In January 1999, the anthropologist Mary Douglas and I were walking across Hampstead Heath to a workshop held at the home of Gerald Mars, now honorary professor of anthropology at University College, London, who lived on Nassington Road. The historian Eric Hobsbawm also lived on Nassington Road, just across the street from where we were going, and Mary suggested that we ring his bell and stop to say hello.
What would I say? I had read some of his work, found it enormously enlightening for my efforts as a historian, and applied it to the study of Judaism in antiquity. But we were also so different: My way of being Jewish was far from Hobsbawm's, and I had chosen to live in Israel—a place Hobsbawm denounced in the strongest terms. I feared the conversation would be unpleasant, so I declined to meet him.
Now Hobsbawm is gone, called to the Academy on High, and I can no longer ring his bell. But I want to reflect on the contributions of this unusual scholar to the study of Judaism—a contribution that has, perhaps because of his anti-Zionism, not always been adequately acknowledged.
Hobsbawm was born in Alexandria in 1917, and his life included stops at Vienna and Berlin before he settled in Britain. His principal academic position throughout his life was at Birbeck, University of London. He was ultimately acknowledged, even by his political enemies (of whom there were not a few) as one of the outstanding historians of the 20th century.
Yet, almost despite himself, despite defining himself as a "non-Jewish Jew," Hobsbawm had an enormous influence on the writing of Jewish history. Perhaps his most important distinct contribution was the collection of essays he edited on The Invention of Tradition, published in 1992. Hobsbawm there introduced the term "invented tradition," which he defined as a kind of false consciousness, an invented connection to the past that is largely factitious. One example was the annual Festival of Nine Lessons and Carols celebrated on Christmas Eve in the Chapel of Kings College, Cambridge, where Hobsbawm was a student. It was supposedly ancient, going back (like all else at Cambridge) to the 13th century, but had in fact been invented only a few years before Hobsbawm came to college.
In considering the problem of tradition we must distinguish between two questions. The first is historical: How did a tradition endowed with religious dignity come to be formed? The other question is: How was this tradition understood once it had been accepted as a religious phenomenon? For the faithful promptly discard the historical question once they have accepted a tradition … yet for the historian the historical question remains fundamental: In order to understand the meaning of what the faithful simply accept, the historian is not bound to accept the fictions that veil more than they reveal concerning the origins of the accepted faith.
For Scholem, however, the veils of fiction that surrounded a tradition arose at a later stage of the process of its acceptance. There was some genuine historical origin, which it was the historian's task to discover. Hobsbawm offered his notion of invented tradition as a radical critique of conventional faith, intended to break the intellectual logjam and achieve a much deeper understanding of the processes by which some traditions were formed.
Invented traditions, according to Hobsbawm, came into being at times of great and rapid change. They were ways of re-establishing a connection with the past, when that connection had been broken by circumstances. As such, they had little or no claim to a "genuine" link to the remote past, as they asserted. Instead, they were heavily loaded symbolic actions that promoted a specific way of bridging over the chasm between past and present, wholly meaningful in the present. The essays in the volume on The Invention of Tradition covered any number of examples of such invented traditions, none of them Jewish, but this notion would have helped resolve Scholem's paradox: Claims of teachings given to Moses at Sinai could now be understood as invented traditions that would grant a measure of legitimacy to rabbinic ideas, which even the rabbis who told the story recognized would not have been acknowledged or understood by the historical Moses.
It is therefore not surprising that this idea has been taken up with enthusiasm by some Jewish historians. Michael Silber of the Hebrew University has proposed understanding ultra-Orthodoxy as an invented tradition. However, perhaps on account of Hobsbawm's anti-Zionism, his article contains no reference to Hobsbawm, even though Hobsbawm's notion of invented tradition is fundamental to Silber's argument. Aaron W. Hughes of the University of Buffalo was more generous, acknowledging Hobsbawm as his source of inspiration when discussing religious tolerance in Muslim Spain, as analyzed by modern historians, as an invented tradition. The understanding of da'at Torah in the world of orthodoxy from the early 20th century until our own time as "rabbinic infallibility," as proposed by Gershon Bacon of Bar-Ilan University and others, also fits Hobsbawm's category of invented tradition.
Hobsbawm was widely criticized for his lifelong adherence to communism, despite all the repression and mass death it brought, and for the effect it may have had on his work as a historian. My perspective is different. If I have any criticism of Hobsbawm's notion of invented tradition it concerns his insistence that, while invented traditions were possible at any time and place, they were especially prominent in the modern world. I disagree and have discussed this point at length in an article co-authored with Marina Rustow, now of Johns Hopkins University, in which we noted many examples of invented traditions from the pre-modern Jewish past. As one instance, I suggested there that the notion of invented tradition helps explain texts such as mSheq 6:1, which described the practice of the families of R. Gamaliel and R. Hananiah, who performed one more prostration than others in the Temple "since they had a tradition [masoret] from their ancestors that the ark was buried" in a particular place. The vessels of the First Temple were now lost and hidden, but their eventual discovery also played a significant role in some ancient Jewish millennial scenarios. This is already the case in 2 Macc 2:7–8, in which Jeremiah supposedly reprimanded the Jews of his time who wanted to mark the way to the place where he had hidden the vessels, because "the place shall remain unknown, he said, until God finally gathers his people together and shows mercy to them." Josephus, similarly, told the story of a Samaritan who promised to find the hidden vessels "where Moses had deposited them." Pontius Pilate ordered Roman troops to attack this Samaritan and many of his followers, and according to Josephus they were killed. Pilate understood that this was not an innocent archeological excavation, but an action fraught with millennial meaning and a challenge to Roman rule. It was a claim about the past with important political implications for the present. The same conclusion holds true for the extra prostration performed at the Temple by the families of R. Gamaliel and R. Hananiah.
Furthermore, Rustow and I argued, the distinction between invented traditions and genuine ones may not be the most useful way to analyze traditions. At least sometimes, the new traditions advanced in response to changing circumstances have a genuine basis in older sources, but these are rescued from relative oblivion or reinterpreted to create the new "modern" tradition (Rustow and I called these "strong" traditions). Therefore, rather than focus on whether a tradition is invented or genuine, it is better to concentrate on the change that has taken place and the ways in which strong traditions respond to that change.
Until the end, Hobsbawm defined himself as a Communist, and not as a Jew. But by his own terms, his disproportionate and remarkable contributions to understanding the human past were expressions of his own Jewishness, and—regardless of his insistence that he was a "non-Jewish Jew"—should find their rightful place in our histories.
Albert I. Baumgarten, Professor Emeritus of Jewish History at Bar-Ilan University, is the author of Elias Bickerman as a Historian of the Jews: A Twentieth Century Tale.
I would assumed that the gift you are talking about is that he's dead. There is nothing good about this man. He never apologized for Stalin's crimes and he even justified them.
I suppose Hobsbawm endorsed Stalin and Communism because they would have made repression universal.
The man was another criminal intellectual who endorsed and supported mass murder.
The 20th c gave rise to many intellectuals who supported all kinds of tyrannies from Stalinist communism, to Fascism,, and Nazism.
Hobsbawm is in the company of Ezra Pound, of Robert Brasillach, of Heidegger and Carl Schmitt.
One can be a non believer and still be proud Jew, but people who endorsed pogroms against Jews shouldn't be considered Jews.
You trivialize what being a Jews means by including people who rejected and even worked against Jews.
Let me get this straight: do you posit that he was not a Jew?
His parents were Jews, and he was born into a Jewish family, but he chose not to be a Jew.
Do you consider people born into Jewish families who converted to Christianity or Islam Jews?
Yes,I don't consider him a Jew.
We are quickly veering into the vexed "Who is a Jew?" question, I guess.
