}]$ and so we conclude
$f(x_{0}+b)=f(x_{0})$ and so $-b\in\bar{S}(f)$, and thus $b\in S(f)$. ∎

##  6\. At least half the lengths

The main ingredients in this section are Hopf’s theorem and the central
symmetry $\ell\to d-\ell$ of the interval $[0,d]$ . Let
$S_{d}(f):=S(f)\cap[0,d]$. The image of $S_{d}(f)$ by this symmetry is the set
$T_{d}(f):=\\{\ell:d-\ell\in S_{d}(f)\\}$. Denote the Lebesgue measure on
$[0,1]$ by $\lambda$. We have the following result:

######  Theorem 6.

Let $f:[0,1]\to\mathbf{R}$ be a continuous function with $f(0)=f(1)=0$. Then
the length of $S_{d}(f)$ satisfies $\lambda(S_{d}(f))\geq\frac{1}{2}d$ for
each $d\in(0,1]$. This inequality is strict if $d\not\in S(f)$. For $d=1$ this
yields

$\lambda(S(f))\geq\frac{1}{2}.$

This last bound is sharp.

###### Proof.

Notice that $1\in S(f)$, so $S(f)\neq\emptyset$. Fix $d\in S(f)$. Since the
map $\ell\to d-\ell$ preserves the Lebsegue measure restricted to $[0,d]$ we
have

$\lambda(T_{d}(f))=\lambda(S_{d}(f)).$

By Hopf’s theorem if $a+b\not\in S^{*}$ then we must have either $a$ or $b$
not in $S^{c}$, i.e., at least one of $a$ or $b$ belongs to $S(f)$. We apply
this to $d=\ell$ and $b=d-\ell$, since $d\in S(f)$ we conclude that either
$\ell$ or $d-\ell$ is contained in $S(f)$, or equivalently either $\ell\in
S_{d}(f)$ or $\ell\in T_{d}(f)$; i.e.,

$S_{d}(f)\cup T_{d}(f)=[0,d].$

If $\lambda(S_{d}(f))<d/2$ the two display equations yield an immediate
contradiction, thus the first statement of the theorem holds for any $d\in
S(f)$.

Next suppose $d\not\in S(f)$, then since $S(f)$ is closed and $1\in S(f)$ we
can find $d_{m}\in S(f)$ such that $(d,d_{m})\subset S^{*}$. But then we have
$S_{d}(f)=S_{d_{m}}(f)\setminus\\{d_{m}\\}$ and so
$\lambda(S_{d}(f))=\lambda(S_{d_{m}}(f))\geq d_{m}/2>d/2$, which finishes the
proof of the first and second statements.

To show that the bound is sharp, we construct an example of a function whose
chord set has length $1/2$, with any desired number $n$ of mountains and
valleys.

For each $n\in\mathbf{N}$, consider the complementary sets

$\displaystyle S_{n}$
$\displaystyle=\left[\frac{0}{n},\frac{1}{n+1}\right],\left[\frac{1}{n},\frac{2}{n+1}\right],\ldots,\left[\frac{n-1}{n},\frac{n}{n+1}\right],\text{
and }$ $\displaystyle{S_{n}}^{*}$
$\displaystyle=\left(\frac{1}{n+1},\frac{1}{n}\right),\left(\frac{2}{n+1},\frac{2}{n}\right),\ldots\left(\frac{n-1}{n+1},\frac{n-1}{n}\right),\left(\frac{n}{n+1},1\right).$

It is clear that ${S_{n}}^{*}$ is an additive set, because all of the
intervals are integer multiples of the first one. Thus since ${S_{n}}^{*}$ is
an open additive set, by Hopf’s theorem it is the complement of the chord set
of some function on $[0,1]$. That chord set is $S_{n}$.

The total length of each $S_{n}$ is $1/2$:

$\displaystyle\sum_{k=1}^{n}\left(\frac{k}{n+1}-\frac{k-1}{n}\right)$
$\displaystyle=\sum_{k=1}^{n}\frac{k}{n+1}-\sum_{k=1}^{n}\frac{k-1}{n}$
$\displaystyle=\frac{1}{n+1}\frac{n(n+1)}{2}-\frac{1}{n}\frac{(n-1)n}{2}=\frac{1}{2}.$

Figure 4. Functions having a horizontal chord set of length exactly $1/2$,
with 1, 2, 5 and 22 mountains, respectively.

