\font\tenbxsl=cmbxsl10
\centerline{\bf Errata for \tenbxsl Axioms and Hulls}
\centerline{\sl (last updated 01 March 2025)}

\font\manfnt=manfnt
\def\ruparrow{{\setbox0=\hbox{$\nearrow$}
 \rlap{$\vcenter{\hbox to\wd0{\hfil$\searrow$\hfil}}$}\box0}}
\def\luparrow{{\setbox0=\hbox{$\nwarrow$}
 \rlap{$\vcenter{\hbox to\wd0{\hfil$\swarrow$\hfil}}$}\box0}}
\def\duparrow{{\setbox0=\hbox{$\swarrow$}
 \rlap{$\vcenter{\hbox to\wd0{\hfil$\searrow$\hfil}}$}\box0}}
\def\uuparrow{{\setbox0=\hbox{$\nwarrow$}
 \rlap{$\vcenter{\hbox to\wd0{\hfil$\nearrow$\hfil}}$}\box0}}
\def\ua{\uparrow}
\def\da{\downarrow}
\def\ra{\rightarrow}
\def\la{\leftarrow}
\def\Lrl{\Longleftrightarrow}
\def\RA{\Longrightarrow}
\def\LA{\Leftarrow}
\input picmac
\newcount\n
\newcount\k
\newcount\m
\def\|#1{\m=\n \advance\m-1
  \beginpicture(1,\m)(0,0)
  \multiput(0,0)(0,1)\n{\line(1,0)1}
  \ifnum#1>0 \module{#1}\fi
  \endpicture}
\def\module#1{\beginpicture(0,0)(0,0)
  \k=\n \advance\k-#1
  \put(.5,\k){\line(0,-1)1}
  \multiput(.5,\k)(0,-1)2{\disk{.2}}
  \endpicture}
\unitlength=10pt
\n=5

\def\becomes{\ifmmode\ \hbox\fi{\manfnt y}\ } % wiggly arrow indicates a change
\def\bugonpage #1, #2:
  {\medbreak\noindent{\bf Page #1, }#2\par\nobreak\smallskip\noindent}

\bugonpage 4, line $-4$:
$\neg tpr$ \dots is a convex \becomes
$\vert tpr\vert\le0$ implies $\vert pqr\vert>0$ by (1.2),
so (1.3) expresses $t$ as a convex

\bugonpage 6, line 13:
(1.4)and \becomes (1.4) and

\bugonpage 6, line 19:
$pux$, $pvx$, $pwy$, $qvy$, $qwy$, $quz$, $rwz$, $ruz$, $rvx$ \becomes
$pux$, $pvz$, $pwy$, $qvy$, $qwx$, $quz$, $rwz$, $ruy$, $rvx$

\bugonpage 8, equation (2.6a):
$abd,dbc,acd,abd$ \becomes $abd,dbc,acd,abc$

\bugonpage 12, line 3:
$str\vee tpq$ \becomes $str\wedge tpq$

\bugonpage 12, line $-2$:
tems \becomes terms

\bugonpage 14, line 18:
Macmahon \becomes MacMahon

\bugonpage 16, line 28:
labelled \becomes labeled

\bugonpage 20, line 10:
$(\sigma_1\vee\cdots\vee\sigma_1)$ \becomes
$(\sigma_1\vee\cdots\vee\sigma_k)$

\bugonpage 20, in (6.1):
$\bar{\sigma_2}$ \becomes $\bar{\sigma}_2$

\bugonpage 21, line 1:
$cf_j$, $'_j$, $c''_j$,  \becomes
$c_j$, $c'_j$, $c''_j$,

\bugonpage 22, line 10:
$\bar{q}rs$ \becomes $\bar{q}r\bar{s}$

\bugonpage 23, lines 1 and 2:
that when \dots we \becomes that $t\in\Delta pqr$
implies $t\ra p\vee q\ra t$, $t\ra q\vee r\ra t$, and $t\ra r\vee p\ra t$.
Hence we

\bugonpage 23, line 12:
$e^{i\theta j}$ \becomes $e^{i\theta_j}$\quad(twice)

\bugonpage 23, line 15:
[61] \becomes [68]

\bugonpage 24, line 6:
$$\vcenter{\halign{%
$\hfil#\hfil\;$&$\hfil#\hfil\;$&$\hfil#\hfil$\quad%
&$\hfil#\hfil$\cr
0&\ra&1\cr
\ua&\luparrow&\ua&\cr
4&\ra&6\cr}}\qquad \becomes \ \qquad
\vcenter{\halign{%
$\hfil#\hfil\;$&$\hfil#\hfil\;$&$\hfil#\hfil$\quad%
&$\hfil#\hfil$\cr
0&\ra&1\cr
\ua&\luparrow&\ua&\cr
4&\la&6\cr}}$$