Well, since you answered my question, I must answer yours: Yes, sort of, for about one generation. For instance, Marx and Disraeli are sort-of Jews in my book; John Kerry or Capriles or Sarkozy aren't. I think this correlates well with their own preoccupation or lack thereof with their Jewish legacy. Come to think of it, I'd also differentiate between Christianity which (in modern time) allows for some vestiges of complex identity and Islam which requires more or less total immersion.
Now, I'd like you to ponder the obverse of your argument: if it's that easy to will oneself out of Jewishness, is it as easy to will oneself in? How do you regard one who was not born into a Jewish family but wishes to be one (say, as a result of marrying a Jew)?
We are not discussing your book, Mr. Saint.
Ugh, ogh. Boy, that hurt. LOL.
Anyway, I thought we were having a nice polite discussion. If you are not it, fine.
"Expressions of his own" jewishness" begs the question of what "jewishness" is.if it cant be differentiated from non-Jewishness it is enigmatic ,Furthermore "humanity" as an ideal or generic category iswithout is empty manifestations of cultural distinctiveness.As for "non-Jewish Jew" it seems to be the only formulation of jewishness accoetable to the left implying that the only accpetable Jew is one who isnt.
Of course it can be differentiated from non-Jewishness.
Though born a Jew, he lived and died an anti-Semite who justified murder and worse in the name of his ideology.
One last point, Hobsbawm was a mediocre thinker precisely because he didn't recognize the tension between the universal and the particular.
In any case, communism is no more universal than any other system because it is only one of many competing universal creeds. The idea that one live in the universal is an illusion. We are all situated in a particular space and time.
When Hobsbawm says: "Historians, however microcosmic, must be for universalism … because it is the necessary condition for understanding the history of humanity, including that of any special section of humanity." that surely includes the history of communism.
Ironically, Hobsbawm's history is also a kind of "identity history" and in his case he just substituted Communist for Jew.
Claims to universality can never abolish the particular which is why universalists tend to repress the particular often with extreme violence.
"Claims to universality can never abolish the particular which is why universalists tend to repress the particular often with extreme violence." Can you elaborate on this, and provide sources for this statement? I agree with you completely; in fact, I'm writing about this in relation to the work of Mircea Eliade and his ilk. I don't want to leave out any major discussions of this issue.
others have also thought and written about it.
its gulags and "re-educations camps: in China and elsewhere.
gained power for example allowed a modicum of freedom in private affairs. The Hungarian Fascists did not.
Here we are in agreement. Complicated things become very simple once one recognizes that Communism is a religion. (There is by the way a fascinating take on this in Гравилет "Цесаревич", an alternate history by the (postmodernist?) V. Rybakov in which Communism became an actual *peaceful* religion; I think it has not been translated into English – only into Esperanto, which is rather ironic).
I would rather say that religions, just like communism or fascism, are ideological systems.
After reading the entire article, I declare it a banal account of a banal person. If it was written on paper I would feel guilty for wasting a tree.
nationhood is just a song.
cars, computers and poetic verse.
without it life would be far worse.
Communism is a guilty as any ideology of manufacturing traditions.
That Hobsbawm didn't write about that shows what a narrow minded parochial thinker he was.
Don't see the relevance of the first comment. And as for the second one, Hobsbawm was biased and blinkered but hardly parochial.
What do you call his defense of the communist invasion of Hungary in 1956, or the invasion of Czechoslovakia in 1968?
Was it a sign of his universalism?
Professor Baumgarten's focus was on Hobsbawm's proposal of the concept of invented tradition. Hobsbawm's political opinions are irrelevant to this innovative concept, which was the rationale of Baumgarten's excellent article..
Did Hobsbawm invent the concept of invented tradition?
I thought that fact was known as long ago as the 18th century when Samuel Johnson exposed the fake Ossian's poems by Macpherson?
Dbtw: did Hobsbawm ever tackle the invention of history by Marxist historians as well as traditions in Communist countries?
To claim that pseudepigraphy such as Ossian's icorresponds to Hobsbawm's concept of "invented tradition" is misleading. Pseudepigraphy i at least as old as the book of Deuteronomy, and may give rise to invented traditions but does not per se constitute them. Ditto for the invention of history by historians throughout the ages, long before Karl Marx invented himself.
I claimed no such thing. What I claimed was that thinkers were always aware of the difference between invented history and culture and ongoing form of social life.
This is what anthropology and the social sciences are about.
Why do you think the work out methods to screen differences between authentic primitive customs and non authentic (borrowed one's) claimed to be authentic.
I'm not aware of a single person who embrace because he is Jewish. So your parochial hypothesis seems to me totally hypothetical and in fact mistaken. Hobsbawm's construct of invented tradition is original and only appears to be unoriginal when misunderstood and/or misinterpreted. In the Jewish tradition Aher (Elisha ben Abayu) is considered to be so evil that a voice from heaven singles him out as being disqualified from olam haba. In spite of this, some of his opinions are quoted approvingly, as in Vot 4:5. The rabbis did not try to write out of the record the positive contributions he made to Jewish thought. I think Hobsbawm deserves the same credit for his thoughts, whether or not they conform to preconceived notions. Jacob Arnon might usefully consider whether Rabbi Meir's approach to Elisha is more commendable that his own approach to Hobsbawm.
of the Jewish tradition otherwise no Rabbi would have quoted him, period.
supported these actions would be quoted by Rabbis.
Show me one legitimate Rabbi who quotes Hobsbawm?
Elisha exited the Jewish tradition as clearly as Hobsbawm. The fact that Meir continued have a a dialogue with him and that he is quoted in Tannaitic texts proves that Hazal welcomed dialogue with counter-traditional Jews and valued what they said, if what they said was valuable. Your argument to the contrary ('otherwise no Rabbi would have quoted him") is defectively circular. I don't care whether any Rabbi quotes Hobsbawn or not. I do in my book Legal Friction,, and though I am no rabbi I no no one who does not consider me to be art of klal yisrael., even though my book radically reinterprets biblical texts in a way that had not been previously proposed..
Not as clearly since Elisha is pretty much a shadowy figure. Not much is known about him which is why you and others can say whatever they want without fear of contradiction.
However we do know about Hobsbawm and what we know about him isn't going to endear him to the Rabbinic community.
In any case, there is no point in continuing this since neither of us will convince the other of Hobsbawm's merit or lack thereof.
I'd like us to be clear about something essential which is missed by those Jews who embrace Hobsbawm because he was born a Jew.
Their embrace of this communist universalist just because he was "a Jew" is itself a sign of parochialism.
Jew, Communist, Islamic. Christian, Buddhist, Shaivist – or any other form of Hindu – , Taoist, Wiccan, Native American, Laumalist, Universalist, – the list goes on and on – or ANY group of human-formed rituals that purposely attempt to define the ladder of 'enlightenment' of a deity greater than ourselves is in itself the definition of tradition and is thereby itself 'invented.' Each BEING brings a variance to the table, so it matters not whether it is 'invented' or 'historical' for by the very nature all of history – it is ALL 'invented'…. whether based in documented fact or not it is real to whomever practices whatever they practice. As the definition of Jew pervades – to struggle with G-D – the ever changing influence of 'social norms' must always be taken into account. It is only when the boundaries dissolve do any of us truly gain what we seek. The irony is that to accept either Communism or Zionism one must limit the other, but in truth… the ideal of all religions – including both of these formations – is in fact Unity of One…. Therefore, Mr. Hobsbawm has no more affect on the whole of Judaism than what is granted by the individual Jew, and to judge him is simply a 'non jewish' act – or should I say a 'non G-D-like act'. How one defines G-D is never more important than the one who formulates the definition. Be at peace and recognize the part of all of humanity (not just the Jew) that is itself divine….. We are ONE.
Incidentally, I posted on EH today: http://manwithoutqualities.com/2012/10/11/eric-hobsbawm/ — in tune with many of the comments here.
Interesting website, "manwithout " and good critique of Hobsbawm, thanks for posting the link.
Why on earth would TABLET eulogize this creep.
And i was amazed to learn that he was born a Jew.. a very common fixture in today's world.. the self-hating Jew..


