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README.html
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<!DOCTYPE html>
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<!-- initial template from http://docs.mathjax.org/en/latest/start.html -->
<!--
Notes:
\; for thick space
\: for medium space
\, for thin space
\! for negative thin space
Can't seem to adjust print font size by user :-(
so messing with it in the style sheet here.
TODO: make interaction purely delta-based, not requiring any absolute positioning at all?
TODO: validator says width/height not allowed on div at that point
TODO: why is it not jumping to #youarehere any more?? (is it a jquery thing?)
TODO: my specified default zoom trigger not working?
TODO: it would be awesome if it were possible to start typesetting
at a particular place in the page, right after the anchor
we started loading the page at
TODO: figure out how to validate?
TODO: can't seem to get asciimath to work? oh wait! it's working, from the web page http://cdn.mathjax.org/mathjax/2.1-latest/test/sample-asciimath.html . EEK, works over the web but not locally?? hmm. wonder if I need to put something locally? or enable something? weird
TODO: css validator is giving me errors on the styles inside the svg sections, wtf?
ERROR: Property stroke doesn't exist : blue
ERROR: Property stroke-width doesn't exist : 2.0
ERROR: Property fill doesn't exist : none
TODO: main validator:
ERROR: Attribute vector-effect not allowed on element path at this point.
oh no! and if I try to change version from 1.1 to 1.2 it says:
Unsupported SVG version specified. This validator only supports SVG 1.1. The recommended way to suppress this warning is to remove the version attribute altogether.
argh!
TODO: can I preprocess the preprocessed stuff so that _x and ^x mean \underline{x} and \overline{x}? YES! but make it nicer
Content:
TODO: maybe r,l should be the edge indices (even) instead of the vert indices (odd)? not sure, maybe it's fine the way it is.
Typesetting:
TODO: typeset in priority order!
TODO: legit equation numbering?
TODO: URL params:
- initial x1,x2,y1,y2 etc.
- ability to typeset only delimited by "you are here" stuff?
-->
<title>The Spiral Moment Lemma: Every Polygonal Lagoon Has At Least Two Good Exits</title>
<style type="text/css">
/* hide anything with class="asciiprevis" */
.asciiprevis {display:none;}
/* hide anything with class="debug" */
.debug {display:none;}
/*
For some reason printing comes out really tiny
(even more in chrome than in firefox)
if I don't make an explicit font size change.
I don't know how to change font size for a particular printout,
so hard-coding sizes in here.
Font sizes:
xx-small
x-small
small
medium
large
x-large
xx-large
ARGH! anything media-specific causes corrupt output of math when printing in chrome!
need to figure out why!
printing in chrome seems pretty funky anyway (snap!) so I'll just print in firefox.
ideal settings (when expressed outside @media) seem to be:
firefox screen: medium (or large)
firefox print: large (or x-large)
chrome screen: large
chrome screen: x-large
but inside @media it acts different!
- in chrome, mathjax max gets corrupt when printed
- in firefox, @media print large
is bigger (and more to my liking)
than regular print large!
OH I think it's because the screen one leaked into
the print one? some subtlety of the print previewer?
it's corrupt, too... didn't notice that before.
*/
@media screen {
body {font-size:large;}
}
@media print {
body {font-size:large;}
}
/*
messing with trying to keep pre's on one page.
these directives probably don't mean anything on screen.
doesn't do much in firefox :-(
and at this point chrome won't even give me a preview screen :-(
*/
pre {
page-break-during: avoid;
}
pre * {
page-break-after: avoid;
page-break-before: avoid;
}
p.breakhere {page-break-before:always;}
text {
/* make it so the svg text doesn't get highlighted
if the mouse is over it during dragging,
since that got pretty distracting.
the user-select one is commented out because it provokes a warning in firefox.
