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Cell["\<\
Cylinders Through Five Points: Complex and Real Enumerative Geometry\
\>", "Title"],
Cell["Daniel Lichtblau", "Address"],
Cell["\<\
Wolfram Research, Inc.
100 Trade Center Dr.
Champaign, IL 61820
danl@wolfram.com\
\>", "Address",
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". It is known that five points in ",
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" generically determine a finite number of cylinders containing those \
points. We discuss ways in which it can be shown that the generic (complex) \
number of solutions, with multiplicity, is six, of which an even number will \
be real valued and hence correspond to actual cylinders in ",
Cell[BoxData[
SuperscriptBox["\[DoubleStruckCapitalR]", "3"]]],
". We partially classify the case of no real solutions in terms of the \
geometry of the five given points. We also investigate the special case where \
the five given points are coplanar, as it differs from the generic case for \
both complex and real valued solution cardinalities.\n\n2000 Mathematics \
Subject Classification: 14N10, 14Q10, 51N20, 52A15, 65C05, 65H10, 68U20.\n\n\
Key words and phrases: Enumerative geometry, Gr\[ODoubleDot]bner bases, \
nonlinear systems, constraint geometry.\n\nThis is a slightly updated version \
of a paper that appears in: Proceedings of the Sixth International Workshop \
on Automated Deduction in Geometry (ADG 2006), Francisco Botana and Tomas \
Recio, editors. Lecture Notes in Artificial Intelligence, \
Springer\[Hyphen]Verlag."
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". Introduction"
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Cell[TextData[{
"Given five generic points in ",
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" it is not hard to show that there are finitely many solutions to the set \
of equations that determine cylinders containing those points. This is to be \
expected because cylinders have five degrees of freedom (a radius and four \
parameters to determine the axial line). Several papers prove that the \
generic number of such solutions, in complex space, is six (counted with \
multiplicity) [",
CounterBox["Reference", "Bottema Veldkamp"],
"] [",
CounterBox["Reference", "Chaperon Goulette"],
"] [",
CounterBox["Reference", "Devillers Mourrain Preparata Trebuchet"],
"] [",
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"]. Of these any even number may be real valued."
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Cell["\<\
This is of importance for several reasons. First we indicate two constraint \
geometry interpretations of the problem.\
\>", "Text",
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Cell[TextData[{
"\[FilledSmallCircle] ",
"Given five points in ",
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", find the smallest positive ",
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" and axis parameters such that the cylinder of radius ",
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" with those parameters tangentially encloses the balls of radius ",
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" centered at the points [",
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"] [",
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"]."
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"\[FilledSmallCircle] ",
"A common need in scene classification [",
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"] is to find a best fitting cylinder for a set of more than five data \
points. To do so one might start with an exact fit to five points, followed \
by optimization methods to get a best fit to the full ensemble. See [",
CounterBox["Reference", "Lichtblau 1"],
"] for several references to applications of this."
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"In order to tackle either of these it is necessary to find all cylinders \
through five given points [",
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"]; clearly it is useful to know the expected size of the solution set (or \
an upper bound, if we restrict to only real valued solutions). In this paper \
we discuss several aspects to this enumeration problem."
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Cell["\<\
Some experimentation indicates that it is not infrequent, when working with \
pseudorandom point coordinates, to have configurations with no real cylinder \
solutions. We investigate this situation in an attempt to understand the \
geometry of the configurations that give rise to it. Another point of \
interest is the case where the five points are coplanar. In this situation we \
will see that the generic number of complex solutions to the cylinder \
parameter equations is four. It turns out, however, that the number of real \
solutions is at most two. In discussing the case of coplanar data we state a \
conjecture regarding the degenerate situation where there is a dimensional \
component to the cylinder solutions. This is of interest insofar as \
configurations near one with such a solution set might exhibit numerical \
instabilities in a computational setting.\
\>", "Text",
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"The related problem of finding cylinders of a given radius through four \
given points in ",
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" is discussed in [",
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"] [",
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"] [",
CounterBox["Reference", "Macdonald Pach Theobald"],
"] [",
CounterBox["Reference", "Schomer Sellen Teichmann Yap"],
"]. The special case of four coplanar points is discussed in [",
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"]. In contrast to cylinders through five coplanar points, all cylinder \
solutions of given radius through four coplanar points can be real valued."