Given any open, additive set $S^{*}$, Hopf gives a construction for a function
$h_{S}(x)$ whose horizontal chord set is exactly $S^{*}$. Examples of
functions $h_{S_{n}}(x)$ with $n$ mountains and valleys, and chord set with
length $1/2$, given by Hopf’s construction are shown in Figure 4. Thus, for
each $n$, this construction gives the desired function with $n$ bumps and
horizontal chord set of length $1/2$. ∎

Our proof leads to the following natural question: does there exist a example
of a function $f$ with countably many mountain ranges and countably many
valley ranges such that $\lambda(S(f))=1/2$?

The following observation suggests that this is harder to achieve than one
might imagine. Let $C$ be the middle thirds Cantor set, and let
$f:[0,1]\to\mathbb{R}$ be any continuous function such that $f(x)=0$ for all
$x\in C$. The intersection of shifted Cantor sets is well studied; in
particular, it is known that $\displaystyle\\{\ell\in[0,1]:(C-\ell)\cap
C\neq\emptyset\\}=[0,1]$ (see for example [7]). Since this set is just
$\\{\ell\in[0,1]:f(x-\ell)=f(x)=0\\}$, it is a subset of the horizontal chord
set; thus we conclude that $f$ has the full chord property, rather than the
desired measure $1/2$.

##  7\. Acknowledgments

We thank CIRM for an excellent working environment during our « recherche en
binomes » collaboration in July 2022. We used Desmos [2] to experiment,
explore all of the many cases, and create our figures. We thank the referees,
whose suggestions improved our article.

## References

  * [1] Keith Burns, Orit Davidovich, Diana Davis, Average pace and horizontal chords, The Mathematical Intelligencer, 39(4) (2017) 41-45.
  * [2] Desmos, https://www.desmos.com/calculator/, accessed 2022–2023.
  * [3] Jacob E. Goodman, Janos Pach and Chee K. Yap Mountain Climbing, Ladder Moving, and the Ring-Width of a Polygon, The American Mathematical Monthly, 96 (6) (1989) 494–510.
  * [4] Heinz Hopf, Über die Sehnen ebener Kontinuen und die Schleifen geschlossener Wege, Comment. Math. Helv., 9 (1936) 303–319.
  * [5] Tamás Keleti, The mountain climbers’ problem , Proc. AMS, 117(1) (1993) 89–97.
  * [6] Paul Lévy, Sur une Généralisation du Théoréme de Rolle, C.R. Acad. Sci., Paris, 198 (1934) 424–425.
  * [7] Steen Pedersen, Jason D. Phillips, On intersections of Cantor sets: Hausdorff measure, Opuscula Math. 33(3) (2013) 575–598.
  * [8] James Whittaker, A mountain-climbing problem, Canad. J. Math 18 (1966) 873–882.

##  Appendix A Two mountain ranges separated by a valley range

It would be nice to give a classification of which functions’ chord sets have
the full chord property, but the general case seems complicated. Thus here we
will only analyze the simplest non-trivial case, the piecewise affine case
with two mountains and one valley in between.

Let $w_{\ell},w_{v},w_{r}>0$ be the widths of the left mountain, valley, and
right mountain, which are normalized to add up to one, and let
$h_{\ell},h_{r}>0$ and $h_{v}<0$ be their heights. Since the chord set does
not depend on vertical scaling we normalize $\max(h_{\ell},h_{r})=1$. We also
use the widths of the ascent and descent of each mountain, which we call
$a_{\ell},d_{\ell},a_{r}$ and $d_{r}$, respectively (see Figure 5).

Figure 5. A narrow tall mountain always intersects a wide short mountain if
their bases intersect. The image also shows the variables defined above.

We will use the notation $f_{{\bf w},{\bf a},{\bf d},{\bf h}}$. Certain
variables are redundant since a width is the sum of its ascent and descent,
and one of the ascents or descents or widths is also redundant because of our
normalization, thus we have a seven dimensional parameter space along with one
discrete parameter. However the constraints in the next theorem live in a
5-dimensional subspace of our 7-dimensional space because of the three
parameters needed to describe a valley only its width plays a role.