text:
Апеляційний суд Кіровоградської області — колишній загальний суд апеляційної (другої) інстанції, розташований у місті Кропивницькому, юрисдикція якого поширювалася на Кіровоградську область.

Суд здійснював правосуддя до початку роботи Кропивницького апеляційного суду, що відбулося 5 жовтня 2018 року.

Керівництво
  Голова суду — Драний Валерій Вікторович
  Заступник голови суду — Кіселик Станіслав Анатолійович
  Заступник голови суду — Осєтров Віктор Іванович
  Керівник апарату — Карауш Тетяна Олегівна.

Показники діяльності у 2015 році
У 2015 році надійшло 4552 справи і матеріалів усіх категорій, серед яких 1616 кримінальних, 2680 цивільних та 256 справ про адміністративні правопорушення. З урахуванням залишку від попереднього періоду, всього перебувало у провадженні 5065 справ і матеріалів.

Із них закінчено провадженням 89 %.

Розглянуто:
 кримінальних — 1613; 49 % апеляційних скарг задоволено;
 цивільних — 2218; 48 % апеляційних скарг задоволено;
 про адміністративні правопорушення — 217; 60 % апеляційних скарг задоволено.

Примітки

Посилання

 Історична довідка про суд
 Голова Апеляційного суду Кіровоградської області В.Мельник: «Професія судді ставить більше вимог, ніж дає переваг»  // Закон і бізнес, № 40 (1079) 06.10—12.10.2012
 У Кіровограді «Правий сектор» заблокував апеляційний суд / ESPRESO.TV, 8 ГРУДНЯ 2015

Колишні суди Кіровоградської області
Суди України, ліквідовані 2018

redpajama_set_name: RedPajamaWikipedia


text:
Q: Calculate BPM from history of hand positions I calculate the beats per minute (BPM) out of Kinect hand positions tracked from a conductor. I do that by finding the last and the second last minimum in my history data. I then calculate the time difference between these two minimums and extrapolate this difference to a minute to get the current BPM.
However I am struggling finding the correct mathematical equation for that. How would I start? It is basically just the following, but how would I use the $history$ instead of first and secondLow? And have a function like $bpm(x) = ?$ or similar.
$$bpm = 60 / (firstLow - secondLow)$$
To show how my data-set does look like I add a simple graphic which should give you a little more details.

Thank you very much for your help.

A: Long story short, you don't want to look for peaks and the distance between them, for many reasons. First, it is heuristically difficult to do this with real data. Second, you will get all sorts of BPM variation due to small effects -- essentially small variations will corrupt your data.
What you want to do is draw a threshold at around 40% of a recent maximum, and then count all rising (or falling) crossings of that threshold. It's much easier to compute the crossing of a threshold than it is to determine a maximum/minimum in real time, and it is far less sensitive to noise.
This works because if you assume a periodic behavior, which is usually a fair assumption, then the wavelength between peaks is exactly the same as the wavelength between rising crossings of a given threshold.
In practice, to get reliable results, you might want an adaptive type estimator, because of possible variations in the baseline of the signal. This is a similar algorithm (almost exactly the same, really) to that used in detection of heart rate based on pulse oximetry data.
However, if you are comfortable with your method, then it is simple to compute BPM.
Let $\Delta t := t_{2}-t_{1}$. Then, you have $BPM = \frac{1 \mathrm{beat}}{\Delta t \mathrm{sec}} \cdot \frac{60 \mathrm{sec}}{1 \mathrm{min}}$ (in other words, your equation was correct).

redpajama_set_name: RedPajamaStackExchange


text:
Goldman's New Tack Trims Risk and Profit
By Liz Rappaport
Updated April 20, 2011 12:01 am ET
Goldman Sachs Group Inc. handily beat analysts' first-quarter estimates, but shares dropped on concerns that the firm will have trouble posting the kind of outsize profits shareholders have come to expect.
The bank reported a $2.74 billion profit, down 21% from the $3.46 billion it reported in the same period last year. The firm's bottom line was hurt as Goldman paid a $1.64 billion dividend while repaying a $5 billion investment from Warren Buffett's Berkshire Hathaway Inc. that was made in 2008. The bank's revenue dropped 7% to $11.89 billion, from $12.78 billion a year earlier.

redpajama_set_name: RedPajamaCommonCrawl


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\section{Results}\label{intro}
Recently David Recio-Mitter (\cite{RM}) introduced the notion of geodesic complexity, which is an analogue of Farber's topological complexity (\cite{Far}), but requires that paths be minimal geodesics. This is a useful requirement for efficient motion-planning algorithms. In \cite{RM} and \cite{DR}, the geodesic complexity of several spaces was determined.

Configuration spaces are of central importance in topological robotics, since they model the situation of several robots moving throughout a region. In this paper, we first consider the case of two distinguished points (or balls) moving in $\R^n$. We obtain explicit formulas for the geodesics and optimal geodesic motion-planning rules. We also consider two indistinguishable points moving in $\R^n$, and show that geodesic motion-planning rules for these correspond to ordinary motion-planning rules in real projective space $RP^{n-1}$.

Let $F(\R^n,2)$ denote the ordered configuration space of two distinct points in $\R^n$. It is a subspace of $\R^{2n}$ and is given the Euclidean metric. This space is not geodesically complete. For example, there is no geodesic from $((1,\overline 0),(\overline 0,1))$ to $((-1,\overline 0),(\overline 0,-1))$ since the linear path $\s(t)=((1-2t,\overline 0),(\overline 0,1-2t))$ has $\sigma(\frac12)\not\in F(\R^n,2)$, but there are paths in $F(\R^n,2)$ between these points arbitrarily close to $\s$. By ``geodesic,'' we will always mean ``minimal geodesic.''

For a positive number $\eps$, we consider the subspace of $F(\R^n,2)$ consisting of points $(x,x')$ for which $d(x,x')\ge\eps$. By scaling, we may assume $\eps=2$, and define
$$F_0(\R^n,2)=\{(x,x')\in F(\R^n,2):\ d(x,x')\ge2\}.$$
This can be viewed as the space of ordered pairs of disjoint open unit balls in $\R^n$. Note that $F_0(\R^n,2)$ is a manifold with boundary $\partial F_0$ consisting of points of the form $(x-u,x+u)$ with $\|u\|=1$.
In the following theorem, we  give explicit formulas for geodesics in $F_0(\R^n,2)$ between any two points.

\begin{thm}\label{mainthm} Let $P=(a,a')$ and $Q=(b,b')$ be points of $F_0(\R^n,2)$. Let
$$h=(a'-a)/2,\quad k=(b'-b)/2,\quad A=(a'+a)/2, \quad B=(b'+b)/2.$$
Let $\delta=\min\{d(tb+(1-t)a,tb'+(1-t)a'):\ 0\le t\le1\}$, the minimal distance between the two components of the linear path between $P$ and $Q$.
\begin{itemize}
\item[a.] If $\delta\ge2$, the linear path from $P$ to $Q$ is the unique geodesic between $P$ and $Q$ in $F_0(\R^n,2)$.
\item[b.] If $0<\delta\le2$, there is a unique geodesic in $F_0(\R^n,2)$ from $P$ to $Q$. It is the path composition $\ell_1\sigma\ell_2$, where $\ell_1$ is the linear path from $P$ to $C_0=(x-u,x+u)$, $\sigma$ the geodesic in $\partial F_0$ from $C_0$ to $C_1=(y-v,y+v)$, described in Proposition \ref{bdrylem}, and $\ell_2$ the linear path from $C_1$ to $Q$.
    Here $u$ and $v$ are  unique unit vectors in $\R^n$ satisfying
\begin{equation}\label{=1}h\cdot u=1\text{ and }k\cdot v=1\end{equation}
with minimal $\|u-v\|$. Let $\b$ be the angle between this $u$ and $v$ with $0\le\b<\pi$. Then
 \begin{equation}\label{xy}x=\frac{\b A+S_0B+S_1A}{\b+S_0+S_1},\quad y=\frac{\b B+S_0B+S_1A}{\b+S_0+S_1},
 \end{equation}
 where $S_0=\sqrt{\|h\|^2-1}$ and $S_1=\sqrt{\|k\|^2-1}$. If $\b=0$, then $C_0=C_1$, $\delta=2$, and $\ell_1\ell_2$ is the linear path from $P$ to $Q$. When $\delta=2$, the linear path in (a) can also be obtained by the method of (b).
\item[c.] If $\delta=0$, then $h$ and $k$ are parallel in opposite directions, and conversely. In this case,
the unit vector solutions $u,v$
 of (\ref{=1}) with minimal $\|u-v\|$ are
 $$u=\frac h{\|h\|^2}+\frac{S_0}{\|h\|}w,\quad v=\frac k{\|k\|^2}+\frac{S_1}{\|k\|}w,$$
 where $w$ ranges over the set of all points satisfying $h\cdot w=0$ and $w\cdot w=1$.
 The geodesics from $P$ to $Q$ are paths as described in (b) for each of these pairs $u,v$, using (\ref{xy}) and Corollary \ref{cor1}.
 \end{itemize}
 \end{thm}
 \noindent We will refer to these as type (a), (b), or (c) paths or Situations. In Proposition \ref{minl} and Corollary \ref{cor1} we give explicit formulas for $u$, $v$, and $\b$  in terms of $h$ and $k$.