}], "Text"],
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"This paper is a companion to [",
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"]. Substantial emphasis therein was placed on various aspects of solving \
systems and related computation for purposes of deducing properties of \
cylinders through five points. Here we rely primarily on elementary arguments \
that cover the theory, with less focus on computational specifics. \
Experiments that gave rise to this use code similar to that presented and \
explained in [",
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"]. The rest of this paper is as follows. Section 2 proves that there are \
generically six (possibly complex valued) solutions to the equations for \
cylinders containing five given points. We begin by formulating polynomial \
conditions that the fourth and fifth points project to the same circle as the \
first three in a given direction, and work with the ideal formed by these \
polynomials. The main tactic is a count of solutions in projective space, \
followed by a simple computation to enumerate solutions at infinity. We then \
cover some implications of this. Section 3 discusses in detail the case of no \
real solutions. Here the main tool is again the cocircularity polynomials; we \
now make observations about the behavior of their respective direction curves \
in real space. In section 4 we look at cases that have six real solutions. \
Section 5 investigates the special case when the points are coplanar. This is \
followed with a summary and some directions of further inquiry."
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"One way to approach some of these problems, from the perspective of \
automated geometry, is to employ comprehensive Gr\[ODoubleDot]bner systems as \
in [",
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"], to classify both generic solutions and specialized configurations e.g. \
where the number of solutions becomes infinite. The difficulty is that, to \
date, the computations involved have been intractable. One could regard the \
alternate approach taken in this paper as a blend of computational tools and \
human guidance, to make progress on problems for which fully automated \
methods seem to falter. That is to say, we do not work with \
full\[Hyphen]blown automated geometric deduction tools, but borrow a bit from \
underlying computational methods."
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"In the sequel we use \"cylinder\" to denote solutions to polynomial \
equations for a cylinder, regardless of whether they are real or complex \
valued. To specify the former we use \"real cylinders\". Typically we will \
use the term \"parameters\" to refer to coordinates in the configuration \
space of the five points (which we may identify with ",
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"). The values that specify a cylinder, to wit, radius and axial line \
features, are generally referred to as \"variables\" (since they are what we \
solve for in finding cylinders) or as \"cylinder parameters\" to distinguish \
them from the coordinate parameters already mentioned."
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Cell["\<\
I thank the anonymous referees of this and a prior draft. Their several \
useful remarks and suggestions improved both exposition and references.\
\>", "Text",
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". Counting (Possibly Complex) Cylinders Through Five Points"
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"We recall a way of setting up the problem that gives rise to two equations \
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remaining points, ",
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"One observes from this that the number of solutions is generically finite, \
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has total degree of three in the variables ",
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\
\>"],
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Cell[TextData[{
StyleBox["PROPOSITION 1.",
FontSlant->"Plain",
FontVariations->{"CompatibilityType"->0}],
" Configurations that give rise to an empty solution set or to a solution \
set of positive dimension lie on a variety in the configuration space."
}], "Theorem"],
Cell[TextData[{
"PROOF (sketch). One obtains, in principle, a description of the generic \
solutions by forming a lexicographic Gr\[ODoubleDot]bner basis for the system \
(",
CounterBox["NumberedEquation", "cocircular equations"],
"). The process of doing this gives rise to the generic basis because at \
steps along the way one is allowed to divide by polynomials in the \
indeterminates. All inputs that fail to give the generic basis thus must \
satisfy conditions among the coordinates that cause these polynomials, upon \
specialization, to vanish. As there are finitely many steps in forming the \
basis, there are finitely many such conditions. As these vanishing conditions \
are defined by polynomials, their union is a variety. We may further refine \
it. Some configurations might fail to give the generic basis but still yield \
a nonzero finite solution set. If we exclude the conditions that give this \
situation, we are still left with a variety for which we get either zero or \
infinitely many (complex valued) solutions. ",
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Cell[TextData[{
StyleBox["DEFINITION",
FontSlant->"Plain",
FontVariations->{"CompatibilityType"->0}],
StyleBox[" 1.",
FontSlant->"Plain"],
" ",
StyleBox["Above we saw that the subset of nongeneric configurations that \
give either zero or infinitely many solutions comprises a variety. We refer \
to this as the \"bad variety\", denoted ",
FontSlant->"Italic"],
Cell[BoxData[
StyleBox[
SubscriptBox["V", "bad"],
FontSlant->"Italic"]]],
StyleBox[". Several results below are stated in terms of configurations that \
miss this variety.",
FontSlant->"Italic"]
}], "Theorem"],
Cell[TextData[{
"We remark that ",
Cell[BoxData[
StyleBox[
SubscriptBox["V", "bad"],
FontSlant->"Italic"]]],
" is a part of the discriminant variety (see [",
CounterBox["Reference", "Lazard Rouillier"],
"])."
