(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 621632, 12135] NotebookOptionsPosition[ 549327, 10504] NotebookOutlinePosition[ 610589, 11792] CellTagsIndexPosition[ 610508, 11787] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}]& ), ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}]& ), ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageLast"]}]& ), ButtonNote->"Last Slide"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}, ButtonFunction:>Null]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ Cylinders Through Five Points: Computational Algebra and Geometry\ \>", "Title", CellChangeTimes->{{3.36407644171621*^9, 3.36407644570487*^9}}], Cell["\<\ Daniel Lichtblau Wolfram Research, Inc. 100 Trade Centre Dr. Champaign, IL 61820 danl@wolfram.com\ \>", "Author", CellChangeTimes->{{3.36407645189594*^9, 3.36407646988575*^9}}], Cell["\<\ ICMS, Castro Urdiales, Spain September 1\[Hyphen]3, 2006\ \>", "Author", CellChangeTimes->{{3.36407645189594*^9, 3.36407649859461*^9}, 3.36407661108185*^9, {3.36422751166418*^9, 3.3642275281864*^9}, { 3.36432541182797*^9, 3.36432541293438*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->"Previous Slide"], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->"Next Slide"] }], "PreviousNext"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}]& ), ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}]& ), ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageLast"]}]& ), ButtonNote->"Last Slide"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}, ButtonFunction:>Null]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Abstract", "Section", CellChangeTimes->{{3.36407688380276*^9, 3.36407688537294*^9}}], Cell[TextData[{ "We address the following question: Given five points in ", Cell[BoxData[ SuperscriptBox["\[DoubleStruckCapitalR]", "3"]]], ", determine a right circular cylinder containing those points. We obtain \ algebraic equations for the axial line and radius parameters and show that \ these give six solutions in the generic case. An even number (0, 2, 4, or 6) \ will be real valued and hence correspond to actual cylinders in ", Cell[BoxData[ SuperscriptBox["\[DoubleStruckCapitalR]", "3"]]], ". We will investigate computational and theoretical matters related to this \ problem. In particular we will show how exact and numeric Gr\[ODoubleDot]bner \ bases, equation solving, and related symbolic\[Hyphen]numeric methods may be \ used to advantage. We will also discuss some applications.\n" }], "Abstract", CellChangeTimes->{{3.36407686232606*^9, 3.3640768664716*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->"Previous Slide"], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->"Next Slide"] }], "PreviousNext"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}]& ), ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}]& ), ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageLast"]}]& ), ButtonNote->"Last Slide"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}, ButtonFunction:>Null]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Outline of the Problem", "Section", CellChangeTimes->{{3.36413448300955*^9, 3.36413448509781*^9}}], Cell[TextData[{ "Given five points in ", Cell[BoxData[ SuperscriptBox["\[DoubleStruckCapitalR]", "3"]]], ", we are to determine all right circular cylinders containing those \ points." }], "Text", CellChangeTimes->{{3.36413450398334*^9, 3.36413453809894*^9}, { 3.36413545053163*^9, 3.36413545152151*^9}}], Cell[CellGroupData[{ Cell["Questions of importance:", "Subsection", CellChangeTimes->{{3.36413450398334*^9, 3.36413453809894*^9}, { 3.36413545053163*^9, 3.36413545152151*^9}}], Cell["\<\ How do we know there are finitely many in generic case? How many are there? \ (Depends...Are we working in real or complex space?)