######  Theorem 7.

The piecewise linear map $f_{{\bf w},{\bf d},{\bf a},{\bf h}}$ does not have
the full chord property if and only if the parameters satisfy one of the
following two conditions

  1. (1)

(i) $w_{\ell}<w_{r}$, (ii) $h_{\ell}<1$, (iii) ${h_{\ell}\cdot
a_{r}}+1-w_{r}-a_{\ell}<w_{v}+w_{r}$ and (iv) $1-h_{\ell}\cdot
d_{r}-a_{\ell}>\max(w_{r};w_{\ell}+w_{v})$, or

  2. (2)

(i) $w_{r}<w_{\ell}$, (ii) $h_{r}<1$, (iii) ${h_{r}\cdot
a_{\ell}}+1-w_{\ell}-a_{r}<w_{v}+w_{\ell}$ and (iv) $1-h_{r}\cdot
d_{\ell}-a_{r}>\max(w_{\ell};w_{r}+w_{v})$.

###### Proof.

There are two different ways to get a chord value. The first possibility is
that either the chord connects the two sides of the same mountain or the two
side of the same valley, for such chord lengths Theorem 1 implies $S_{\rm
init}:=[0,w_{\rm max}]\subset S(f)$ where $w_{\rm
max}:=\max(w_{\ell},w_{v},w_{r})$.

Because of the special form of our graph there is only one other type of chord
length possible, namely the lengths of chords that arise between the two
mountains, or equivalently when the left mountain shifted by such a length
intersects the right mountain.

Clearly if the narrower mountain is at least as high as the wider mountain,
then the mountains always intersect when their bases overlap (Figure 5), so
such a function has the full chord property. Thus in order not to have the
full chord property, one mountain must be narrower and shorter. We can
restrict to two cases:

Case (1) (Figure 6) Suppose first that the narrower and shorter mountain is on
the left ((i) $w_{\ell}<w_{r}$ and (ii) $h_{\ell}<h_{r}=1$).

Figure 6. The case when a narrower and shorter mountain is on the left, along
with the definitions of $s_{*}$ and $s^{*}$.

No matter which parameters we choose, chord lengths between the two mountains
arise if the bases of the shifted left mountain and the right mountain
intersect, but neither base is contained in the other base. This set consists
of two intervals described below.

When a parameter value belongs to the interval $S_{\rm
mid}:=[w_{v},w_{\ell}+w_{v}]$, the shift of the left mountain intersects the
ascent of the right mountain, and so $S_{\rm mid}\subset S(f)$ (Figure 7).

When a parameter value belongs to the interval $S_{\rm fin}:=[w_{r}+w_{v},1]$,
the shift of the left mountain intersects the descent of the right mountain,
so $S_{\rm fin}\subset S(f)$.

Figure 7. (Left) a shift by the parameter $w_{v}$ (middle) a shift by the
parameter $w_{l}+w_{v}$ (right) the horizontal chord lengths in $S_{\rm mid}$

Summarizing, we have shown that $S_{\rm init}\cup S_{\rm mid}\cup S_{\rm
fin}\subset S(f)$ no matter which parameter values we choose, as long as the
narrower shorter mountain is on the left.

We will now analyze the complementary set $T:=(S_{\rm init}\cup S_{\rm
mid}\cup S_{\rm fin})^{c}$. The intervals $S_{\rm init}$ and $S_{\rm mid}$
abut or overlap since $w_{v}\leq w_{\rm max}$. The larger of the two right
endpoints of these two intervals is

$\max(w_{\rm
max},w_{\ell}+w_{v})=\max(w_{\ell},w_{v},w_{r},w_{\ell}+w_{v})=\max(w_{r},w_{\ell}+w_{v}).$

Thus we have

(1) $T=(\max(w_{r};w_{\ell}+w_{v}),w_{v}+w_{r}).$

Note that the set $T$ is contained in the set of parameters for which the base
of the shifted left mountain is inside the base of the right mountain, and is
a strict subset if $S_{\text{mid}}\subset S_{\text{init}}\setminus[0,w_{v})$,
or equivalently if $w_{r}>w_{\ell}+w_{v}$.

If the ascent of the smaller mountain is steeper than the ascent of the larger
mountain, or if the descent of the smaller mountain is steeper than the
descent of the larger mountain, then we get an additional interval of chord
values in $S(f)$ that are in $T$, one for each such case.