}], "Text"],
Cell[TextData[{
StyleBox["PROPOSITION 2.",
FontSlant->"Plain",
FontVariations->{"CompatibilityType"->0}],
" There is a nonempty open set in our configuration space (which is, in \
effect, ",
Cell[BoxData[
SuperscriptBox["\[DoubleStruckCapitalR]", "15"]]],
") for which we obtain no cylinders in ",
Cell[BoxData[
SuperscriptBox["\[DoubleStruckCapitalR]", "3"]]],
"."
}], "Theorem"],
Cell[TextData[{
"PROOF. If one point is contained strictly within the convex hull of the \
other four then, as cylinders are convex, we have no real cylinders \
containing the five points. Small perturbations in real configuration space \
do not alter the situation that one is inside the hull of the other four. \
Hence there is an open set in real configuration space for which we obtain no \
real cylinders. ",
Cell[BoxData["\[Square]"]]
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CellChangeTimes->{{3.391368775866609*^9, 3.39136877622669*^9}}],
Cell[TextData[{
StyleBox["COROLLARY 1. ",
FontSlant->"Plain",
FontVariations->{"CompatibilityType"->0}],
"The maximum number of cylinders, already shown to be bounded by nine, is in \
fact no larger than eight."
}], "Theorem"],
Cell[TextData[{
"PROOF. This is a consequence of the following facts. (i) Restricting to \
real inputs does not move us out of the generic case because this restriction \
is not algebraic. (ii) Given real data, complex solutions appear in pairs. \
(iii) The case where one point lies in the hull of the other four contains an \
open subset in the real part of the parameter space. Hence there is an open \
set in parameter space for which there are only complex solutions. So in \
general there must be an even number of solutions. ",
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CellChangeTimes->{{3.39137037898957*^9, 3.391370379864995*^9}}],
Cell["\<\
This also shows that the the number cannot be seven. So we know it is either \
six or eight.\
\>", "Text"],
Cell[TextData[{
StyleBox["THEOREM 1.",
FontSlant->"Plain",
FontVariations->{"CompatibilityType"->0}],
" ",
"Five generic points in ",
Cell[BoxData[
SuperscriptBox["\[DoubleStruckCapitalR]", "3"]]],
" determine six distinct sets of cylinder parameters, of which an even \
number are real valued."
}], "Theorem"],
Cell[TextData[{
"Proofs may be found in [",
CounterBox["Reference", "Bottema Veldkamp"],
"] [",
CounterBox["Reference", "Chaperon Goulette"],
"] [",
CounterBox["Reference", "Devillers Mourrain Preparata Trebuchet"],
"] [",
CounterBox["Reference", "Lichtblau 1"],
"]. They make use of various algebraic or geometric features particular to \
this problem. An algorithmic approach in [",
CounterBox["Reference", "Lichtblau 2"],
"] proves this blindly, that is, without use of geometry\[Hyphen]specific \
features. (As with many algorithms in geometry, one can argue as to whether \
this is a good or bad thing, insofar as automated proofs often convey little \
insight. Regardless, algorithmic technology should not be ignored.) Proofs in \
[",
CounterBox["Reference", "Lichtblau 1"],
"] count roots based on either a Gr\[ODoubleDot]bner basis or resultant \
computation from (",
CounterBox["NumberedEquation", "cocircular equations"],
"). We give an independent proof below."