XXX actually this doesn't happen any more even without this, I don't think... ?? figure out why or why not
*/
-webkit-touch-callout: none;
-webkit-user-select: none;
-khtml-user-select: none;
-moz-user-select: none;
-ms-user-select: none;
/* user-select: none; */
/* fill: green; */
}
/*
* text-decoration:underline and overline have problems:
* - underline too close to text
* - overline gets split, e.g. when overlining x_d
* so use box borders instead.
* http://stackoverflow.com/questions/1734618/css-underline-possible-to-increase-gap
*/
.ul {
border-bottom: 1px solid black;
padding-bottom: 1px;
}
.ol {
border-top: 1px solid black;
padding-top: 0px;
}
.uol {
border-top: 1px solid black;
padding-top: 0px;
border-bottom: 1px solid black;
padding-bottom: 1px;
}
/*
* note, svg needs to be wrapped alone in a div for dragging scheme to work.
* also we'd like to center them, so each figure is wrapped
* in a <div class="figure">
* which we present by centering it
* XXX THIS DOESN'T WORK YET
*/
div.figure {
width:60%; /* XXX ARGH! don't want to have to do this! */
margin:auto;
outline: 1px dashed #d0d0d0;
}
svg.figure {
outline: 1px dashed black;
/* this makes it an arrow cursor throughout the svg
(if we don't do this, it turns into the text cursor
when hovering over text elements */
cursor: default;
}
.movable {
cursor: pointer;
}
</style>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
tex2jax: {
// make '$' delimit inline math
inlineMath: [['$','$'], ['\\(','\\)']],
// make \$ give a real dollar sign, even in non-math mode
processEscapes: true,
},
TeX: {
Macros: {
// just a record of how to make a macro, for my reference.
// (note, this apparently works for _ but not for ^)
_: ["\\underline{#1}",1],
},
},
asciimath2jax: {
delimiters: [['`','`']],
},
menuSettings: {
// make it so that hovering over an equation zooms it
// (right click on an equation to see how to alter this
// within a session)
// XXX this isn't working?
zoom: 'hover',
},
"HTML-CSS": {
// oh hmm, it's complicated... see http://docs.mathjax.org/en/latest/options/HTML-CSS.html . not sure how to get a font I want.
preferredFont: "STIX",
},
// we're going to make a pass over the document using jquery first,
// so skip typesetting during startup, and we'll call MathJax.Hub.Typeset()
// explicitly when we want it.
// (actually MathJax.Hub.Queue(["Typeset",MathJax.Hub])).
skipStartupTypeset: true,
});
</script>
<!-- note, the particular config used below allows
both tex (using $...$)
and asciimath (using `...`) on same page -->
<script type="text/javascript"
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-MML-AM_HTMLorMML.js">
</script>
<script
src="http://code.jquery.com/jquery-1.7.1.min.js">
</script>
<script>
// make $ not be an alias for jQuery
// since I think that's ridiculous...
jQuery.noConflict();
</script>
<script src="initFigureInteraction.js"></script> <!-- what the hell? do NOT self-close! it makes everything fail in mysterious ways-->
<script src="vertexColoredPath.js"></script>
<script>
/*
To make the source of the proofs
more readable and maintainable,
the following preprocessing is done
on all latex math (anything inside $...$'s or $$..$$'s)
before feeding it to mathjax's tex processor.
(In fact, rather than trying to detect where the $'s begin and end,
we just do this on the whole darn document,
hopefully excluding this explanation)
1.
Any of the following:
_ ^ ^_ _^
followed by any thing of the following:
A B C D
tp tm td
rp rm rd
xp xm xd
yp ym yd
p m d
get transformed into the respective one of
\underline{thing}
\overline{thing}
\underline{\overline{thing}}
\overline{\underline{thing}}
(the latter two are effectively the same).
(The _ and ^ go before the object rather than after,
to avoid complication in parsing, for example, rd^2)
2.
The following:
tp tm td
rp rm rd
xp xm xd
yp ym yd
get turned into, respectively:
{\theta_p} {\theta_m} {\theta_d}
{r_p} {r_m} {r_d}
{x_p} {x_m} {x_d}
{y_p} {y_m} {y_d}
How to do this?