\ \>", "Text", CellDingbat->"\[ClubSuit]", CellChangeTimes->{{3.36413456062874*^9, 3.36413461697853*^9}, { 3.36413473681751*^9, 3.36413477003013*^9}}], Cell["How do we find the axial line and radius parameters?", "Text", CellDingbat->"\[DiamondSuit]", CellChangeTimes->{{3.36413456062874*^9, 3.36413464430908*^9}}], Cell["\<\ Given the cylinder parameters, how do we obtain its implicit equation?\ \>", "Text", CellDingbat->"\[HeartSuit]", CellChangeTimes->{{3.36413456062874*^9, 3.36413458063678*^9}, { 3.36413465902249*^9, 3.36413466576896*^9}}], Cell["\<\ Reversing this, how can one obtain parameters from the implicit form?\ \>", "Text", CellDingbat->"\[SpadeSuit]", CellChangeTimes->{{3.36413456062874*^9, 3.36413457942826*^9}}], Cell["How might we display them graphically?", "Text", CellDingbat->"\[ClubSuit]", CellChangeTimes->{{3.36413456062874*^9, 3.36413457237071*^9}, 3.36413464929186*^9}], Cell[TextData[{ "Given six or more points, how do we find the coordinates of a cylinder in \ ", Cell[BoxData[ SuperscriptBox["\[DoubleStruckCapitalR]", "3"]]], " that \"best\" fits those points?" }], "Text", CellDingbat->"\[DiamondSuit]", CellChangeTimes->{{3.36413456062874*^9, 3.3641345842583*^9}, { 3.36413468567692*^9, 3.36413468639553*^9}}], Cell["\<\ Given five points chosen with random uniform distribution in a cube, what is \ the expected probability that one lies inside the convex hull of the other \ four. Related to \"no real cylinder\" case . Also to an old recently solved \ problem in integral geometry.\ \>", "Text", CellDingbat->"\[HeartSuit]", CellChangeTimes->{{3.36413456062874*^9, 3.36413458631113*^9}, { 3.36413490906855*^9, 3.36413495203718*^9}}], Cell["\<\ To what extent can computational methods be used to prove enumerative \ geometry or other types of results related to this problem?\ \>", "Text", CellDingbat->"\[SpadeSuit]", CellChangeTimes->{{3.36413456062874*^9, 3.36413458063678*^9}, { 3.36413465902249*^9, 3.36413466576896*^9}, {3.3641350395005*^9, 3.3641350895528*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->"Previous Slide"], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->"Next Slide"] }], "PreviousNext"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}]& ), ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}]& ), ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageLast"]}]& ), ButtonNote->"Last Slide"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}, ButtonFunction:>Null]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Basics", "Section", CellChangeTimes->{{3.36413448300955*^9, 3.36413448509781*^9}, { 3.36413512543128*^9, 3.36413512860691*^9}, {3.36413580231783*^9, 3.36413580624819*^9}, 3.3641360561742*^9}], Cell[CellGroupData[{ Cell["Terminology", "Subsection", CellChangeTimes->{{3.36413456062874*^9, 3.36413458063678*^9}, { 3.36413465902249*^9, 3.36413466576896*^9}, {3.3641350395005*^9, 3.3641350895528*^9}, {3.36413514221242*^9, 3.36413530740587*^9}, 3.36413541856212*^9}], Cell["\<\ We say \"cylinder\" for any solution, and \"real cylinder\" for the real \ valued solutions.\ \>", "Text", CellChangeTimes->{{3.36413456062874*^9, 3.36413458063678*^9}, { 3.36413465902249*^9, 3.36413466576896*^9}, {3.3641350395005*^9, 3.3641350895528*^9}, {3.36413514221242*^9, 3.3641353185432*^9}}], Cell["\<\ We call configurations \"generic\" if they do not have multiple solutions and \ if all sufficiently small perturbations give same number of solutions. \ Usually we assume this of our configurations.