\bugonpage 25, line 3:
contains $p\ra q$ \becomes contains $r\ra p$

\bugonpage 29, line 1:
adjaciences \becomes adjacencies

\bugonpage 29, replacement for (8.1):
$$\unitlength=8pt
\vcenter{\halign{\hfil#\hfil\quad\qquad&\hfil#\hfil\quad\qquad&\hfil#\hfil\cr
\|0\|4\|3\|2\|1\|4\|3\|2\|4\|3\|4\|0&
\|0\|4\|3\|2\|1\|2\|3\|4\|3\|2\|3\|0&
\let\\=\module \|0\\1\|3\\2\|4\\1\|3\\2\|4\\1\|3\|0\cr
\noalign{\smallskip}
bubble&cocktail-shaker&odd-even\cr}}
\eqno(8.1)$$

\bugonpage 30, lines $-5$ and $-2$:
(5.1) \becomes (5.2)

\bugonpage 37, line 10:
$B^n$ \becomes $B_n$

\bugonpage 37, line 16:
can also be written \becomes leads to others such as

\bugonpage 37, in (9.7):
$(\alpha''_m+1)$ \becomes $\alpha''_m$

\bugonpage 37, line $-10$:
Equations \becomes Inequalities

\bugonpage 39, line $-15$:
The quantity \dots\ at least, this is \becomes
Can the quantity $3^n$ in (9.8) be reduced to $n2^{n-2}$? This, at least, is

\bugonpage 39, line $-1$:
large. \becomes large. (And alas, the conjecture is false; see section 21.)

\bugonpage 40, line $-7$:
one circuit $C'$ \becomes on circuit $C''$

\bugonpage 42, line $-16$:
enough imply \becomes enough

\bugonpage 42, line $-10$:
960 \becomes 1920

\bugonpage 46, line 10:
paths \becomes points

\bugonpage 46, line $-5$:
(11.3) \becomes (11.4)

\bugonpage 49, line $-13$:
afer \becomes after

\bugonpage 50, line 3:
$pp't$ \becomes $pp't_1$

\bugonpage 50, line 6:
$t_rp't_{r-1}$ \becomes $t_{r-1}p't_r$

\bugonpage 50, line 10:
$t_3t_2p'$ \becomes $t_3p't_2$

\bugonpage 50, in (12.5):
$e_{<k}$ \becomes $e_{<j}$

\bugonpage 51, line 3:
$x_{10}$ \becomes $c_{10}$

\bugonpage 53, line 1:
changed \becomes charged

\bugonpage 57, in (14.4):
$^(14.4)$ \becomes $(14.4)$

\bugonpage 61, Step 3:
$x_r-x_p$ \becomes $x_r=x_p$\smallskip\noindent
[However, Lyle Ramshaw told me on 19 Nov 1993 that we can omit the clause `or
$x_p=x_q$ and $y_p=y_q$ and $x_r=x_p$ and $y_r\ge y_p$,' since a valid CC
system is obtained without this complication.]

\bugonpage 64, replacement for the bottom 12 lines:
$$\vcenter{\halign{#\hfil\qquad\qquad&$#$\hfil\cr
{\bf if} $(x<y)$\cr
\quad{\bf if} $(y<z)$ output $(xyz)$&x<y<z\cr
\quad{\bf else if} $(z<y)$\cr
\quad\quad{\bf if} $(x<z)$ output $(xzy)$&x<z<y\cr
\quad\quad{\bf else} output $(zxy)$&z\leq x<y\cr
\quad{\bf else} output $(xzy)$&x<y\leq z\leq y\cr
{\bf else if} $(y<x)$\cr
\quad{\bf if} $(x<z)$ output $(yxz)$&y<x<z\cr
\quad{\bf else if} $(y<z)$ output $(yzx)$&y<z\leq x\cr
\quad{\bf else} output $(zyx)$&z\leq y<x\cr
{\bf else if} $(x<z)$ output $(xyz)$&x\leq y\leq x<z\cr
{\bf else} output $(zyx)$.&z\leq x\leq y\leq x\,.\cr}}$$

\bugonpage 72, line $-11$:
same sign \becomes same sign as

\bugonpage 73, line 9:
$\bar a,q,r,s\}$ \becomes $\{\bar a,q,r,s\}$

\bugonpage 73, line 24:
allpoints \becomes all points

\bugonpage 78, line $-10$:
$pt_jt_{j-1}t_{j-2}$ \becomes $pt_{j-1}t_{j-2}$

\bugonpage 81, line $-14$:
1000 $n$-gon \becomes 10000 $n$-gon

\bugonpage 86, line $-6$:
labelled \becomes labeled

\bugonpage 87, line 8:
$p_1,\ldots,p_k$ \becomes $p_1\,\ldots\,p_k$

\bugonpage 87, line 10:
labelled \becomes labeled

\bugonpage 88, line $-13$:
$p'_1,\ldots\,p'_{r-2}$ \becomes $p'_1\,\ldots\,p'_{r-2}$