}], "Text",
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Cell[TextData[{
"PROOF. We will count the solutions at infinity for the polynomials shown in \
(",
CounterBox["NumberedEquation", "cocircular equations"],
"). We do this by homogenizing and setting the homogenizing variable to zero \
to get the initials (that is, the degree forms). They are"
}], "Text",
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Cell[TextData[{
"The solution set for ",
Cell[BoxData[
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" consists of the three cases ",
Cell[BoxData[
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", ",
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", and ",
Cell[BoxData[
FormBox[
RowBox[{"b", "\[Rule]", "0"}], TraditionalForm]]],
". We thus obtain three solutions at infinity for the homogenized system \
(these are simply the directions of the three lines between any pair of the \
first three points). The number of solutions from the Bezout theorem, nine, \
counts these three, and hence there are six solutions in affine space. ",
Cell[BoxData["\[Square]"]]
}], "Text"],
Cell[TextData[{
"In [",
CounterBox["Reference", "Lichtblau 1"],
"] a mixed volume computation is done in order to bound the number of \
solutions using a method presented in [",
CounterBox["Reference", "Sturmfels"],
"]. The value obtained in this computation is ",
Cell[BoxData[
FormBox["8", TraditionalForm]]],
", and from [",
CounterBox["Reference", "Sturmfels"],
"] we moreover know that that will be the generic number of solutions for \
problems with the same Newton polytope for monomial exponent vectors. To \
understand why this cylinder problem is not generic for the mixed volume \
computation, note that the initials are identical up to a constant multiple; \
any random perturbation (say, add ",
Cell[BoxData[
FormBox[
SuperscriptBox["b", "3"], TraditionalForm]]],
" to the first polynomial) will give only one solution at infinity, and \
therefore yield ",
Cell[BoxData[
FormBox["8", TraditionalForm]]],
" finite solutions."
}], "Text",
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Cell[TextData[{
StyleBox["PROPOSITION 3.",
FontSlant->"Plain",
FontVariations->{"CompatibilityType"->0}],
" There is a closed set with nonempty interior in ",
Cell[BoxData[
RowBox[{
SuperscriptBox["\[DoubleStruckCapitalR]", "15"], "-",
SubscriptBox["V", "bad"]}]]],
" for which we obtain six real cylinders provided we count solutions by \
multiplicity."
}], "Theorem"],
Cell[TextData[{
"PROOF (sketch). We may count the number of real roots using the Rule of \
Signs [",
CounterBox["Reference", "Mishra"],
"] on the univariate polynomial in any generic lexicographic \
Gr\[ODoubleDot]bner basis. This gives a closed condition for the boundary of \
the set of configurations that yield six sets of real valued cylinder \
parameters. To show it has nonempty interior it suffices to demonstrate one \
such configuration that has no multiple solutions. But this is the case for \
the two six\[Hyphen]real\[Hyphen]cylinder examples shown in [",
CounterBox["Reference", "Lichtblau 1"],
"] (one of which is formed from two regular tetrahedra sharing a common \
face). ",
Cell[BoxData["\[Square]"]]
}], "Text"],
Cell[TextData[{
"Similar argument shows that the sets in ",
Cell[BoxData[
RowBox[{
SuperscriptBox["\[DoubleStruckCapitalR]", "15"], "-",
SubscriptBox[
StyleBox["V",
FontSlant->"Italic"], "bad"]}]]],
" that give rise to two or four cylinders real cylinders also have nonempty \
interiors. From Proposition 2 we already knew this to be the case for \
configurations that give no real cylinders."
}], "Text",
CellChangeTimes->{3.391994154557924*^9}],
Cell[TextData[{
"It may be useful to look at Theorem 1 in the context of what are known as \
comprehensive Gr\[ODoubleDot]bner bases [",
CounterBox["Reference", "Becker Weispfenning Kredel"],
"] [",
CounterBox["Reference", "Manubens Montes 2"],
"] [",
CounterBox["Reference", "Weispfenning"],
"]. This construction in effect allows one to circumvent the problem that Gr\
\[ODoubleDot]bner bases are not continuous in their input data; indeed it \
seems designed to address that defect. Such a basis contains encoded all Gr\
\[ODoubleDot]bner bases, for a given ideal with respect to a specified term \
order, under all specializations of the parameters. It does so in essence by \
doing multiple polynomial reductions on a given polynomial in the basis, to \
simultaneously allow for the possibility that any nonnumeric leading \
coefficient might or might not be zero. The upshot is that the coefficients \
of the comprehensive Gr\[ODoubleDot]bner basis vary continuously in the \
parameters of the configuration. The typical use of such a basis is in \
concrete examples when one wishes to make case distinctions based on \
parameter values. When a lexicographic term ordering is utilized we can say a \
bit about the structure of such bases in the (generic) case of finite \
solution sets, using insight gained from our examples."