Well, the following says there's a way to register a function
as a preprocessor:
https://groups.google.com/forum/?fromgroups=#!topic/mathjax-users/lWmbG9FClJs
It says there's currently no way to get my preprocessor
to run before the others, however mml2jax is the only extension
that currently loads a preprocessor so it should be safe
(DANGER-- might this change?)
Davide also suggests making it a config file
and loading it before original combined config file.
*/
//http://stackoverflow.com/questions/901115/how-can-i-get-query-string-values/901144#901144
function getParameterByName(name) {
name = name.replace(/[\[]/, "\\\[").replace(/[\]]/, "\\\]");
var regex = new RegExp("[\\?&]" + name + "=([^&#]*)"),
results = regex.exec(location.search);
return results == null ? null : decodeURIComponent(results[1].replace(/\+/g, " "));
}
// queue calling the following code
// when the document is loaded
jQuery(document).ready(function() {
console.log("in jquery document-ready function!");
var nTimesToBeVerbose = 0;
var myCallback = function(index,element) {
if (nTimesToBeVerbose-- >= 1)
{
console.log(" in myCallback");
console.log(" element=",element);
console.log(" element.tagName=",element.tagName);
console.log(" element.nodeValue=",element.nodeValue);
console.log(" element.innerText=",element.innerText);
console.log(" element.outerText=",element.outerText);
console.log(" element.innerHTML=",element.innerHTML);
console.log(" element.outerHTML=",element.outerHTML);
}
if (element.nodeValue !== null)
{
element.nodeValue = (element.nodeValue
.replace(/\b(([trxy]?_?[dmp]|[ABCD])[_^][^2])/g,'\\color{red}{\\fbox{BAD0: $1}}')
// force always using td instead of t_d, etc.
.replace(/\b(([trxy]_[dmp])\b)/g,'\\color{green}{\\fbox{BAD1: $1}}')
.replace(/(\\theta_)/g,'\\color{blue}{\\fbox{BAD2: $1}}')
.replace(/(\^_xd)\b/g,'\\color{red}{\\fbox{$$$1$$}}')
.replace(/(_\^xd)\b/g,'\\color{red}{\\fbox{$$$1$$}}')
.replace(/\^_([trxy]?[dmp]|[ABCD]'?)\b/g,'\\overline{\\underline $1}')
.replace(/_\^([trxy]?[dmp]|[ABCD]'?)\b/g,'\\underline{\\overline $1}')
.replace(/_([trxy]?[dmp]|[ABCD]'?)\b/g,'\\underline $1')
.replace(/\^([trxy]?[dmp]|[ABCD]'?)\b/g,'\\overline $1')
.replace(/\b(t)([dmp])\b/g,'{\\theta_$2}')
.replace(/\b([rxy])([dmp])\b/g,'{$1_$2}')
.replace(/\\</g,'\\langle')
.replace(/\\>/g,'\\rangle')
);
}
}
// the logic I want is:
// if under class='doMyThing'
// and there's no ancestor <PRE> strictly closer than the class='doMyThing',
// then do the substitutions.
// I don't think there's any jQuery selector that does such a thing... ?
var myCallbackAll = function(element,callback,parentEnabled) {
var enabled = ((parentEnabled
&& element.tagName !== 'PRE')
|| element.className == 'doMyThing');
if (enabled)
{
callback(0,element);
}
for (var child = element.firstChild; child !== null; child = child.nextSibling)
{
myCallbackAll(child,callback,enabled);
}
}
if (true)
{
var figure4div = jQuery('#figure4div');
if (figure4div.length != 1) throw "oh no! can't find figure4div!"
var figure4svg = figure4div.children();
var dudleyNeighborsPath = figure4svg.find('.dudleyNeighborsPath');
var priscillaNeighborsPath = figure4svg.find('.priscillaNeighborsPath');
if (dudleyNeighborsPath.length != 1) throw "oh no! can't find dudleyNeighborsPath in figure 4!"