\bugonpage 91, line 8:
fours \becomes form

\bugonpage 91, line 23:
$t_n$ \becomes $t_m$

\bugonpage 91, line 27:
$t_mt_{m-1},\ldots,t_1$ \becomes $(t_m,t_{m-1},\ldots,t_1)$

\bugonpage 95, line 11:
48, 50 \becomes 48, 51

\bugonpage 97, lines 15 and 16:
Laffaile \becomes Laffaille\quad(twice)

\bugonpage 97, new material just before section 22:
{\it Addendum, added March 2025\/}:\enspace
A paper by Stephan Felsner and Pavel Valtr,
``Coding and counting arrangements of pseudolines''
[{\sl Discrete \& Computational Geometry\/ \bf46} (2011), 405--416]
proves that the constant 3 in the
theorem of section 9 can be improved to 2.487. It also mentions that
Ond\v{r}ej B{\'\i}lka has disproved the conjecture following that
theorem, by constructing reflection networks with
$\Omega(2.076^n)$ cutpaths.

A paper by Xavier Goaoc and Emo Welzl,
``Convex hulls of random order types''
[{\sl Journal of the ACM\/ \bf70} (2023), 8:1--8:47]
shows among other things that the average number of points on the convex
hull of a random $n$-point CC system is $4-8/(n^2-n+2)$.

\bugonpage 100, near the top:
I meant to cite also the paper by Franz Aurenhammer, ``Voronoi
 diagrams --- a survey of a fundamental geometric data structure,''
{\sl ACM Computing Surveys\/ \bf23} (1991), 345--405.

\bugonpage 100, reference [5]:
(1992) \becomes (1993)

\bugonpage 100, reference [7]:
in preparation. \becomes unpublished preprint.

\bugonpage 100, reference [13]:
Krystallographie \becomes Kristallographie

\bugonpage 101, reference [19]:
{\sl SIAM} \dots\ 108--123. \becomes
{\sl SIAM Journal on Computing\/ \bf22} (1993), 418--429.

\bugonpage 102, reference [32]:
1992), to appear. \becomes 1993), 103--134.

\bugonpage 102, reference [34]:
London Interscience \becomes London: Interscience

\bugonpage 102, reference [37]:
{\sl Algorithmica}, to appear. \becomes
{\sl Algorithmica\/ \bf9} (1993), 534--560.

\bugonpage 103, reference [45]:
Addison-Wesley \becomes Addison\kern.1em--Wesley

\bugonpage 103, reference [47]:
, book in preparation. \becomes (New York: ACM Press, 1994), $\rm viii+576$~pp.

\bugonpage 103, reference [54]:
{\sl Discrete} \dots\ 1990. \becomes
{\sl Syomposium on Computational Geometry\/ \bf6} (1990), 16--22.

\bugonpage 104, reference [59]:
331-332 \becomes 331--332

\bugonpage 104, reference [60]:
probl\'eme \becomes probl\`eme

\bugonpage 104, reference [61]:
realisierbarkeitskriterien \becomes Realisierbarkeitskriterien\hfil\break
{\sl Giessen\/} \becomes {\sl Gie{\ss}en\/}

\bugonpage 104, reference [69]:
{\bf133} (1907) \becomes {\bf133} (1908)

\bugonpage 104, reference [71]:
Eidgenossische \becomes Eidgen\"ossische

\bugonpage 105, new entry:
Aurenhammer, Franz, 100.

\bugonpage 105, new entry:
B{\'\i}lka, Ond\v{r}ej, 97.

\bugonpage 105, entry for Bokowski:
95, 96 \becomes 95--97

\bugonpage 106, left column, line 1:
vii \becomes vii.

\bugonpage 106, left column:
Dress, Andreas \becomes Dress, Andreas Walter Martin

\bugonpage 106, new entry:
Felsner, Stephan, 97.

\bugonpage 106, new entry:
Goaoc, Xavier, 97.

\bugonpage 107, entry for Lawrence:
95 \becomes 95--96

\bugonpage 107, left column:
Macmahon \becomes MacMahon

\bugonpage 108, entry for perturbations:
59-60 \becomes 59--60

\bugonpage 109, top line of left column:
instructions! 74 \becomes instructions, 74

\bugonpage 109, new entry:
Valtr, Pavel, 97.

\bugonpage 109, entry for Welzl:
103 \becomes 97, 103

\bye