There are three solutions to the equation $f(s)=h_{\ell}$,
$a_{\ell}<s_{*}<s^{*}$ (pictured in Figure 6). The zero, one or two additional
intervals of chord values in $T$ are then
$T_{1}:=[\max(w_{r};w_{\ell}+w_{v});s_{*}-a_{\ell}]$ and
$T_{2}:=[s^{*}-a_{\ell};w_{v}+w_{\ell}]$, where here an interval $[a,b]$ with
$b<a$ is interpreted as the empty interval.

To calculate $s_{*}$ and $s^{*}$ consider the equations of the ascent and
descent of the right mountain, recalling that $h_{r}=1$:

$y=\frac{1}{a_{r}}(s-(1-w_{r}))\quad\hbox{ and }\quad
y=\frac{-1}{d_{r}}(s-1).$

Replacing $y$ by $h_{\ell}$ in these two equations yields

$s_{*}={h_{\ell}\cdot a_{r}}+1-w_{r}\hbox{ and }s^{*}=1-h_{\ell}\cdot d_{r}.$

Thus the two additional intervals are

(2) $\displaystyle T_{1}$ $\displaystyle:=$
$\displaystyle[\max(w_{r};w_{\ell}+w_{v});{h_{\ell}\cdot
a_{r}}+1-w_{r}-a_{\ell}]\text{ and }$ (3) $\displaystyle T_{2}$
$\displaystyle:=$ $\displaystyle[1-h_{\ell}\cdot
d_{r}-a_{\ell};w_{v}+w_{\ell}].$

The intervals $T_{1}$ and $T_{2}$ never overlap since $s_{*}<s^{*}$, but
either one of them can cover $T$. Thus, the full chord property in the case
under consideration is equivalent to having either $T\subset T_{1}$ or
$T\subset T_{2}$.

Comparing equations (1) and (2)–(3), we see that $T_{1}$ covers $T$ if and
only if

${h_{\ell}\cdot a_{r}}+1-w_{r}-a_{\ell}\geq w_{v}+w_{r}$

and the right end point of $T_{2}$ is larger than the right endpoint of $T$,
thus $T_{2}$ covers $T$ if and only if

$1-h_{\ell}\cdot d_{r}-a_{\ell}\leq\max(w_{r};w_{\ell}+w_{v}).$

If neither of these last two equations, which correspond to the negations of
(iii) and (iv), hold, then we do not have the full chord property.

Case (2) (Figure 8) Now suppose that the narrower and shorter mountain is on
the right ((i) $w_{r}<w_{\ell}$ and (ii) $h_{r}<h_{\ell}=1$).

Figure 8. The case when the right mountain is narrower and shorter

We take the walk backwards, in other words, we consider the function
$g(s):=f\circ i(s)$ where $i$ is the isometry $i:s\mapsto 1-s$. The function
$g$ has the same chord set as $f$, and furthermore if $f$ is in case (2) then
$g$ is in case (1). Applying the characterization proven in case (1) to $g$
and reinterpreting this characterization in terms of the parameters of $f$
yields: (i) $w_{r}<w_{\ell}$, (ii) $h_{r}<1$, (iii) ${h_{r}\cdot
a_{\ell}}+1-w_{\ell}-a_{r}<w_{v}+w_{\ell}$, and (iv) $1-h_{r}\cdot
d_{\ell}-a_{r}>\max(w_{\ell};w_{r}+w_{v}).$ ∎

A Monte Carlo simulation suggests that the percentage of such functions with
the full chord property is approximately $70.4\%$. In principle an exact
calculation is possible, but as the constraints are not linear, it is not
elementary.


Figure 1. (1) and (2) Robot pointing using its body (3) Red path visualised on
the ground (4) Snapshot from the user study with three occluding robots.

# Excuse me! Perception of Abrupt Direction Changes
Using Body Cues and Paths on Mixed Reality Avatars

Nicholas Katzakis  0000-0002-1120-5648 Department of Informatics
Universität Hamburg<EMAIL_ADDRESS>and  Frank Steinicke
Department of Informatics
Universität Hamburg<EMAIL_ADDRESS>
###### Abstract.