if (priscillaNeighborsPath.length != 1) throw "oh no! can't find priscillaNeighborsPath in figure 4!"
setupVertexColoredPaths(dudleyNeighborsPath);
setupVertexColoredPaths(priscillaNeighborsPath);
}
var interactParam = getParameterByName('interact');
//console.log("interact param = ",interactParam);
if (interactParam != "no")
{
console.log(" initializing figures interaction...");
initFigures567Interaction(function(){});
console.log(" done initializing figures interaction.");
}
if (true)
{
console.log(" doing my preprocessing...");
myCallbackAll(document,myCallback,false);
console.log(" done my preprocessing.");
}
// say "typeset=no" in url to suppress typesetting
var typesetParam = getParameterByName('typeset');
//console.log("typeset param = ",typesetParam);
if (typesetParam !== "no")
{
console.log(" calling mathjax typeset...");
MathJax.Hub.Queue(["Typeset",MathJax.Hub]);
//MathJax.Hub.Queue(["Typeset",MathJax.Hub,"typesetJustMe"]);
console.log(" returned from mathjax typeset.");
}
console.log("out jquery document-ready function!");
}); // jquery document ready function
</script>
</head>
<body>
<div class="debug">
<hr> <!--============================================================-->
<div class="doMyThing">
first thing: $xp$
</div>
<div class="doMyThing">
second thing: $_^yp$
</div>
<div class="doMyThing">
$ \angle (x,y)
$ 0 , 1 $
$a^b {a}^{b}$
$rp_$
$_rp$
$\underline{hey}$ $xp$ $yp$ $tp$ $_xp$ $_^xp$ $^xp$
... when `x \lt y^2` we have ...
... when $x \lt y^2$ we have ...
`(:
(a,=,b),
( ,=, c)
:)`
will this work? $_x$
how bout this? $^x$
ok how about a simple little regex
that turns those into \underline and \overline?
$\underline{\overline{x}}$
$\underoverline{x}$
$\_x$
$\^x$
$a ^ b$
</div> <!-- doMyThing -->
<pre>
</pre>
<hr> <!--============================================================-->
</div>
<h1>The Spiral Moment Lemma: Every Polygonal Lagoon Has At Least Two Good Exits</h1>
Visualizing an infinite spiral
to try to prove the every-lagoon-has-at-least-two-good-exits conjecture.
<p>
Priscilla and Dudley take a walk counterclockwise
all the way around the primal and dual perimeter respectively,
in such a way that whenever Priscilla is moving along an edge of the primal,
Dudley is stopped at the corresponding vertex of the dual,
and when Dudley is moving along an edge of the dual,
Priscilla is stopped at the corresponding vertex of the primal.
Exactly one of Priscilla and Dudley are moving at any particular time.
<div class="doMyThing">
<p>
Denote Priscilla's and Dudley's positions,
in rectangular and polar coordinates,
by the following, all continuous functions of time $t$:
<p>
\begin{align}
rp,tp, & xp, yp \\
rd,td, & xd, yd \\
xp = rp \cos(tp) \;\; & \;\; yp = rp \sin(tp) \\
xd = rd \cos(td) \;\; & \;\; yd = rd \sin(td)
\end{align}
<p>
Priscilla's and Dudley's paces are such that the one who's moving
always does so at unit rotation speed around the origin
while the other stays still.
So their positions are continuous piecewise differentiable functions of time,
with
$$(d tp + d td) / dt = 1 \; \text{wherever both are differentiable.}$$
<p>
Since the combined rotational speed is 1 and the combined rotational distance is
$4 \pi$,
it will take
$4 \pi$
units of time for Priscilla and Dudley
to complete one revolution and arrive back at their starting configuration;
if we fix Priscilla's starting point
$ \langle xp,yp \rangle (t=0)$
in the interior (not endpoint) of some primal side, this forces Dudley
to start at the dual vertex
$ \langle xd,yd \rangle (t=0)$
corresponding to that initial primal edge,
and the entire walk is thus determined.