\ \>", "Text", CellChangeTimes->{{3.36413456062874*^9, 3.36413458063678*^9}, { 3.36413465902249*^9, 3.36413466576896*^9}, {3.3641350395005*^9, 3.3641350895528*^9}, {3.36413514221242*^9, 3.3641353185432*^9}, { 3.36413536250761*^9, 3.36413540103895*^9}, {3.36414892188662*^9, 3.36414894036611*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Easy to show", "Subsection", CellChangeTimes->{{3.36413456062874*^9, 3.36413458063678*^9}, { 3.36413465902249*^9, 3.36413466576896*^9}, {3.3641350395005*^9, 3.3641350895528*^9}, {3.36413514221242*^9, 3.36413521559493*^9}, 3.36413541937702*^9}], Cell["\<\ There are finitely many solutions (plausible, because we get one equation for \ each data point and require five parameters to describe a cylinder).\ \>", "Text", CellChangeTimes->{{3.36413456062874*^9, 3.36413458063678*^9}, { 3.36413465902249*^9, 3.36413466576896*^9}, {3.3641350395005*^9, 3.3641350895528*^9}, {3.36413514221242*^9, 3.36413521819912*^9}}], Cell["If points are real valued then complex solutions pair off.", "Text", CellChangeTimes->{{3.36413456062874*^9, 3.36413458063678*^9}, { 3.36413465902249*^9, 3.36413466576896*^9}, {3.3641350395005*^9, 3.3641350895528*^9}, {3.36413514221242*^9, 3.36413530371812*^9}}], Cell["\<\ One expects six solutions. Reason: Take five \"random\" points. Solve for \ cylinder parameters. You \"always\" get six solutions. This is the Shape \ Lemma at work for you.\ \>", "Text", CellChangeTimes->{{3.36413456062874*^9, 3.36413458063678*^9}, { 3.36413465902249*^9, 3.36413466576896*^9}, {3.3641350395005*^9, 3.3641350895528*^9}, {3.36413514221242*^9, 3.36413530371812*^9}, { 3.36413569385438*^9, 3.3641357554142*^9}, {3.36417110713293*^9, 3.36417113053116*^9}}] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}]& ), ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}]& ), ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageLast"]}]& ), ButtonNote->"Last Slide"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}, ButtonFunction:>Null]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Related Work", "Section", CellChangeTimes->{{3.36413448300955*^9, 3.36413448509781*^9}, { 3.36413512543128*^9, 3.36413512860691*^9}, {3.36413580231783*^9, 3.36413580624819*^9}, {3.36413590720543*^9, 3.36413591019657*^9}}], Cell[CellGroupData[{ Cell["Already known", "Subsection", CellChangeTimes->{{3.36413456062874*^9, 3.36413458063678*^9}, { 3.36413465902249*^9, 3.36413466576896*^9}, {3.3641350395005*^9, 3.3641350895528*^9}, {3.36413514221242*^9, 3.36413521559493*^9}, 3.36413541937702*^9, {3.36413550691008*^9, 3.36413551034494*^9}}], Cell["\<\ Number of solutions in generic case is indeed six. First shown 1977 (Bottema \ and Veldkamp). Various other proofs appearing in recent years. We will show \ two simple computational proofs at end of this report.\ \>", "Text", CellChangeTimes->{{3.36413552897393*^9, 3.36413564173324*^9}, { 3.3641357621107*^9, 3.36413578292521*^9}, {3.36413606625188*^9, 3.36413610017882*^9}, {3.36417107015722*^9, 3.36417108313331*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Related to this presentation", "Subsection", CellChangeTimes->{{3.36413456062874*^9, 3.36413458063678*^9}, { 3.36413465902249*^9, 3.36413466576896*^9}, {3.3641350395005*^9, 3.3641350895528*^9}, {3.36413514221242*^9, 3.36413521559493*^9}, 3.36413541937702*^9, {3.36413550691008*^9, 3.36413551034494*^9}, { 3.36413594397354*^9, 3.36413596250396*^9}}], Cell["\<\ This is a companion to \"Cylinders Through Five Points: Complex and Real \ Enumerative Geometry\", which is really part 2 but was presented already at \ ADG 2006. The focus here is more on computational methods and related \ problems.