We evaluate two methods of signalling abrupt direction changes of a robotic
platform using a Mixed Reality avatar. The “Body” method uses gaze, gesture
and torso direction to point to upcoming waypoints. The “Path” method
visualises the change in direction using an angled path on the ground. We
compare these two methods using a controlled user study and show that each
method has its strengths depending on the situation. Overall the “path”
technique was slightly more accurate in communicating the direction change of
the robot but participants overall preferred the “Body” technique.

Robot, Interaction, Perception, Intentions, Gait, Externalizing, AR, VR

††copyright: none††ccs: Human-centered computing Empirical studies in
visualization††ccs: Human-centered computing User studies††ccs: Computing
methodologies Robotic planning

##  1\. Introduction

Humans have the expectation that robotic agents behave with a degree of social
intelligence (Dautenhahn, 2004; Kidd and Breazeal, 2005) so that bystanders
can interpret their state and predict their action. The diverse forms and
locomotion methods (Murata et al., 2002) of modern robots, however, make it
challenging to impart social characteristics. Some research to date has
attempted to make robots more predictable. This was done primarily by
exploring humanoid robot designs, by adapting their gestures and gaze (Szafir
et al., 2015), and by adapting their locomotion to be more expressive and
predictable (Hoque, 2012; Hoffman et al., 2015; Breazeal et al., 2005). All of
these approaches however, impose a number of constraints on the design of the
robot or on its locomotion method. i.e. If the robot must perform expressive
movements to cater for perception from bystanders, then it cannot travel to
its next waypoint using the shortest trajectory possible, or with the most
energy efficient locomotion method.

_Mixed Reality_ (MR) offers a possible solution to this problem. Mixed reality
agents (Holz et al., 2011) seperate the physical form of the robot from its
outward appearance. The MR approach assumes that robots broadcast their future
trajectories and state etc. to bystanders. Bystanders can then use their
choice of MR display technology to display the robots – and their actions – in
a way that is easier to perceive, thus enhancing their comfort in the
vincinity of robots. An example of the envisioned scenario can be seen in
Figure 2.

Figure 2. Illustration of an envisioned application of MR agents. A humanoid
avatar is superimposed on top of a real, physical wheeled robot.

This work specifically examines direction changes because they are a potential
cause of discomfort to bystanders. When humans make significant changes in
walking direction they convey this change with a number of body cues including
gaze (Hollands et al., 2002), slowing down, etc. Contrary to humans, robots
can change the parameters of their locomotion instantly (Watanabe et al.,
1998) to account for a new heading or to avoid obstacles. Therefore, even if
bystanders trust that robots will not collide with them, these abrupt
direction changes might be a source for distraction and discomfort.

The proliferation of autonomous agents and vehicles raises the question: Is it
possible, with MR agents, to warn bystanders of abrupt direction changes? We
explore the design space and present two potential solutions: One uses body
cues of a humanoid MR avatar, the other displays the direction change as a
path on the ground (Figure 1). The two approaches are evaluated in a
controlled user study that manipulates _cue onset timing_ , _cue
expressiveness_ and _robustness to occlusion_.

##  2\. Related Work

Shoji et al. superimposed an avatar on a robot to increase attractiveness
(Shoji et al., 2006) while Shimizu et al. (Shimizu et al., 2008) superimposed
an avatar that attempts to mimic the movements of the robot. Aspects of intent
or predictability, however, remain largely unexplored.

The approach of Dragone and O’Hare (Dragone et al., 2007, 2006; Holz et al.,
2011) makes some headway towards overcoming these limitations. In the context
of a Mixed Reality Agent (MiRA), they display a humanoid avatar that appears
sitting on a rhoomba robot to communicate the state of the robot. Such an
avatar can indicate direction changes by pointing yet the authors have not
evaluated their proposed system in this context.

Young et al. explored a similar concept using cartooning (Young et al., 2007).
They separated _behavioural likeness_ from _visual likeness_ , in an attempt
to avoid the “uncanny valley” (Mori, 1970). These cartoon-like illustrations
offer a simple, yet powerful means of expressing various states of the robot.
It remains unclear how this approach could be applied to express spatial
intent and direction changes.