They will arrive back at
$ \langle xp,yp\rangle,\langle xd,yd\rangle $ at time
$t = 4 \pi$.
<p>
Priscilla and Dudley's walk is defined (modulo the choice of starting point)
no matter what the polygon.
Let's apply it to the case of interest: a polygon
with only one good lagoon exit.
(We'll end up with a contradiction,
showing that, in fact, no single-exit single-polygon lagoon exists.)
<p>
Since the primal polygon is bounded by straight lines,
points on the boundary at local maximum and local minimum distance
from the polygon's cc are isolated,
and they alternate, with exactly the same number of local maxima
as local minima.
Furthermore by assumption there's exactly one local maximum
(at some farthest vertex);
therefore there's exactly one local minumum
(somewhere along the nearest side, *not* at a vertex),
which is therefore the unique global minimum.
We'll work in a coordinate system such that the cc
is at the origin
$\langle 0,0 \rangle$,
and the nearest point on the perimeter
is one unit below it, at
$\langle 0,-1\rangle$.
For the dual (at unrelated scale),
place the central vertex at the origin.
Then the dual edge corresponding to the primal edge closest to the origin
in primal space starts at the origin in dual space
and lies along the
$-y$
axis;
choose a scale so that it ends at
$\langle 0,-1\rangle$.
Having chosen the position and scale of the dual,
the entire dual diagram (triangle complex) is thus determined.
It consists of
$n$
triangles, each having one vertex at the origin.
<p>
XXX do we really want to nail down the rotation like that? I don't think it's necessary
</div> <!-- doMyThing -->
<p>
We'll number the primal features in CCW order:
primal edges have even indices and primal vertices have odd indices:
\begin{align}
e_0 = & \text{the bottom horizontal edge passing through $\langle 0,-1 \rangle$} \\
v_1 = & \text{right endpoint of $e_0$} \\
e_2 = & \text{next edge CCW} \\
v_3 = & \text{next vertex CCW} \\
... & \\
e_{2 n - 2} = & \text{edge CW from e_0} \\
v_{2 n - 1} = & \text{left endpoint of e_0} \\
\end{align}
<p>
Let
$m$
be the (odd) index of the vertex
that is the global maximum distance from the origin
of the entire primal polygon perimeter.
So
$v_m$
is the only good exit,
and distance-from-origin is strictly increasing
from
$\langle 0,-1\rangle$
around CCW to
$v_m$,
and then strictly decreasing from
$v_m$
back around CCW to
$\langle 0,-1\rangle$.
<p>
Consider the directions of the "quill" emanating
from each vertex of this primal polygon.
Since
$v_m$
is the only good exit,
every quill *except* the one at
$v_m$
points more towards the origin than away from it
(more precisely, the quill
$i$'s direction's dot product with
$v_i$ is
$\lt 0$), so it can't point directly out from the origin,
it must lean
$\gt 90^{\circ}$
from that, CW or CCW.
For
$v_i$
with
$i \lt m$,
distance from origin to perimeter is increasing,
so the quill at
$v_i$
can't lean CCW
(or it would point into the polygon's interior)
so it must lean at least
$90^{\circ}$
CW from straight out.
Similarly for
$v_i$
with
$i>m$,
the quill at
$v_i$
must lean at least
$90^{\circ}$
CCW from straight out.
<p>
There is no apparent contradiction in anything described so far;
some plausible examples are shown in Figures 1(a-e).
<br>
XXX add the missing ones! generate them from shephards play applet
<br>
XXX arrange them side by side
<br>
XXX show duals too
<p>
<pre class="asciiprevis">
Figure 1(a): 212
|
*v5
/ \
/ \
/e6 e4\
/ \
v7* *v3
/ \e8 e2/ \
\ e0 /
v9*---*v1
\ /
On paper, I seemed to think the following was nice:
.....|.....