 The geodesics in $\partial F_0$ to which we just referred are described in the following result.
 \begin{prop}\label{bdrylem} Let $u$ and $v$ be unit vectors in $\R^n$, and $\a$ be the angle from $u$ to $v$ with $0\le\a<\pi$. For $0\le t\le 1$, let
$$u(t)=\frac{\sin((1-t)\a)u+\sin(t\a)v}{\sin\a}$$
if $\a>0$. If $\a=0$, then $u(t)=u=v$ for all $t$.
For $x,y\in\R^n$, the unique geodesic in $\partial F_0$ from $(x-u,x+u)$ to $(y-v,y+v)$ is the curve
$$\sigma(t)=((1-t)x+ty-u(t),(1-t)x+ty+u(t)).$$
Its length is \begin{equation}\label{len}\sqrt{2(\|x-y\|^2+\a^2)}.\end{equation}
 \end{prop}

 Recall that the geodesic complexity $\GC(X)$ is the smallest $k$ such that $X\times X$ can be partitioned into ENRs $E_0,\ldots,E_k$ such that on each $E_i$ there is a continuous map $s_i$ from $E_i$ to the free path space $PX$ such that $s_i(x_0,x_1)$ is a geodesic from $x_0$ to $x_1$. (\cite{RM}) The topological complexity $\TC(X)$ is defined similarly without requiring that the paths be geodesics.(\cite{Far}) Our second result is the determination of $\GC(F_0(\R^n,2))$.
 \begin{thm}\label{GCthm} For $n\ge2$,
 $$\GC(F_0(\R^n,2))=\TC(F_0(\R^n,2))=\TC(S^{n-1})=\begin{cases}1&n\text{ even}\\ 2&n\text{  odd.}\end{cases}$$
 \end{thm}

The unordered configuration space $C(\R^2,2)$ is the quotient of $F(\R^n,2)$ by the involution which reverses the order of the two points. Points of $C(\R^n,2)$ are sets $\{a,a'\}$ with $a, a'\in\R^n$. Surprisingly, $C(\R^n,2)$ is, in some sense, easier for these considerations than $F(\R^n,2)$.
\begin{thm}\label{Cthm} With $d$ denoting the Euclidean metric in $\R^n\times \R^n$, defining
$$d_U(\{ a,a'\},\{b,b'\})=\min(d((a,a'),(b,b')),\ d((a,a'),(b',b)))$$
gives a metric on $C(\R^n,2)$ which has linear geodesics between any two points.\end{thm}
So, we need not bother with the intricacies for geodesics in $F(\R^n,2)$ caused by the need to keep points at least a certain distance apart.
The space $C(\R^n,2)$ has the homotopy type of $RP^{n-1}$, and so the following result, which we prove  in Section \ref{unorsec}, may not be surprising. The proof will show that the geodesics in $C(\R^n,2)$ are obtained from not-necessarily-geodesic paths in $RP^{n-1}$.
\begin{thm}\label{GCCthm} For $n\ge2$, $\GC(C(\R^n,2))=\TC(RP^{n-1})$.\end{thm}
\noindent By \cite{FTY}, $\TC(RP^n)$ equals the immersion dimension of $RP^n$ unless $n=1$, 3, or 7, but this does not enter into our proof.

In Section \ref{sec4}, we consider a different metric on $F(\R^n,2)$ in which it is geodesically complete, and discuss geodesics in that metric.
\section{Proof of Theorem \ref{mainthm}}\label{pfsec}
The following proof of Proposition \ref{bdrylem} benefited from ideas of David L.~Johnson.
\begin{proof}[Proof of Proposition \ref{bdrylem}]
With $u\cdot v=\cos\a$ and $u\cdot u=1=v\cdot v$, $u(t)=c_0u+c_1v$ is obtained by solving $u\cdot(c_0u+c_1v)=\cos(t\a)$ and $(c_0 u+c_1 v)\cdot(c_0u+c_1v)=1$. We have
\begin{eqnarray*}\sigma'(t)&=&(-x+y-\tfrac{\a}{\sin\a}(-\cos((1-t)\a)u+\cos(t\a)v),\\
&&-x+y+\tfrac{\a}{\sin\a}(-\cos((1-t)\a)u+\cos(t\a)v)).\end{eqnarray*}
Expanding $\cos(\a-t\a)$ yields
$\|-\cos((1-t)\a)u+\cos(t\a)v\|^2=\sin^2\a,$ and hence $\|\sigma'(t)\|^2=2(\|x-y\|^2+\a^2)$, which implies the claim about the length of the curve.

A constant-speed curve $\sigma$ with $\sigma''$ orthogonal to the surface is a geodesic. We have
$$\sigma''(t)=\tfrac{\a^2}{\sin\a}(\sin((1-t)\a)u+\sin(t\a)v,-(\sin((1-t)\a)u+\sin(t\a)v)).$$
The surface $\partial F_0$ is parametrized by
$X(x,u)=(x-u,x+u)$ with $x\in\R^n$ and $u\in S^{n-1}$. Then $\sigma''(t)$ is orthogonal to the $x$-directions and is orthogonal to the spherical parameter $u$ since it is a multiple of the radius vector at each point.

Since $\sigma'(0)=(-x+y-\frac\a{\sin\a}(-\cos(\a) u+v),-x+y+\frac\a{\sin\a}(-\cos(\a) u+v))$, every tangent direction from the initial point $(x-u,x+u)$ is obtained for one of our geodesics, showing that they are unique.
\end{proof}

The following lemma will be very important to our analysis.
\begin{lem}\label{anglelem} Let  $(a,a')\in F_0(\R^n,2)$, and let $h=(a'-a)/2\in\R^n$. If $u$ is a unit vector in $\R^n$ and $x\in\R^n$, the segment between $(a,a')$ and $(x-u,x+u)$ lies in $F_0(\R^n,2)$ iff $h\cdot u\ge1$.
\end{lem}
\begin{proof} We require that for $t\in[0,1]$
$$d(ta+(1-t)(x-u),ta'+(1-t)(x+u))\ge2.$$
Halving and squaring, this becomes
\begin{eqnarray*}1&\le&\|th+(1-t)u\|^2\\
&=&t^2\|h\|^2+2t(1-t)h\cdot u+(1-t)^2.\end{eqnarray*}
By assumption, $\|h\|^2\ge1$, so this quadratic function $f(t)$ satisfies $f(0)=1$ and $f(1)\ge1$. It is $\ge1$ for all $t\ge0$ iff $f'(0)\ge0$. Since $f'(0)=-2+2h\cdot u$, the result follows.
\end{proof}

 Because of Lemma \ref{anglelem}, paths of the form $\ell_1\sigma\ell_2$ in Theorem \ref{mainthm}(b) exist as long as $h\cdot u\ge1$ and $k\cdot v\ge1$ for unit vectors $u$ and $v$, for any $x$ and $y$. The proof of Theorem \ref{mainthm} will show that minimal length of such paths is achieved when $h\cdot u=1=k\cdot v$ and $\|u-v\|$ is minimized. The following result relates intersections of the hyperplanes $h\cdot u=1$ and $k\cdot u=1$ inside and on the unit sphere to the value of $\delta$ in Theorem \ref{mainthm}, which in turn determines the types of geodesics.

\begin{prop}\label{HKprop} Let $h$, $k$, and $\delta$ be as in Theorem \ref{mainthm}. Let $H=\|h\|^2\ge1$, $K=\|k\|^2\ge1$, and $D=h\cdot k$.
\begin{itemize}
\item If $\min(H,K)\le D$, then $\delta^2=4\min(H,K)\ge4$, so $\delta\ge2$.
\item If $\min(H,K)\ge D$ and $h\ne k$, then $$\delta^2=4(HK-D^2)/(H+K-2D).$$ In this case, regarding solutions of $h\cdot u=1$ and $k\cdot u=1$, we have
    \begin{itemize}
   \item [i.] There exist solutions with $\|u\|<1$ (and more than one solution with $\|u\|=1$) iff $\delta>2$.
 \item[ii.] There exists a unique solution with $\|u\|=1$ iff $\delta=2$.
 \item[iii.] There exist no solutions with $\|u\|\le1$ iff $\delta<2$.
 \end{itemize}
 \end{itemize}
 \end{prop}
 \begin{proof}  Note that  $HK\ge D^2$ by Cauchy-Schwarz, and $H+K>2D$, since $\|h\|^2+\|k\|^2>2\|h\|\,\|k\|\cos\a$ when $h\ne k$.

Let $d^2(t)=\|2tk+2(1-t)h\|^2=4(t^2K+(1-t)^2H+2t(1-t)D)$. Then $\delta^2=\min(d^2(t):\ t\in [0,1])$.
The minimum of $d^2(t)$ over all $t\in\R$ occurs when $t=t_0:=\frac{H-D}{H+K-2D}$, and has value $4\frac{HK-D^2}{H+K-2D}$.
Note that $t_0\in[0,1]$ iff $\min(H,K)\ge D$. If $\min(H,K)\le D$, then $d^2(t)$ does not have a relative minimum for $0<t<1$ so its absolute minimum on $[0,1]$ occurs at an endpoint.