The idea of externalizing the internal state of a robot using AR
visualisations has been explored by Collett et al. (Collett and Macdonald,
2010) in the context of debugging the various sensors of the robot. More
recently Hönig et al. (Hönig et al., 2015) also used an augmented reality
setup with the goal of creating a safer environment for debugging algorithms
where robots are to be used in close proximity to humans. Other works have
explored path planning (Zollmann et al., 2014; Giesler et al., 2004; Stilman,
2005), also in an attempt to assess accuracy and diagnose waypoint errors.

Hoffman and Ju (Hoffman and Ju, 2014) have suggested that rather than visual
likeness to humanoids, motion can be a powerful tool to communicate the state
of the robot. Although this approach is ideal for communication when the robot
is stationary, instant waypoint change situations leave little time for
expressive movements.

##  3\. Design

The design goal of the two proposed methods is to inform bystanders that the
robot will change its direction and allow them to predict _where the robot is
going next_. The assumption being that bystanders will feel more comfortable
knowing where the robot is heading. Rather than superimposing a realistic
human avatar, we chose the “Kyle” robot model in unity because of its torso.
The slim torso reduces limb occlusion from various angles and the elbow and
shoulder guards make arm rotations more salient.

###  3.1. Body

When choosing cues for the _body_ method, we opted for cues that are easy to
interpret by humans. Previous studies on human behavior in steering situations
found that participants typically align their head (Hollands et al., 2002;
Patla and Vickers, 1997; Patla, 1999) with the next waypoint immedately
following delivery of a direction change cue. This head alignment occurs
before torso orientation (Patla, 1999). The fact that people expect such cues
from other humanoids in their vincinity, inspired the use of a humanoid avatar
and the first two expressivity levels (Figure 3).

The medium expressivity level is inspired by how in some cultures people
extend their forearm forward as a sign of politeness. A third level of
expressivity was added by exaggerating the turn of the avatar torso, in the
hope that it will make the waypoint change cues more salient.

(a)

(b)

(c)

(d)

Figure 3. Expressivity levels.

###  3.2. Path

Earlier works have explored visualising the path of the robot (Stilman, 2005;
Omidshafiei et al., 2015), yet these designs extend lines far into the future
and this means that in an MR setting where all the robots in the vincinity
broadcast their paths the visual field would quickly become cluttered. We were
therefore interested in exploring the potential of a shorter path with a
pointed tip to indicate direction.

A path displayed at the feet of the avatar is possibly difficult to perceive.
i.e. when a person is fixating forward while walking, the path of an adjacent
robot will be outside his or her field of view. Bringing the path up to the
view frustum of the person might introduce unwanted view obstruction. A
further limitation of a path or arrow displayed along the line of sight is
that it is difficult to judge their magnitude due to perspective (Ogle et al.,
1949).

The displayed path begins between the feet of the robot, joins the point where
the robot will change direction and extends towards the next waypoint _without
reaching_ it (Figure 3d). The path maintains its length the entire time it is
visible i.e. it extends towards the next waypoint with the same speed the
robot is walking. We manipulate the length of the path as a means of
manipulating _expressivity_ in the _path_ method.

##  4\. Evaluation

We conducted a controlled study to evaluate the robustness of the two proposed
methods with regards to:

  * $\circ$

Cue onset timing: How long do these methods need to be displayed in order to
be effective in communicating the robot’s waypoint?

  * $\circ$

Cue expressiveness: How subtle do these cues need to be?

  * $\circ$

Robustness to occlusion: How do these cues perform in a situation with
numerous occluding robots?

Conducting this study in Virtual Reality (VR) allows for studying the
perception of the robotic avatar isolated from unwanted environmental
confounding factors.

###  4.1. Participants - Apparatus

Fourteen unpaid participants (Ages 22-38, mean age 31) were recruited from the
faculty and students of ¡anonymized¿. 75% of participants were right handed
and 25% left handed. 91% of participants had used a virtual reality headset
before. In post-experiment questionnaires participants reported a high-level
of attention during the experiment. Written and informed consent was obtained
from each participant and the experiment was approved by a local ethics
committe. Participants stood in the center of a 4x4 meter space and mounted
the HTC Vive on their head while holding one of the Vive controllers on their
dominant hand as pointer (Figure 4). Auditory cues were delivered through
noise cancelling headphones. The experiment lasted approximately 40 minutes
and participants were free to rest at any point during the experiment.

Figure 4. A participant during the experiment.