.....*.....
...........
...........
..../.\....
...........
...........
...*...*...
...........
../.*-*.\..
...........
...........
.....X.....
But the dual gets kind of extreme.
Maybe the following is nicer (hexagon angles):
In ShephardsPlayApplet:
nVirtual=6
1,1 1/1
1,1 1/1
and tweak the primal upwards on plane
so that spiral becomes proper with respect to origin
(maybe tweak by exactly half of dual edge)
I'll enter it the simple way here,
and then get ShephardsPlayApplet to spit out
better coords.
</pre>
<p>
<div class="figure">
<svg class="figure" xmlns="http://www.w3.org/2000/svg"
version="1.1"
height="360"
width="360"
>
<!-- 1 unit = 20 pixels -->
<g transform="translate(180,180) scale(30)">
<!-- Q: how to scale the coords but not the stroke width??? -->
<!-- A: supposedly non-scaling-stroke was introduced in SVG Tiny 1.2, not sure if that's what's being used or what -->
<path
id="fig1apath"
vector-effect="non-scaling-stroke"
style="stroke: blue; stroke-width: 2.0; fill: none"
d="
M 1 4
L 2 2
L 0 -4
L -2 2
L -1 4
L 1 4
M 1 4
l -.5 1.5
M 2 2
l 1 2
M 0 -4
l 0 -2
M -2 2
l -1 2
M -1 4
l .5 1.5
"
/>
<g transform="translate(0,3.8) scale(.033333) scale(6)">
<path
vector-effect="non-scaling-stroke"
style="stroke:black; stroke-width:1.0; fill:none"
d="
M -1 0
L 1 0
M 0 -1
L 0 1
"
/>
</g>
<!-- final local scale must undo effect of global scale! otherwise we get wrong sized text -->
<g transform="translate(0,4.6) scale(.033333) translate(-7.5,-7) "> <text>e<tspan dy="5" font-size="70%">0</tspan></text> </g>
<g transform="translate(1,4) scale(.033333) translate(5,5) "> <text>v<tspan dy="5" font-size="70%">1</tspan></text> </g>
<g transform="translate(1.5,3) scale(.033333) translate(-20,5) "> <text>e<tspan dy="5" font-size="70%">2</tspan></text> </g>
<g transform="translate(2,2) scale(.033333) translate(5,0) "> <text>v<tspan dy="5" font-size="70%">3</tspan></text> </g>
<g transform="translate(1,-1) scale(.033333) translate(-15,20) "> <text>e<tspan dy="5" font-size="70%">4</tspan></text> </g>
<g transform="translate(0,-4) scale(.033333) translate(5,0) "> <text>v<tspan dy="5" font-size="70%">5</tspan></text> </g>
<g transform="translate(-1,-1) scale(.033333) translate(0,20) "> <text>e<tspan dy="5" font-size="70%">6</tspan></text> </g>
<g transform="translate(-2,2) scale(.033333) translate(-20,0) "> <text>v<tspan dy="5" font-size="70%">7</tspan></text> </g>
<g transform="translate(-1.5,3) scale(.033333) translate(5,5) "> <text>e<tspan dy="5" font-size="70%">8</tspan></text> </g>
<g transform="translate(-1,4) scale(.033333) translate(-20,5) "> <text>v<tspan dy="5" font-size="70%">9</tspan></text> </g>
</g>
</svg>
</div>
<br>
Figure 1(a): 2 exits tilted too far CW, 1 good exit, 2 exits tilted too far CCW
<p>
<pre class="asciiprevis">
Figure 1(b): 211
\
*v5
| \_
| \e4
| \_
e6| *v3
| | \
| |
| |
| |e2
| |
| |
| |
v7*---+---*v1
| e0 |
(XXX Need to tweak it a bit from this to give quills slop)
or maybe:
\
|
*v5
|\
| \
| \
| \e4
| \
| \
| \
e6| *v3
| | \
| |e2
| |
v7*---+---*v1
| e0 |
In ShephardsPlayApplet:
nVirtual=5
0,1 1/1
1,1 1/1
\
*
\
\
\ \
\
\
\ \
\
\
\ \
\
\