If $k$ is a scalar multiple of $h$, the result is easily verified, so we assume this is not the case, and have $HK-D^2>0$.
Now let $u=\a h+\b k+\ell$ with $\ell$ orthogonal to $h$ and $k$. If $n=2$, omit $\ell$.
The equations $h\cdot u=1$ and $k\cdot u=1$ become $\a H+\b D=1$ and $\a D+\b K=1$, whose solution is
$$\a=\frac{K-D}{HK-D^2},\qquad \b=\frac{H-D}{HK-D^2},$$
yielding
\begin{equation}\label{uu}u\cdot u=\frac{H+K-2D}{HK-D^2}+\ell\cdot\ell.\end{equation}

Thus when $k$ is not a scalar multiple of $h$, we have $\|u\|^2=\frac4{\delta^2}+\|\ell\|^2$. The conclusions follow. By the complementary nature of the cases, it suffices to show implication in one direction. For each hypothesis on $\delta$, the conclusion about $\|u\|$ for solutions is clear. In case (i), the solutions are obtained by varying $\ell$.
\end{proof}

By Proposition \ref{HKprop}, the next result applies exactly when $\delta\le2$.
 \begin{prop}\label{minl} Let $h$ and $k$ satisfy $\|h\|>1$ and $\|k\|>1$. Assume there does not exist $u$ with $h\cdot u=1=k\cdot u$ with $\|u\|<1$. The solutions of $h\cdot u=1=k\cdot v$ and $\|u\|=1=\|v\|$ with minimal $\|u-v\|$ are
 \begin{itemize}\item[i.] If $k$ is a scalar multiple of $h$, then
 $$u=\frac h{\|h\|^2}+\frac{S_0}{\|h\|}w,\quad v=\frac k{\|k\|^2}+\frac{S_1}{\|k\|}w,$$
 where $w$ ranges over the set of vectors satisfying $h\cdot w=0$ and $w\cdot w=1$. Here $S_0$ and $S_1$ are as in Theorem \ref{mainthm}.
 \item[ii.] If $k$ is not a scalar multiple of $h$, then there is a unique solution, using notation of Proposition \ref{HKprop},
 \begin{eqnarray*}u&=&\frac hH+\frac{S_0(k-\frac DHh)}{\sqrt{HK-D^2}}\\
 v&=&\frac kK+\frac{S_1(h-\frac DKk)}{\sqrt{HK-D^2}}.\end{eqnarray*}
     \end{itemize}
     \end{prop}
\begin{proof} (i.) The vectors $u$ and $v$ lie on the intersections with the unit sphere of parallel hyperplanes.
The vector $u$ can be written uniquely as $u=c_0h+c_1w$ satisfying $h\cdot u=1$, $u\cdot u=1$, $w\cdot w=1$, and $h\cdot w=0$. These equations yield the above formula for $u$, and similarly for $v$. The vector $v$ closest to $u$ will be the one with the same unit vector $w$, and for all vectors $w$, the values of $\|u-v\|$ are the same.

(ii.) Let $\ell=k-\frac DHh$, which is orthogonal to $h$. Then $v$ can be written uniquely as $ah+b\ell+m$, with $m$ orthogonal to $h$ and $\ell$. The point $v$ closest to the hyperplane $h\cdot u=1$ will be the one with the largest $a$. From $v\cdot k=1$, we deduce $1=aD+b(HK-D^2)/H$. Use this to eliminate $b$. Then $v\cdot v=1$ implies
$$1=a^2H+\biggl(\frac{H(1-aD)}{HK-D^2}\biggr)^2\biggl(\frac{HK-D^2}H\biggr)+M,$$
where $M=\|m\|^2$. This simplifies to
$$0=HK\,a^2-2Da+1-(1-M)(HK-D^2)/H,$$
which has solution
$$a=\frac{D\pm\sqrt{D^2+K((1-M)(HK-D^2)-H)}}{HK}.$$
The maximum of this occurs when $M=0$ and has $a=(D+\sqrt{(K-1)(HK-D^2)})/HK$. One easily finds $b$ now and obtains the formula for $v$. The formula for $u$ is obtained similarly.
\end{proof}

\begin{cor}\label{cor1} The angle $\b$ in Theorem \ref{mainthm}(b) satisfies
$$\cos(\b)=\frac{(S_0+S_1)\sqrt{HK-D^2}+(1-S_0S_1)D}{HK}.$$
The angle $\b$ in Theorem \ref{mainthm}(c) satisfies
$$\cos(\b)=\frac D{HK}+\frac{S_0S_1}{\sqrt{HK}}.$$
\end{cor}
\begin{proof} We compute $\cos(\b)=u\cdot v$ from Proposition \ref{minl}.\end{proof}
\begin{proof}[Proof of Theorem \ref{mainthm}]
The conclusion of part (a) is immediate. Next we reduce consideration of part (c) to that of part (b).

First note that $\delta=0$ iff $tb+(1-t)a=tb'+(1-t)a'$ for some $t\in[0,1]$ iff $b'-b$ and $a'-a$ are negative multiples of one another. Then the analysis of type-(b) paths which follows applies, with minor modifications which are discussed below, to all of the pairs $u,v$ obtained in Proposition \ref{minl}(i), listed again in Theorem \ref{mainthm}(c).

Geodesics in a manifold with boundary are path compositions of geodesics in the manifold and geodesics in the boundary.(e.g., \cite{Hu}.) In our case, this will consist of at most one geodesic in $\partial F_0$. [\![If it were $\ell_1\sigma_1\ell_2\sigma_2\ell_3$, then $\ell_2$ would be a line segment connecting two points of $\partial F_0$. Similarly to the proof of Lemma \ref{anglelem}, the line segment connecting two points $(x-u,x+u)$ and $(y-v,y+v)$ of $\partial F_0$ will lie outside $F_0(\R^n,2)$ unless the two points have $u=v$, in which case it is a line segment lying in $\partial F_0$. When path-multiplied by an angle-changing geodesic in $\partial F_0$, the result will not be a geodesic.]\!]
So we need just consider path compositions of the form $\ell_1\sigma\ell_2$.

If $\|h\|=1$, then $P\in\partial F_0$. By the argument just described, we may then choose $\ell_1$ to be the constant path. This is consistent with (\ref{xy}) since we would have $S_0=0$, $x=A$, and $u=h$. Similarly, if $\|k\|=1$, the path $\ell_2$ may be ignored. Thus we shall assume $\|h\|>1$ and $\|k\|>1$, so Proposition \ref{minl} applies.

With the notation of the theorem, let $\Dhat_1$ denote the length of the linear path $\ell_1$ in $F_0(\R^n,2)$ from $P$ to any point $(x-u,x+u)$ with $\|u\|=1$. This equals
\begin{eqnarray}&&\sqrt{\|x-u-a\|^2+\|x+u-a'\|^2}\nonumber\\
&=&\nonumber\sqrt{\|a\|^2+\|a'\|^2+2\|x\|^2+2-4x\cdot A-4h\cdot u}\\
&=&\sqrt2\sqrt{\|h\|^2+1+\|x-A\|^2-2h\cdot u}.\label{D1}\end{eqnarray}

A path  $\ell_1\sigma\ell_2$ has length $\Dhat_1+\Dhat_3+\Dhat_2$, where $\Dhat_3$ is the length of the curved path $\sigma$ described in Proposition \ref{bdrylem}, and $\Dhat_2$ is a formula similar to (\ref{D1}) for a linear path $\ell_2$ from $(y-v,y+v)$ to $Q$. Let $D_i=\Dhat_i/\sqrt2$ and $T=D_1+D_3+D_2$. If $h\cdot u=1=k\cdot v$ and $\a$ is the angle between $u$ and $v$, the formulas for $D_1$ and $D_2$ simplify nicely, and we have
\begin{equation}\label{T3}T=\sqrt{S_0^2+\|x-A\|^2}+\sqrt{\|x-y\|^2+\a^2}+\sqrt{S_1^2+\|y-B\|^2}.\end{equation}

Setting $\partial T/\partial x_i=0$ gives
\begin{equation}\label{px}\frac{x_i-A_i}{\sqrt{S_0^2+\|x-A\|^2}}+\frac{x_i-y_i}{\sqrt{\|x-y\|^2+\a^2}}=0,\end{equation}
so
$$(x_i-A_i)^2(\|x-y\|^2+\a^2)=(x_i-y_i)^2(S_0^2+\|x-A\|^2).$$
Summing over $i$ and cancelling yields $\a^2\|x-A\|^2=S_0^2\|x-y\|^2$. Now (\ref{px}) says $\a(x-A)=(y-x)S_0$. Similarly $\a(y-B)=(x-y)S_1$. Solving these equations yields (\ref{xy}), with $\b$ replaced by $\a$. This is a consequence of $\partial T/\partial x_i=0=\partial T/\partial y_i$ and (\ref{=1}).