###  4.2. Stimuli and procedure

The virtual environment consisted of a 100 x 100 meter plain tiled floor on
which participants were free to roam. At the beginning of the experiment
participants were allowed to familiarize themselves with the VR environment
and the controls.

When the experiment started the avatar walked segments between 1.5 and 3.0
meters. At three preset onset distances, before reaching the end of these
segments, the waypoint change cue would appear (body or path). Participants
were instructed to point using their wand to indicate where they thought the
next waypoint of the robot will be (following the direction change – Figure
5). Participants received visual feedback about their pointing location by a
ray extending from their wand in addition to a sphere at the location where
the ray intersected with the floor. Participants were free to walk around and
teleport using the thumb button (pad) of the Vive controller using a standard
VR teleportation technique.

Figure 5. Experiment task. Participants had to guess where is the next
waypoint of the robot.

The robot continued cycling waypoints with a walking speed of 1 meter/sec and
participants had to keep guessing waypoints. If participants failed to
indicate the correct waypoint in time the trial was queued at the end of the
trials queue to be repeated later. Method was treated as one factor with two
levels, _Body_ and _Path_.In addition to the cue presentation method there
were three factors with three levels each.

  * $\circ$

_Cue onset distance_ , manipulated with three levels: 0.8 m, 1m and 1.5m. In
addition we manipulate

  * $\circ$

_Cue Expressivity_ , manipulated with three levels: low, medium and high.

  * $\circ$

_Number of occluding robots_ , manipulated with three levels: no robots, 3
robots and 6 robots.

  * $\circ$

_Angle_ at which the robot turned, manipulated at 6 levels:
-30,-20,-10,10,20,30.

The occluding robots spawned on a random location in a circle of 3m radius
around the main robot avatar. The robots chose random waypoints in a circle
with a 3m radius centered two waypoints ahead of the main robot avatar. The
result was random occlusion and intersection of the main avatar’s path. Rather
than passing through one another without collisions, in a real situation the
robots should perform collision avoidance. Preventing collisions was a
conscious design decision on our part which aimed to disentangle visual cues
from behavioral prediction cues formed by prior lifetime experiences of the
participants. i.e. We did not want participants to be able to predict that
robots will change direction to avoid collision. They should only rely on the
proposed direction change cues. The main avatar had a white outline attached
in all conditions so that it can be distinguished from the rest of the robots
(Figure 6).

Figure 6. Six Occluding robots in the _Path_ condition. The main robot avatar
has a faint white outline for disambiguation.

All factors were randomized except cue method and number of occluding robots.
Those were repeated in counter balanced blocks. e.g. Body method was tested
with 3 distracting robots, followed by 0 robots, followed by 9 robots etc.
Following one Body x Distracting Robots block the Path method was tested again
with all Distracting robot levels. This was counter balanced across
participants.

This design was chosen because the experiment was too challenging if robot
numbers and technique were alternated with every trial (robots
appearing/disappearing etc.). Given that the robot travelled randomly between
1.5m-3m in every path segment this resulted on an average of 2.25 seconds per
segment. The occluding robots travelled at the same speed (1 m/sec) as the
main avatar. Because of this, if they randomly disappeared and spawned with
every single trial, there would be insufficient time for them to walk around
and form a random pattern while the appearance and disappearance of robots
would be too distracting and detached from the envisioned application scenario
of this work.

We recorded the _reaction time_ (RT), i.e. time between cue onset and trigger
press by the participant. We also recorded the prediction accuracy of
participants, by recording the _Error_. i.e. The euclidean distance between
the actual robot waypoint and the location the participant indicated with the
wand.

The experiment was a within-subjects design with 2 methods x 3 onset distances
x 3 expressivity levels x 3 occluding robot levels x 6 angles x 2 trials = 648
trials. 14 participants x 648 trials per participant = 9072 total trials
collected. The data was analyzed using a repeated-measures ANOVA test in R.

##  5\. Results

As expected, method had a significant effect on both Error ($F_{1,13}=20.78$,
$p<0.01$) and RT ($F_{1,13}=6.12$, $p<0.01$).

There was a significant effect of number of distracting robots (Figure 7) on
Error ($F_{1,13}=6.12$, $p=0.02$). Number of robots had no effect on RT
($F_{1,13}=8$, $p=0.6$).