\ \>", "Text", CellChangeTimes->{{3.36413552897393*^9, 3.36413564173324*^9}, { 3.3641357621107*^9, 3.36413578292521*^9}, {3.36413595058607*^9, 3.36413603605441*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->"Previous Slide"], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->"Next Slide"] }], "PreviousNext"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}]& ), ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}]& ), ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageLast"]}]& ), ButtonNote->"Last Slide"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}, ButtonFunction:>Null]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Computing cylinders through five points", "Section", CellChangeTimes->{{3.36413448300955*^9, 3.36413448509781*^9}, { 3.36413512543128*^9, 3.36413512860691*^9}, {3.36413580231783*^9, 3.36413580624819*^9}, {3.36413590720543*^9, 3.36413591019657*^9}, { 3.36414886187831*^9, 3.36414887751073*^9}}], Cell[CellGroupData[{ Cell["Our set up", "Subsection", CellChangeTimes->{{3.36413456062874*^9, 3.36413458063678*^9}, { 3.36413465902249*^9, 3.36413466576896*^9}, {3.3641350395005*^9, 3.3641350895528*^9}, {3.36413514221242*^9, 3.36413521559493*^9}, 3.36413541937702*^9, {3.36413550691008*^9, 3.36413551034494*^9}, { 3.36414888404236*^9, 3.36414888611734*^9}, {3.36414920903305*^9, 3.36414921000254*^9}}], Cell[TextData[{ "Parametrize the axis line as", StyleBox[" ", "InlineInput"], Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ RowBox[{"{", RowBox[{ RowBox[{"y", "=", RowBox[{ RowBox[{"a", " ", "x"}], "+", "b"}]}], ",", RowBox[{"z", "=", RowBox[{ RowBox[{"c", " ", "x"}], "+", "d"}]}]}], "}"}], "Text"], StyleBox[" ", "InlineInput"]}], TraditionalForm]]], ". So we need to solve for ", Cell[BoxData[ FormBox["a", TraditionalForm]]], ", ", Cell[BoxData[ FormBox["b", TraditionalForm]]], ", ", Cell[BoxData[ FormBox["c", TraditionalForm]]], ", ", Cell[BoxData[ FormBox["d", TraditionalForm]]], ", and a radius ", Cell[BoxData[ FormBox["r", TraditionalForm]]], ". One might argue that this only captures \"generic\" cases. But it avoids \ issues with double counting if we allow most general form of direction vector \ (because its negative gives same cylinder)." }], "Text", CellChangeTimes->{{3.36413552897393*^9, 3.36413564173324*^9}, { 3.3641357621107*^9, 3.36413578292521*^9}, {3.36413606625188*^9, 3.36413610017882*^9}, {3.36414888919884*^9, 3.36414889990518*^9}, { 3.36414894646219*^9, 3.36414898512187*^9}, {3.36414922282247*^9, 3.36414926808419*^9}, {3.3641493227209*^9, 3.36414936049289*^9}, { 3.36417281161532*^9, 3.36417289524014*^9}}], Cell[TextData[{ "Project points onto axis. For ", Cell[BoxData[ FormBox[ SubscriptBox["j", "th"], TraditionalForm]]], " point ", Cell[BoxData[ RowBox[{"(", RowBox[{ SubscriptBox["x", "j"], " ", ",", SubscriptBox["y", "j"], ",", SubscriptBox["z", "j"]}], ")"}]], FontFamily->"Utopia"], " we need length of orthogonal projection ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["perp", "j"], FontSlant->"Italic"], TraditionalForm]]], ". \nWork with equation ", StyleBox[" ", FontFamily->"Utopia"], Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"\[LeftDoubleBracketingBar]", StyleBox[ SubscriptBox["perp", "j"], FontSlant->"Italic"], "\[RightDoubleBracketingBar]"}], "2"], "=", SuperscriptBox["r", "2"]}], TraditionalForm]], FontFamily->"Utopia"], ".