When $x$ and $y$ are as in (\ref{xy}),
\begin{equation}\label{three}x-y=\tfrac{(A-B)\a}{\a+S_0+S_1},\quad x-A=\tfrac{(B-A)S_0}{\a+S_0+S_1},\quad y-B=\tfrac{(A-B)S_1}{\a+S_0+S_1},\end{equation}
and we obtain the dramatic simplification
$$T=\sqrt{\|A-B\|^2+(\a+S_0+S_1)^2},$$
showing clearly that we should choose $\b$ to minimize $\a$.

Note that $\b<\pi$ (so Lemma \ref{bdrylem} applies), since the only way to have $\b=\pi$ would be with the hyperplanes $h\cdot u=1$ and $k\cdot v=1$ tangent to the unit sphere, and parallel, so $\|h\|=1=\|k\|$, which we have removed from our consideration.

We must also consider changes of $T$ caused by changes in $u$ or $v$. The more general formula for $T$ at any point is
\begin{eqnarray}\nonumber T&=&\sqrt{H+1+\|x-A\|^2-2h\cdot u}+\sqrt{\|x-y\|^2+(\arccos(u\cdot v))^2}\\
&&+\sqrt{K+1+\|y-B\|^2-2k\cdot v}.\label{Te}\end{eqnarray}
At our claimed critical point, which was derived from $\frac{\partial T}{\partial x_i}=0=\frac{\partial T}{\partial y_i}$, (\ref{=1}), and minimal $\|u-v\|$, $u$ and $v$ lie in the $h$-$k$ plane by Proposition \ref{minl}(ii). Changes in $u$ or $v$ orthogonal to the $h$-$k$ plane will not affect (\ref{Te}) at this point. For Situation (c), each pair $u,v$ is determined by a choice of $w$. They lie in the $h$-$w$ plane. The analysis here applies with $h$-$k$ replaced by $h$-$w$.

 Letting $\a$ denote the angle from $v$ to $u$, and using $\sqrt{\|x-y\|^2+\a^2}$ for the  middle term of (\ref{Te}), we obtain
\begin{eqnarray}\nonumber\frac{\partial T}{\partial\a}&=&\frac{-h\cdot\frac{du}{d\a}}{\sqrt{S_0^2+\|x-A\|^2}}+\frac\a{\sqrt{\|x-y\|^2+\a^2}}\\
\label{num}&=&\frac{\a+S_0+S_1}{\sqrt{(\a+S_0+S_1)^2+\|B-A\|^2}}\biggl(\frac{-h\cdot\tfrac{du}{d\a}}{S_0}+1
\biggr),\end{eqnarray}
incorporating (\ref{three}).
We can parametrize $\R^n$ so that $v=(1,\overline0)$, $u=(\cos\a,\sin\a,\overline0)$,
and $h=(h_1,h_2,\overline0)$. We obtain $h\cdot\frac{du}{d\a}=-h_1\sin\a+h_2\cos\a$, so
\begin{eqnarray*}S_0^2-(h\cdot\tfrac{du}{d\a})^2&=&h_1^2+h_2^2-1-(h_1^2\sin^2\a+h_2^2\cos^2\a-2h_1h_2\cos\a\sin\a)\\
&=&(h_1\cos\a+h_2\sin\a)^2-1\\
&=&0.\end{eqnarray*}
Noting that $h\cdot\frac{du}{d\a}\ge0$ since $h\cdot u$ had minimal allowable value (1) at our point, we obtain $h\cdot\frac{du}{d\a}=S_0$, and so $\frac{\partial T}{\partial \a}=0$ in (\ref{num}).

We have shown that the assumption (\ref{=1}) leads to the unique critical point of $T$ described in Theorem \ref{mainthm}(b) when $h$ and $k$ are not parallel, and to any of the claimed points in Situation (c), and Lemma \ref{anglelem} says that we must have $h\cdot u\ge1$ and $k\cdot v\ge1$. If $h\cdot u=t>1$,  corresponding to a larger value of $\a$, then we can find values of $x$ and $y$ that make $\partial T/\partial x_i=0=\partial T/\partial y_i$ with formulas similar to (\ref{xy}) except that now $S_0=\sqrt{\|h\|^2+1-2t}$. An analysis similar to the above paragraph will lead to $\partial T/\partial\a>0$ for changes in the $h$-$k$ plane. So points with $h\cdot u>1$ or $k\cdot v>1$ cannot be critical points. We conclude that our critical point is a unique minimum of $T$ in Situation (b), and our claimed points are the only critical points in Situation (c).

Next we justify the next-to-last sentence of part (b) by noting that if $\b=0$, so $u=v$ and then clearly $x=y$ in (\ref{xy}), and then showing that
$$(x-u,x+u)=\tfrac{S_1}{S_0+S_1}P+\tfrac{S_0}{S_0+S_1}Q,$$
so the unique point where the lines from $P$ and $Q$ meet $\partial F_0$ is on the line connecting $P$ and $Q$.

This requires showing that
$$\tfrac{S_0B+S_1A}{S_0+S_1}\pm u=\tfrac{S_1}{S_0+S_1}(A\pm h)+\tfrac{S_0}{S_0+S_1}(B\pm k),$$
hence we need to prove
\begin{equation}\label{213}u=\frac{S_1h+S_0k}{S_0+S_1}.\end{equation}
By Proposition \ref{minl}(ii), $u$ is in the $h$-$k$ plane. We parametrize that plane so that $h=(h_1,h_2)$, $k=(k_1,k_2)$, and $u=(\cos\theta,\sin\theta)$. Since $h\cdot u=1$,
$h_2=(1-h_1\cos\theta)/\sin\theta$, so
$$S_0=\sqrt{h_1^2+\bigl(\tfrac{1-h_1\cos\theta}{\sin\theta}\bigr)^2-1}=\frac{\sqrt{\cos^2\theta-2h_1\cos\theta+h_1^2}}{|\sin\theta|}=\bigg|\frac{\cos\theta-h_1}{\sin\theta}\bigg|,$$
and similarly $S_1=|(\cos\theta-k_1)/\sin\theta|$.

Since $\theta$ is the common endpoint of (otherwise disjoint) intervals on which $h_1\cos\theta+h_2\sin\theta\ge1$ and $k_1\cos\theta+k_2\sin\theta\ge1$, the derivatives of these expressions must have opposite signs at $\theta$. The derivative of $h_1\cos\theta+h_2\sin\theta$ is
$$-h_1\sin\theta+\tfrac{1-h_1\cos\theta}{\sin\theta}\cos\theta=\tfrac{\cos\theta-h_1}{\sin\theta}.$$
Thus $\cos\theta-h_1$ and $\cos\theta-k_1$ have opposite signs. Thus
$$\frac{S_1h_1+S_0k_1}{S_0+S_1}=\frac{(\cos\theta-k_1)h_1-(\cos\theta-h_1)k_1}{-(\cos\theta-h_1)+(\cos\theta-k_1)}=\cos\theta.$$
Similarly, $(S_1h_2+S_0k_2)/(S_0+S_1)=\sin\theta$, proving (\ref{213}) and hence the next-to-last sentence of part (b) of Theorem \ref{mainthm}.

Finally, regarding the last sentence of part (b):
 If $\delta=2$, there are points $c=(1-t)a+tb$ and $c'=(1-t)a'+tb'$ such that $d(c,c')=2$. There is a unique unit ball having $cc'$ as a diameter, and the method of (b) will yield $\ell_1$ the linear path from $(a,a')$ to $(c,c')$, $\sigma$ a constant path, and $\ell_2$ the linear path from $(c,c')$ to $(b,b')$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{GCthm}] Let $E_0$ denote the set of all $(P,Q)\in F_0(\R^n,2)\times F_0(\R^n,2)$ of type (c) in Theorem \ref{mainthm}, and $E_1$ it complement.
First we show that the unique geodesics at points $(P,Q)$ of $E_1$  vary continuously with $(P,Q)$, giving a geodesic motion planning rule on $E_1$.