Figure 7. Effect of number of distractor robots on Error (lower is better).
The body method was more sensitive to the number of distractors.

Cue onset distance did not have a significant effect on Error
($F_{1,13}=0.01$, $p=0.9$) but, as expected, a highly significant effect on RT
($F_{1,13}=384$, $p<0.01$). Figure 8 contains a graph of cue onset distance on
RT.

Figure 8. Effect of cue onset distance on RT.

_Expressivity level_ had no statistical effect on RT ($F_{2,26}=0.3$, $p=0.7$)
but had a significant effect on Error ($F_{2,26}=25.5$, $p<0.001$). A graph is
displayed in Figure 9).

Figure 9. Effect of cue expressivity level on Error.

In a post-experiment questionnaire users were asked to rate both methods in
terms of ”Ease of Perception” on a 5-point likert scale. A summary of the
results can be seen in figure 10.

Figure 10. Participant’s ratings of the methods on a 5-point likert scale. The
_Body_ method was preferred.

There was no significant effect of _Angle_ on Error or RT ($F_{2,26}=1.37$,
$p<0.26$).

##  6\. Discussion

As expected, cueing method (Body or Path) affected the participant’s ability
to accurately sense prediction. Using the _Body_ method participants were on
average 1.21m away from the robot’s waypoint vs. 1.04 m from the path. A
difference of approximately 17cm. On the other hand, on the low expressivity
level, when the path was shortest and the avatar only turned its head, the
Body technique performed significantly better ($F_{2,26}=21.3$, $p<0.001$ \-
Figure 9). Summarizing the responses from the questionnaires shows that
participants preferred the _Body_ technique (Figure 10).

It is also notable that the length of the path (cue expressivity) resulted in
a significant improvement in performance from the low to the high condition.
The 3m path resulted in an average _Error_ of 0.84 m vs. 1.32 m for the 1m
path. Despite this the change is not a multiple of the path length (non-
linear) and this suggests that MR interface designers should not extend the
line too far into the future of the robot’s planned path. This makes
additional sense when considering that paths might be updated in response to
external factors.

The _path_ method also proved more robust to visual occlusions than the _body_
method. Specifically, the path technique’s Error remained around 1.03 m
despite the number of robots present on the scene (Figure 7). In a busy
environment with many robots present designers might use the path technique
whereas when there are fewer robots they could signal direction changes using
the body cues alone. This is especially important considering that when robots
are near, the path might be outside the person’s field of view.

Finally, cue onset distance results can be directly translated to time (in
seconds) because the robot was travelling with a 1 meter/second speed. Results
from our experiment suggest that the path method had a marginally shorter
reaction time than the body method. This is another result that can be
considered by designers when choosing the appropriate cue for the moment.
Although we assume that robots would always cater to avoid collisions with
humans, mechanical errors could occur. In such a situation a robot might be
aware of its own malfunction and broadcast its direction change for bystanders
to avoid it. When the robot is very near the participant’s visor could chose
the path technique when reaction time is of the essence.

One participant commented that when there were many occlusions from the other
robots, they occasionally observed the shadow to find the direction. This
makes sense because an arm extended from the robot’s shadow forms a perfectly
aligned 2D vector on the ground, pointing to the desired waypoint and is
perhaps an additional design space for this domain.

The results from this experiment suggest that each technique has its own
strengths and robotics MR interface designers can take these strenghts and
weaknesses into account when choosing which method to use.

A limitation of this study is that direction changes were considered only for
locomotion across the ground plane. What if the robot is traveling in other
axes? Can these cues be extended to convey direction changes when robots are
performing actions other than walking like drones or other, more exotic robots
that jump around (like a spider robot)?

##  7\. Conclusion - Future Work

We have presented two methods for communicating direction changes using
superimposed MR avatars. Our findings suggest that both these methods have
strong points and that interface designers should consider these strenghts
when choosing which technique to use for their application.

This study is a pilot exploration of a design space that we believe offers
potential for rich communication. Many interesting questions remain open for
future work: Can we detach the MR avatar from the actual robot so that even
though the physical robot is abruptly changing position, the MR avatar starts
turning in advance? Can the MR avatar follow a curved turning path even though
the physical robot turns in a sharp angle? How would such a mapping be
accomplished? These remain as open questions for future work.

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