}], "Text",
CellChangeTimes->{{3.391388565544318*^9, 3.391388586121063*^9},
3.391994099368101*^9}],
Cell[TextData[{
"For instance, suppose we have six distinct solutions in a situation where \
the Shape Lemma [",
CounterBox["Reference", "Becker Marinari Mora Traverso"],
"] does not apply (see [",
CounterBox["Reference", "Lichtblau 1"],
"] for two such examples involving six distinct real valued cylinders). We \
consider the basis for such a numeric specialization of the general problem. \
From Gr\[ODoubleDot]bner basis theory we know it contains a univariate \
polynomial in the lexicographically last variable (say, ",
Cell[BoxData[
FormBox["a", TraditionalForm]]],
"). The degree of this polynomial must be less than six as we have assumed \
the Shape Lemma does not hold for this particular ideal and term ordering. We \
refer to the remaining variables as \"higher\"."
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CellChangeTimes->{{3.391994177016472*^9, 3.391994179274629*^9}, {
3.392071163531798*^9, 3.392071194916609*^9}}],
Cell[TextData[{
"We return for the moment to the general case wherein coefficients are again \
indeterminate parameters that vary in our configuration space. Note that a \
comprehensive Gr\[ODoubleDot]bner basis encodes, in vanishing conditions of \
leading coefficients, the basis for specializations of the sort just \
described. It also encodes the generic basis. By the Shape Lemma this latter \
contains linear polynomials in each of the higher variables with lead \
coefficients that generically do not vanish. By [",
CounterBox["Reference", "Kalkbrenner 1"],
"] all parametrized coefficients of at least one such linear polynomial must \
become identically zero for the nongeneric type of specialization under \
consideration (otherwise we would have fewer than six solutions). Hence the \
comprehensive basis must also contain nonlinear polynomials in those \
variables."
}], "Text",
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Cell[TextData[{
"Now consider a generic specialization of the configuration parameters. \
Again invoking results from [",
CounterBox["Reference", "Kalkbrenner 1"],
"] we know that solutions are obtained from the subset of the comprehensive \
basis that encodes the generic case (that is, the generic degree six \
univariate polynomial and the polynomials that are linear in each of the \
higher variables). These are shown also to satisfy those polynomials that are \
nonlinear in the higher variables. So such a linear polynomial (say, in the \
variable ",
Cell[BoxData[
FormBox["b", TraditionalForm]]],
") must have the form ",
Cell[BoxData[
FormBox[
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" where the second factor divides any corresponding polynomial(s) of higher \
degree in ",
Cell[BoxData[
FormBox["b", TraditionalForm]]],
". In other words, when the first factor, which involves only parameters, is \
nonzero, then where the second is satisfied all those in higher degree must \
also be satisfied; moreover they cannot all vanish when the first factor \
vanishes, so they must be divisible by the second factor."
}], "Text"]
}, Open ]],
Cell[CellGroupData[{
Cell[TextData[{
CounterBox["Section"],
". Configurations With no Real Solutions"
}], "Section"],
Cell[TextData[{
StyleBox["THEOREM 2.",
FontSlant->"Plain",
FontVariations->{"CompatibilityType"->0}],
" Suppose we have four noncoplanar points in ",
Cell[BoxData[
SuperscriptBox["\[DoubleStruckCapitalR]", "3"]]],
". They are the vertices of some tetrahedron. Then there is an open set ",
Cell[BoxData["S"]],
" containing the open tetrahedron and a dense subset of its boundary in the \
configuration space, such that if the fifth is chosen in ",
Cell[BoxData["S"]],
" there will be no real cylinders containing all five points."