\ \
\
\
\ *
\ / \
\ / \
*---*
\ /
</pre>
<div class="figure">
<svg class="figure" xmlns="http://www.w3.org/2000/svg"
version="1.1"
height="320"
width="320"
>
<!-- 1 unit = 20 pixels -->
<g transform="translate(160,160) scale(30)">
<!-- Q: how to scale the coords but not the stroke width??? -->
<!-- A: supposedly non-scaling-stroke was introduced in SVG Tiny 1.2, not sure if that's what's being used or what -->
<path
id="fig1bpath"
vector-effect="non-scaling-stroke"
style="stroke: blue; stroke-width: 2.0; fill: none"
d="
M 1 3.5
L 2 .5
L -2.5 -4
L 0 3.5
L 1 3.5
M 1 3.5
l -.25 1
M 2 .5
l 1 .5
M -2.5 -4
l -.75 -1
M 0 3.5
l .25 1
"
/>
<g transform="translate(0.5,3.4) scale(.033333) scale(6)">
<path
vector-effect="non-scaling-stroke"
style="stroke:black; stroke-width:1.0; fill:none"
d="
M -1 0
L 1 0
M 0 -1
L 0 1
"
/>
</g>
<!-- final local scale must undo effect of global scale! otherwise we get wrong sized text -->
<g transform="translate(.5,4) scale(.033333) translate(-5,-5) "> <text>e<tspan dy="5" font-size="70%">0</tspan></text> </g>
<g transform="translate(1,3.5) scale(.033333) translate(5,5) "> <text>v<tspan dy="5" font-size="70%">1</tspan></text> </g>
<g transform="translate(1.5,2) scale(.033333) translate(-20,0) "> <text>e<tspan dy="5" font-size="70%">2</tspan></text> </g>
<g transform="translate(2,.5) scale(.033333) translate(0,-5) "> <text>v<tspan dy="5" font-size="70%">3</tspan></text> </g>
<g transform="translate(0,-1.75) scale(.033333) translate(0,0) "> <text>e<tspan dy="5" font-size="70%">4</tspan></text> </g>
<g transform="translate(-2.5,-4) scale(.033333) translate(2.5,-2.5) "> <text>v<tspan dy="5" font-size="70%">5</tspan></text> </g>
<g transform="translate(-1.25,-.25) scale(.033333) translate(5,0) "> <text>e<tspan dy="5" font-size="70%">6</tspan></text> </g>
<g transform="translate(0,3.5) scale(.033333) translate(-20,5) "> <text>v<tspan dy="5" font-size="70%">7</tspan></text> </g>
<!-- final local scale must undo effect of global scale! otherwise we get wrong sized text -->
</g>
</svg>
</div>
<br>
Figure 1(b): 2 exits tilted too far CW, 1 good exit, 1 exit tilted too far CW
<p>
<pre class="asciiprevis">
Figure 1(c): 310
__/
__/
__*v5
__/ \
e6__/ \
__/ \
\e4
\
\
\
*v3
| \
__/ |e2
__/ |
v7*---- ... -------+---*v1
___/ e0 |
/
In ShephardsPlayApplet:
nVirtual=5 (good for just primal... although primal/dual equalized at about 5.75 or so)
0,1 1/1
1,2 1/1
</pre>
<div class="figure">
<svg class="figure" xmlns="http://www.w3.org/2000/svg"
version="1.1"
height="190"
width="500"
>
<!-- 1 unit = 20 pixels -->
<g transform="translate(250,95) scale(10)">
<!-- Q: how to scale the coords but not the stroke width??? -->
<!-- A: supposedly non-scaling-stroke was introduced in SVG Tiny 1.2, not sure if that's what's being used or what -->
<path
id="fig1cpath"
vector-effect="non-scaling-stroke"
style="stroke: blue; stroke-width: 2.0; fill: none"
d="
M 14 7.5
L 15 4.5
L 0 -7.5
L -18.75 7.5
L 14 7.5
M 14 7.5
l -.25 2
M 15 4.5
l 2 1
M 0 -7.5
l 1.5,-1.5
M -18.75 7.5
l -3 1.5
"
/>
<g transform="translate(13.7,7.3) scale(.1) scale(6)">
<path
vector-effect="non-scaling-stroke"