For the type-(b) geodesics, the issue is whether $u$ and $v$, hence $\b$, vary continuously with $(P,Q)$. Small changes in $(P,Q)$ cause small changes in $h$ and $k$, and hence small changes in $u$ and $v$ of norm 1 satisfying $h\cdot u=1=k\cdot v$. If $(u,v)$ has minimal positive $\|u-v\|$ for such vectors, there will be a neighborhood of $(u,v)$ on which this is true. Alternatively, $u$ and $v$ vary continuously with $h$ and $k$ by Corollary \ref{cor1}.

The linear paths of type (a) vary continuously with the parameters. By the last two sentences of Theorem \ref{mainthm}(b), the paths in the intersection of types (a) and (b) agree, and so by the Pasting Lemma, we have a continuous choice of geodesics on $E_1$.

If $n$ is even, let $V$ be a unit-length vector field on $S^{n-1}$. In Theorem \ref{mainthm}(c), let $w=V(\frac h{\|h\|})$. This leads to a continuous choice of geodesics on $E_0$. Hence $\GC(F_0(\R^n,2))\le 1$ if $n$ is even. Since \begin{equation}F_0(\R^n,2)\approx \R^n\times\{x\in\R^n:\|x|\ge2\}\simeq S^{n-1}\label{he}\end{equation} and $\TC$ is a homotopy invariant, we obtain, for $n$ even,
$$\GC(F_0(\R^n,2))\ge\TC(F_0(\R^n,2))=\TC(S^{n-1})=1\ge\GC(F_0(\R^n,2)).$$
Hence we have equality.

If $n$ is odd, let $V$ be a unit-length vector field on $S^{n-1}-\{(1,\overline0)\}$. Then $w=V(\frac h{\|h\|})$ in \ref{mainthm}(c) leads to a continuous choice of geodesics on $E_0-Z$, where $Z$ is the set of $((a,a'),(b,b'))\in E_0$ such that $a'-a$ and $b'-b$ are scalar multiples of $(1,\overline0)$. On $Z$, you could use the geodesics obtained from using $w=(0,1,\overline 0)$ in \ref{mainthm}(c). Thus $\GC(F_0(\R^n,2))\le 2$, and since $\TC(S^{n-1})=2$, we have equality as in the previous paragraph.

\end{proof}
\section{Examples when $n=2$}\label{2sec}
We illustrate two examples of geodesics when $n=2$.

 Let $P=(P_1,P_2)=((-6,4),(6,8))$ and $Q=(Q_1,Q_2)=((8,-6),(2,-10))$. We have $h=(6,2)$ and $k=(-3,-2)$.
From (\ref{xy}), we obtain $x=(3.1596,-2.8468)$ and $y=(3.2474,-3.0927)$. From Proposition \ref{minl}, we obtain $u=(.4622,-.8867)$ and $v=(.3022,-.9533)$, and from Corollary \ref{cor1}, $\b=.1736$.  In this example, our path has length 25.2455, whereas the straight line path from $P$ to $Q$ (which is not in $F_0(\R^2,2)$) has length 25.2190.
Although we cannot quite draw the short middle part of the paths, in Figure \ref{fig3} we picture the paths in this example.

\bigskip
\begin{minipage}{6in}
\begin{fig}\label{fig3}

{\bf Example of geodesic.}

\begin{center}

\begin{\tz}[scale=.35]
\draw (3.16,-2.847) circle [radius=1];
\draw (3.247,-3.093) circle [radius=1];
\draw (-6,0) -- (8,0);
\draw (0,-10) -- (0,8);
\draw (6,8) -- (3.622,-3.733);
\draw [fill=red] [thick] (-6,4) -- (2.7,-1.96);
\draw (2,-10) -- (3.55,-4.046);
\draw [fill=red] [thick] (8,-6) -- (2.945,-2.14);
\node at (6,8) {$\ssize\bullet$};
\node at (-6,4) {$\ssize\bullet$};
\node at (2,-10) {$\ssize\bullet$};
\node at (8,-6) {$\ssize\bullet$};
\node at (7,8) {$P_2$};
\node at (3,-10) {$Q_2$};
\node at (-5,4) {$P_1$};
\node at (7,-6) {$Q_1$};
\end{\tz}
\end{center}
\end{fig}
\end{minipage}

Now we change the $8$ in $P_2$ of the above example to $12$, so that $\vec{P_1P_2}$ and $\vec{Q_1Q_2}$ are parallel in opposite directions; i.e., $h=(6,4)$ and $k=(-3,-2)$.

The two equal paths are depicted in Figure \ref{fig4}. We have $w=\pm(2,-3)/\sqrt{13}$ in part (c) of Theorem \ref{mainthm}. Also, $x=(3.2385,-2.3633)$ and $y=(3.4291,-2.9730)$ in (\ref{xy}) and $\b=.4202$ in Corollary \ref{cor1}. The length of each of these paths in $F_0(\R^n,2)$ is 28.375, compared with 28.213 for the straight-line path from $P$ to $Q$, which is not in $F_0(\R^n,2)$.

\bigskip
\begin{minipage}{6in}
\begin{fig}\label{fig4}

{\bf Example of two geodesics.}

\begin{center}

\begin{\tz}[scale=.35]
\draw (3.2385,-2.3633) circle [radius=1];
\draw (3.4291,-2.9730) circle [radius=1];
\draw (-6,0) -- (8,0);
\draw (0,-10) -- (0,8);
\draw [fill=red] [thick] (6,12) -- (3.9033,-3.1104);
\draw (6,12) -- (2.8046,-1.4624);
\draw [fill=red] [thick] (-6,4) -- (2.5738,-1.6162);
\draw (-6,4) -- (3.6725,-3.2643);
\draw [fill=red] [thick] (2,-10) -- (3.7312,-3.9263);
\draw (2,-10) -- (2.6654,-2.3275);
\draw [fill=red] [thick] (8,-6) -- (3.1269,-2.0198);
\draw (8,-6) -- (4.1928,-3.6186);
\node at (6,12) {$\ssize\bullet$};
\node at (-6,4) {$\ssize\bullet$};
\node at (2,-10) {$\ssize\bullet$};
\node at (8,-6) {$\ssize\bullet$};
\node at (7,12) {$P_2$};
\node at (3,-10) {$Q_2$};
\node at (-5,4) {$P_1$};
\node at (7,-6) {$Q_1$};
\end{\tz}
\end{center}
\end{fig}
\end{minipage}

\section{Unordered configuration space}\label{unorsec}
As we stated in Theorem \ref{Cthm}, the unordered configuration space $C(\R^n,2)$ has a natural, Euclidean-related metric, and it is geodesically complete.

\begin{proof}[Proof of Theorem \ref{Cthm}] We show that $d_U$ satisfies the triangle inequality. Without loss of generality, assume
$$d((a,a'),(b,b'))\le d((a,a'),(b',b))\quad\text{and}\quad d((b,b'),(c,c'))\le d((b,b'),(c',c)).$$
Then
\begin{eqnarray*}d_U(\{a,a'\},\{c,c'\})&\le&d((a,a'),(c,c'))\\
&\le&d((a,a'),(b,b'))+d((b,b'),(c,c'))\\
&=&d(\{a,a'\},\{b,b'\})+d(\{b,b'\},\{c,c'\}).\end{eqnarray*}
The quotient topology on $C(\R^n,2)$ comes from $\R^n\times(\R^n-\{0\})/\sim$ under $\{a,a'\}\mapsto\bigl(\frac{a+a'}2,\bigl[\frac{a-a'}2\bigr]\bigr)$. Our metric $d_U$ corresponds to the metric $d([x],[x'])=\min(d(x,x'),d(x,-x'))$ on $(\R^n-\{0\})/\sim$, which gives the quotient topology.