}], "Theorem"],
Cell["\<\
REMARK. If the fifth point is inside the convex hull of the other four then \
we already know this result. Now take the tetrahedron formed by the four \
points. Through each vertex the planes containing the three coincident faces \
form a cone with triangular base. If the fifth point lies within that cone \
then it obscures that vertex, i.e. the vertex lies inside the new tetrahedron \
formed by the fifth point and the remaining three. Hence this case is also \
covered by the \"one point in the hull of the others\" situation. Note that \
in this case the fifth point need not be near in distance to the other four.\
\>", "Text"],
Cell[TextData[{
"PROOF 1. Suppose the fifth point lies on a face of the tetrahedron formed \
by the other four. Then the convexity argument still tells us that no \
cylinder in ",
Cell[BoxData[
SuperscriptBox["\[DoubleStruckCapitalR]", "3"]]],
" can contain all five points. As we assumed the tetrahedron coordinates are \
generic, we are in one of two situations: either having the fifth point lie \
on a face formed by three others puts the configuration in ",
Cell[BoxData[
FormBox[
SubscriptBox["V",
StyleBox["bad",
FontSlant->"Italic"]], TraditionalForm]]],
" or it does not. We show that generically it does not, or in other words, \
the algebraic condition that four points are coplanar is not a condition for \
the bad variety. That this is so follows from the trivial observation \
(verified computationally) that there are configurations with four coplanar \
points that give rise to lexicographic Gr\[ODoubleDot]bner basis with the \
generic \"shape\"; were this a condition to lie in the bad variety then every \
configuration with four coplanar points would be in it."
}], "Text",
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Cell[TextData[{
"The preceding argument shows that generically the fifth point is not on a \
tetrahedral face of the other four, so the set of such fifth points is dense \
in the set of all boundary points of the tetrahedron. By genericity we may \
assume that we have a univariate polynomial of degree six for one of the \
cylinder parameters. As there are no real cylinders containing this \
configuration, this polynomial has exclusively nonreal roots. These roots \
vary continuously with the configuration, hence the imaginary parts remain \
nonzero under small perturbations of the five points. Thus there is an open \
set around this point on the boundary for which we still obtain no real \
roots. As we require real roots in order to obtain cylinders in ",
Cell[BoxData[
SuperscriptBox["\[DoubleStruckCapitalR]", "3"]]],
" this suffices to finish the proof. ",
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"PROOF 2.",
" This line of reasoning was suggested by [Richard Bishop, private \
communication]. In order for all five points to lie on a cylinder there must \
be a plane tangent to the unit sphere, and a circle in that plane, such that \
they all project onto that circle. Suppose the fifth point is inside or on \
the boundary of the tetrahedron formed by the other four. Then it is clear \
that the projection of the five points onto any such plane will have the \
projection of this last point contained in the quadrilateral formed by the \
projection of the other four. Hence any quadratic in the plane that contains \
all five projected points must be a hyperbola (because all other quadratic \
curves are convex). Moreover the parameters of the hyperbola equation are \
continuous in the locations of the five points. As the set of projection \
planes is compact, a small perturbation of the fifth point beyond the hull of \
the other four will not alter the situation that the five points project onto \
hyperbolas in all such planes, hence they cannot lie on any cylinder in ",
Cell[BoxData[
SuperscriptBox["\[DoubleStruckCapitalR]", "3"]]],
". Hence from every boundary point on the tetrahedral hull of the four \
points, we may perturb outward some minimal distance (depending on that \
boundary point) and still have no real cylinders. As the tetrahedron boundary \
is compact we deduce that there is a minimum positive distance we can move \
outside and still not get real cylinders. ",
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" ",
"There is an open set ",
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" containing the closed tetrahedron, such that if the fifth is chosen in ",
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" there will be no cylinders in ",
Cell[BoxData[
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" containing all five points. In other words, the \"bad\" variety in \
configuration space is not an issue."