style="stroke:black; stroke-width:1.0; fill:none"
d="
M -1 0
L 1 0
M 0 -1
L 0 1
"
/>
</g>
<!-- final local scale must undo effect of global scale! otherwise we get wrong sized text -->
<g transform="translate(-2,7.5) scale(.1) translate(0,-5) "> <text>e<tspan dy="5" font-size="70%">0</tspan></text> </g>
<g transform="translate(14,7.5) scale(.1) translate(1,9) "> <text>v<tspan dy="5" font-size="70%">1</tspan></text> </g>
<g transform="translate(14.5,6) scale(.1) translate(1,6) "> <text>e<tspan dy="5" font-size="70%">2</tspan></text> </g>
<g transform="translate(15,4.5) scale(.1) translate(0,-5) "> <text>v<tspan dy="5" font-size="70%">3</tspan></text> </g>
<g transform="translate(7.5,-1.5) scale(.1) translate(-20,10) "> <text>e<tspan dy="5" font-size="70%">4</tspan></text> </g>
<g transform="translate(0,-7.5) scale(.1) translate(-15,-5) "> <text>v<tspan dy="5" font-size="70%">5</tspan></text> </g>
<g transform="translate(-9,0) scale(.1) translate(15,-5) "> <text>e<tspan dy="5" font-size="70%">6</tspan></text> </g>
<g transform="translate(-18.75,7.5) scale(.1) translate(-10,-5) "> <text>v<tspan dy="5" font-size="70%">7</tspan></text> </g>
<!-- final local scale must undo effect of global scale! otherwise we get wrong sized text -->
</g>
</svg>
</div>
<br>
Figure 1(c): 3 exits tilted too far CW, 1 good exit
<p>
Figure 1(d): gradual double spiral
<p>
Figure 1(e): steep double spiral
<p>
Figure 1(f): gradual double spiral
<p>
Figure 1(g): steep double spiral
<p>
These examples satisfy all the properties
of a single-exit single-polygon lagoon with the cc at the origin,
*except* that when we actually compute the cc,
we'll find it is not at the origin.
We now show that every attempt at constructing a single-exit single-polygon lagoon
is doomed in this way--
that is, given any candidate primal polygon
and a point in its interior such that there is only one good exit
with respect to that point,
that point is *not* the cc of the lagoon.
<p>
The heavy lifting will be done by a lemma which we'll call
the Spiral Moment Lemma,
which states, roughly, that if a contiguous section of a polygon perimeter
is outwardly-spiral-shaped with respect to the origin
and all exits on this section are bad,
then the cc of this section is offset from the origin
in a direction lying strictly between that of the final (largest) position vector
and the corresponding final outward normal vector
(which are strictly less than
$90^{\circ}$
apart).
<p>
How that helps us is as follows:
a polygon with only a single good exit with respect to the origin
is naturally composed of two such sections
(outwardly spiraling in opposite directions)
plus the good exit.
Roughly, the Spiral Moment tells us
that the cc of each of these two sections
is offset from the origin roughly in the same direction as
(i.e. has strictly positive dot product with)
the good exit direction (or position).
Therefore the cc of the whole polygon,
being a weighted average of these two cc's
and the position of the good exit vertex,
is likewise offset from the origin in that same direction,
and so cannot be the origin.