If $d_U(\{a,a'\},\{b,b'\})=d((a,a'),(b,b'))$, then the linear path $(1-t)(a,a')+t(b,b')$ lies in $F(\R^n,2)$, so its equivalence class is in $C(\R^n,2)$. [\![As observed at the beginning of the proof of Theorem \ref{mainthm}, the only thing that would prevent the path from being in $F(\R^n,2)$ is if $b'-b$ is a negative multiple of $a'-a$. If this is the case, then
\begin{eqnarray}&&d((a,a'),(b,b'))^2-d((a,a'),(b',b))^2\nonumber\\
&=&\|a-b\|^2+\|a'-b'\|^2-\|a-b'\|^2-\|a'-b\|^2\nonumber\\
&=&-2a\cdot b-2a'\cdot b'+2a\cdot b'+2a'\cdot b\nonumber\\
&=&2(a'-a)\cdot(b-b')\label{bb}\\
&>&0,\nonumber\end{eqnarray} contradicting the assumption that $d_U(\{a,a'\},\{b,b'\})=d((a,a'),(b,b'))$.]\!]
\end{proof}

\begin{rmk} {\rm The analogue of Theorem \ref{Cthm} is valid for $C(\R^n,k)$ for any $n$ and $k$.}\end{rmk}

\begin{prop} For $a$, $a'$, $b$, and $b'$ in $\R^n$, $d((a,a'),(b,b'))=d((a,a'),(b',b))$ iff $(a'-a)\cdot (b'-b)=0$. Let $$E_0=\{(\{a,a'\},\{b,b'\})\in C(\R^n,2)\times C(\R^n,2):(a'-a)\cdot (b'-b)\ne0\}.$$ There is a continuous geodesic motion-planning rule on $E_0$.\label{E0}\end{prop}
\begin{proof} The first part follows as in (\ref{bb}).
At each point of $E_0$, there is a unique choice of linear geodesic whose length equals $d_U(\{a,a'\},\{b,b'\})$, varying continuously with the point of $E_0$.
\end{proof}

Now we can prove Theorem \ref{GCCthm} about $\GC(C(\R^n,2))$.
\begin{proof}[Proof of Theorem \ref{GCCthm}]
Let $E_1=C(\R^n,2)\times C(\R^n,2)-E_0$, with $E_0$ as in Proposition \ref{E0}. We need to describe subsets of $E_1$ on which we can make a continuous choice of whether to go from $(a,a')$ to $(b,b')$, or  to $(b',b)$. Let $t_n=\TC(RP^{n-1})$. One motion-planning rule for $RP^{n-1}$ is on the domain $D_0$ consisting of all pairs $(\ell_0,\ell_1)$ such that $\ell_0\cdot\ell_1\ne0$. On $D_0$, rotate from $\ell_0$ to $\ell_1$ in their plane through the smaller arc ($<\pi/2$).
Suppose $D_i$ is one of the other $t_n$ subsets of $RP^{n-1}\times RP^{n-1}$ on which there is a motion-planning rule $s_i$. For $(\ell_0,\ell_1)\in D_i$, $s_i$ associates to an orientation (i.e., direction) on $\ell_0$ an orientation on $\ell_1$ by using the path from $\ell_0$ to $\ell_1$ specified by $s_i$. This association of orientations is continuous on $D_i$.

Let $E_i\subset E_1$ denote all $(\{a,a'\},\{b,b'\})$ such that, if $\ell_0$ is the line through $a$ and $a'$, translated to pass through $\overline 0$, and $\ell_1$ the translated line passing through $b$ and $b'$, then $(\ell_0,\ell_1)\in D_i$.
If $a>a'$ under the orientation of $\ell_0$, choose  the linear geodesic from $(a,a')$ to $(b,b')$, where $b>b'$ under the associated orientation of $\ell_1$. This is a continuous choice on $E_i$.

Thus we have partitioned $C(\R^n,2)\times C(\R^n,2)$ into $t_n+1$ subsets on which we have geodesic motion planning rules, so $\GC(C(\R^n,2))\le\TC(RP^{n-1})$. Since, from (\ref{he}), $C(\R^n,2)$ has the homotopy type of $RP^{n-1}$, we have the following string of inequalities, which imply the claimed equality.
$$\GC(C(\R^n,2))\ge\TC(C(\R^n,2))=\TC(RP^{n-1})\ge\GC(C(\R^n,2)).$$
\end{proof}
Note that our argument did not use that the motion-planning rules of \cite{FTY} can be chosen to be geodesics.

\section{A different metric}\label{sec4}
There is an obvious homeomorphism $F(\R^n,2)\to \R^n\times S^{n-1}\times \R^+$, where $\R^+=(0,\infty)$, given by
$$(a,a')\mapsto \biggl(\frac{a'+a}2, \frac{a'-a}{\|a'-a\|}, \frac{\|a'-a\|}2\biggr).$$
In the notation of Theorem \ref{mainthm}, with $\hh=h/\|h\|$, it is
$(a,a')\mapsto(A,\hh,\|h\|)$.
Here $A$ is the midpoint, and $h$ the directed segment from the midpoint to the second point.
The inverse sends $(A,u,r)$ back to $(A-ru,A+ru)$.

We use the Euclidean metric on $\R^n$ and $\R^+$, and arclength metric $d_S$ on $S^{n-1}$, and the product metric on their product to obtain a metric $d'$ on $F(\R^n,2)$ which is geodesically complete. The formula, with $B$ and $k$ also as in Theorem \ref{mainthm}, is
$$d'((a,a'),(b,b'))=\sqrt{\|B-A\|^2+d_S(\hh,\kk)^2+(\|k\|-\|h\|)^2}.$$
The unique geodesic from $(a,a')$ to $(b,b')$, if $d_S(\hh,\kk)<\pi$, is
$$t\mapsto ((1-t)A+tB-((1-t)\|h\|+t\|k\|)u(t),(1-t)A+tB+((1-t)\|h\|+t\|k\|)u(t)),$$
where, similarly to Proposition \ref{bdrylem}, with $\a=d_S(\hh,\kk)$, $$u(t)=\frac{\sin((1-t)\a)\hh+\sin(t\a)\kk}{\sin\a}.$$
If $d_S(\hh,\kk)=\pi$, we use vector fields on $S^{n-1}$ or $S^{n-1}-\{x_0\}$ to choose geodesics, and obtain an analogue of Theorem \ref{GCthm} for $F(\R^n,2)$ in this metric.
Figure \ref{paths} shows the path obtained in this way between the points that we used in the first example of Section \ref{2sec}.

\bigskip
\begin{minipage}{6.5in}
\begin{fig}
\label{paths}
{\bf Geodesic in $F(\R^2,2)$ using ``different'' metric}
\begin{center}
\includegraphics[width=4.5in,height=4.5in]{path.eps}
\end{center}
\end{fig}
\end{minipage}
\bigskip

There is an analogous metric on $C(\R^n,2)$, but we will not discuss it here because the Euclidean-related metric considered earlier was already geodesically complete and highly satisfactory.

\def\line{\rule{.6in}{.6pt}}

redpajama_set_name: RedPajamaArXiv


text:
extern crate dotenv;
#[macro_use]
extern crate quicli;

extern crate isymtope_ast_common;
extern crate isymtope_generate;

use std::collections::HashMap;
use std::env;
use std::fs::File;
use std::path::PathBuf;
use std::io::Write;

use dotenv::dotenv;
use quicli::prelude::*;

use isymtope_generate::*;

/// Compile Isymtope files to static pages
#[derive(Debug, StructOpt)]
struct Cli {
    #[structopt(long = "base-url")]
    base_url: String,

    #[structopt(long = "template-path")]
    template_path: Option<String>,

    #[structopt(long = "app-name")]
    app_name: Option<String>,

    #[structopt(long = "output", short = "o")]
    output: String,
}

main!(|args: Cli| {
    dotenv().ok();

    let mut template_cache: HashMap<String, DefaultTemplateContext> = Default::default();
    let app_dir: PathBuf = env::var_os("APP_DIR").expect("APP_DIR must be provided").into();

    let app_name = args.app_name.as_ref().map(|s| s.to_owned())
        .unwrap_or_else(|| env::var_os("DEFAULT_APP").expect("DEFAULT_APP must be provided if --app-name is not supplied").to_string_lossy().to_string());

    let app_root = app_dir.join(&app_name);
    let template_path = args.template_path.as_ref().map(|s| s.to_owned()).unwrap_or_else(|| "/app.ism".to_owned());
    let template_key = format!("[templatePath={}]", template_path);

    let base_url = args.base_url;
    let path = "/";

    let template_context = template_cache
        .entry(template_key.clone())
        .or_insert_with(|| DefaultTemplateContext::create(&app_root, &template_path).unwrap());

    let req = TemplateRequestMsg::RenderAppRoute(
        base_url.to_owned(),
        app_name.to_owned(),
        template_path.to_owned(),
        path.to_owned(),
    );

    let response = template_context.handle_msg(req)?;
    let TemplateResponseMsg::RenderComplete(result) = response;
    let body = result.into_inner();

    let mut f = File::create(args.output)?;
    f.write_all(body.as_bytes())?;

    // eprintln!("{}", body);
});

redpajama_set_name: RedPajamaGithub