}], "Theorem"],
Cell["\<\
We now wish to show that all configurations that give no real cylinders arise \
in the setting of Theorem 2. Specifically we state the following conjecture.\
\>", "Text",
CellChangeTimes->{3.391994104816707*^9}],
Cell[TextData[{
StyleBox["CONJECTURE 1.",
FontSlant->"Plain"],
" ",
StyleBox["Suppose we have a configuration of five points in ",
FontSlant->"Italic"],
Cell[BoxData[
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StyleBox[" for which no real cylinders exist, and moreover assume that no \
point lies in the hull of the others. Then one of the points can be moved \
anywhere inside the convex hull of the full set and still we will get no real \
cylinders. In particular we could move this point along a line segment from \
outside to inside the hull of the other four, and at no point on that path \
would we get real cylinders. Thus we could regard the given configuration as \
a perturbation of one that has one point inside the hull of the other four, \
effectively providing a converse to Theorem 2.",
FontSlant->"Italic"]
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Cell[TextData[{
"We make observations of sufficient conditions for a proof and then state as \
a theorem a special case wherein we can fullfill the conditions. First \
consider the two curves that are solution sets to the two polynomials in (",
CounterBox["NumberedEquation"],
"). They are cubics that have one or more closed topological components in \
two dimensional real projective space, and because they are cubic they each \
have at least one real component that goes to infinity. (It is well known \
that they are connected as complex curves; by \"components\" we mean the \
obvious thing with respect to intersections with real space.)"
}], "Text"],
Cell["\<\
We regard each point on such a curve as a solution to the direction parameter \
equations given by the four points in configuration space lying on a cylinder \
with axis in that direction. In other words, each point on the solution curve \
in parameter space defines a cylinder through the four points in \
configuration space that were used to form that equation. Suppose that at a \
solution on one such component, the fifth point lies inside the cylinder thus \
obtained (we are being loose with terminology but trust the meaning of \
\"inside a cylinder\" is clear).\
\>", "Text"],
Cell["\<\
Claim: For generic configurations it must then lie inside all cylinders \
defined by points on that affine component of the solution curve.\
\>", "Text"],
Cell["\<\
The proof of this claim has a small complication. Specifically, we must show \
that in order for the fifth point to \"escape\" outside the cylinder \
containing the other four, it must cross that cylinder (in contradiction to \
our hypothesis that there are no real cylinders containing all five points). \
A priori there is another way it might escape: the cylinder containing four \
can degenerate to a plane and subsequently reverse its \"open\" side. This \
degeneration can arise if the four points project onto a line for some \
direction. Such a direction must then lie in all four planes containing three \
of the four points. This gives an overdetermined and generically inconsistent \
set of linear conditions. We thus have verified the claim for generic \
configurations: if the fifth point is inside a cylinder containing the other \
four, then it stays inside the cylinders defined by all points on that \
component of the solution curve, and these cylinders each contain the other \
four points.\
\>", "Text"],
Cell["\<\
Next we observe that, were this true not just on one topological component of \
the direction solution curve, but on all of them, then that fifth point works \
in the conjecture. We see this as follows. When the fifth point lies inside \
all cylinders defined by a solution curve component, then it projects along \
the cylinder axis to a point inside the circle that intersects the \
projections of the other four. The same must hold for any other point in the \
interior of the convex hull of the five points. This is because such a point, \
written as a convex combination of the five, must either be in the interior \
of the tetrahedron defined by the first four (and thus project to the \
interior of the circle they define), or else have a nontrivial component of \
that fifth point and again thereby project to that circle interior.\
\>", "Text"],
Cell[TextData[{
"We now proceed to construct a solution on one direction curve, that is, a \
cylinder containing four of the points, such that the fifth is inside it. We \
can arrange our four so that three are in the ",
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intersects the triangle defined by the first three (using a rotation, such an \
arrangement can always be found for a configuration of five points in ",
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"). We place the fourth point on the ",
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" axis beneath the origin. Projecting from the fourth point onto a plane in \
the direction of the segment between the fourth and fifth points gives a \
unique circle containing the first three. The cylinder along that direction \
and containing that circle thus encloses the fourth and fifth points. Now we \
simply move one of the direction coordinates, forming new projections and \
cylinders containing the first three points, until one of the remaining \
points (say, the fourth) hits that cylinder. What we have done is to arrive \
on one of the two direction solution curves we set up in (",
CounterBox["NumberedEquation"],
"). Thus we obtain a cylinder containing four points and enclosing the \
fifth. From the discussion above we know that this holds for all cylinders \
defined by this affine component of the curve of directions."
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At this point we have a sufficient condition for the conjecture to hold. We \
simply require that each of the solution curves have only one affine \
component.\
\>", "Text"],
Cell[TextData[{
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FontSlant->"Plain"],
" Given five points in ",
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" for which there are no real solutions to the cylinder equations, suppose \
there are three such that\n(i) The segment joining the remaining two \
intersects the triangle bounded by those three.\n(ii) The two curves of \
solution directions for cylinders containing those three and either the \
fourth or fifth respectively, each have only one component in real projective \
space.\nThen either the fourth or fifth point can be moved anywhere inside \
the hull of the five and there will be no real cylinder containing this new \
point and the other four."
}], "Theorem"],
Cell[TextData[{
"As remarked above we can always order the points in such a way that the \
first condition holds. But then in general the second condition will not \
hold. We believe the conjecture to be true all the same, though we do not \
have a proof at this time. We also mention that extensive graphical evidence \
suggests that most often these curves have one component in two dimensional \
real projective space. This is found by taking random examples with three \
points in the ",
Cell[BoxData[
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" coordinate plane and the fourth and fifth above and below respectively, \
throwing away those that have real solutions, throwing away from the rest \
those for which the segment between fourth and fifth points does not go \
through the triangle bounded by the first three, and plotting the zero level \
sets for the two cylinder equations in remaining cases. \"Most often\", in \
this setting, refers to specifics of how we select our five points; we use \
pseudorandom values in a unit interval for each of the free parameters. In \
any case it would seem that this method applies frequently to configurations \
that give rise to no real cylinders. Finally, we remark that such curves have \
either one infinite component, three infinite components, or one infinite and \
one finite component (homotopic to a circle). This is easily shown by a root \
counting argument."
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"We illustrate with a configuration that meets the conditions of the \
hypotheses. There are no real cylinders containing the set of points ",
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the triangle formed by the other three."
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Here are the direction parameter polynomials obtained by requiring that the \
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the first three points and a given direction vector.\
\>", "Text"],
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" plots of their respective zero sets. It is clear from the way the \
respective affine parts will meet at infinity that each has one topological \
component in real space, and moreover they do not intersect at finite points. \
Hence Theorem 3 applies to this configuration."
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Note that we can weaken the second hypothesis of Theorem 3 so that the curves \
may have multiple components, provided the components for one are not \
separated by any component of the other. Graphical evidence supports the \
belief that this weaker requirement is always satisfied. Clearly a proof to \
this effect would suffice to prove the conjecture.\
\>", "Text",
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Cell[CellGroupData[{
Cell[TextData[{
CounterBox["Section"],
". Configurations that have six real solutions"
}], "Section",
CellChangeTimes->{{3.381771803572245*^9, 3.381771810795013*^9}}],
Cell[TextData[{
"We describe in brief two cases that have six solutions. Further detail may \
be found in [",
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"]."
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Cell[TextData[{
"(1). Start with four points forming vertices of a square in the ",
Cell[BoxData[
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RowBox[{"x", " ", "y"}], TraditionalForm]]],
" plane. This is the base of a pyramid with the fifth point as its apex \
above the centroid of this square. We obtain two horizontal cylinders each \
passing through a pair of opposite triangular faces of the pyramid. The \
remaining four each pass through a triangular face, angled upward, and an \
edge of the base. This case was first posted in [",
CounterBox["Reference", "Rusin"],
"]."
}], "Text",
CellChangeTimes->{{3.381771822022022*^9, 3.381771861401336*^9}, {
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"(2). Take as five points the vertices of two regular tetrahedron sharing a \
common face. This presents an interesting set of symmeries. Any cylinder will \
have a \"mirror image\" obtained by reflection across the joined face, and \
also have two \"conjugates\" obtained by rotation of ",
Cell[BoxData[
FormBox[
RowBox[{"\[Pi]", "/", "3"}], TraditionalForm]]],
". We show a visualization of this in Figure ",
CounterBox["NumberedFigure", "six cylider from double tetrahedra picture"],
"; the common face lies in the horizontal plane.\n"
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