\nAfter some minor algebra, we have five polynomials of form" }], "Text", CellChangeTimes->{{3.36413552897393*^9, 3.36413564173324*^9}, { 3.3641357621107*^9, 3.36413578292521*^9}, {3.36413606625188*^9, 3.36413610017882*^9}, {3.36414888919884*^9, 3.36414889990518*^9}, { 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Take general implicit equation and specific one for cylinder at hand. \ Equate coefficients.\ \>", "Text", CellChangeTimes->{{3.36413552897393*^9, 3.36413564173324*^9}, { 3.3641357621107*^9, 3.36413578292521*^9}, {3.36413606625188*^9, 3.36413610017882*^9}, {3.36414888919884*^9, 3.36414889990518*^9}, { 3.36414894646219*^9, 3.36414898512187*^9}, {3.36414922282247*^9, 3.36414926808419*^9}, {3.3641493227209*^9, 3.36414936049289*^9}, { 3.36415209373923*^9, 3.36415210494351*^9}, {3.36415226232868*^9, 3.36415243493195*^9}, {3.36415247740181*^9, 3.36415273518206*^9}, { 3.36415305970315*^9, 3.36415306136211*^9}, {3.36416206913689*^9, 3.36416216907128*^9}}], Cell["Step 2. This gives equations in the parameters. 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Pick five points,", "Text", CellChangeTimes->{{3.36413552897393*^9, 3.36413564173324*^9}, { 3.3641357621107*^9, 3.36413578292521*^9}, {3.36413606625188*^9, 3.36413610017882*^9}, {3.36414888919884*^9, 3.36414889990518*^9}, { 3.36414894646219*^9, 3.36414898512187*^9}, {3.36414922282247*^9, 3.36414926808419*^9}, {3.3641493227209*^9, 3.36414936049289*^9}, { 3.36415209373923*^9, 3.36415210494351*^9}, {3.36415226232868*^9, 3.36415243493195*^9}, {3.36415247740181*^9, 3.36415273518206*^9}, { 3.36415305970315*^9, 3.36415306136211*^9}, {3.36416206913689*^9, 3.36416216907128*^9}, {3.36416353286938*^9, 3.36416354939093*^9}}], Cell["\<\ Step 2. Solve for cylinder parameters, obtain candidate solutions. Discard \ complex ones.\ \>", "Text", CellChangeTimes->{{3.36413552897393*^9, 3.36413564173324*^9}, { 3.3641357621107*^9, 3.36413578292521*^9}, {3.36413606625188*^9, 3.36413610017882*^9}, {3.36414888919884*^9, 3.36414889990518*^9}, { 3.36414894646219*^9, 3.36414898512187*^9}, {3.36414922282247*^9, 3.36414926808419*^9}, {3.3641493227209*^9, 3.36414936049289*^9}, { 3.36415209373923*^9, 3.36415210494351*^9}, {3.36415226232868*^9, 3.36415243493195*^9}, {3.36415247740181*^9, 3.36415273518206*^9}, { 3.36415305970315*^9, 3.36415306136211*^9}, {3.36416206913689*^9, 3.36416216907128*^9}, {3.36416353286938*^9, 3.36416357990571*^9}}], Cell["\<\ Step 3. Form sum of squares of distances of all points to the remaining \ candidates. Take the one with the smallest sum of squares.\ \>", "Text", CellChangeTimes->{{3.36413552897393*^9, 3.36413564173324*^9}, { 3.3641357621107*^9, 3.36413578292521*^9}, {3.36413606625188*^9, 3.36413610017882*^9}, {3.36414888919884*^9, 3.36414889990518*^9}, { 3.36414894646219*^9, 3.36414898512187*^9}, {3.36414922282247*^9, 3.36414926808419*^9}, {3.3641493227209*^9, 3.36414936049289*^9}, { 3.36415209373923*^9, 3.36415210494351*^9}, {3.36415226232868*^9, 3.36415243493195*^9}, {3.36415247740181*^9, 3.36415273518206*^9}, { 3.36415305970315*^9, 3.36415306136211*^9}, {3.36416206913689*^9, 3.36416217263507*^9}, {3.36416358436888*^9, 3.36416361474008*^9}}], Cell["\<\ Step 4. 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There are 12 solutions to the equations that result from this \ problem. All solutions can be real. In the example above all cylinders had \ the same radius of 9/10. Consider the top tetrahedron to give the four \ points. Gluing another onto each face gives six cylinders that go through \ those four points and all have radius of 9/10. Seems like 24 cylinders, but \ they pair off for a total of 12.\ \>", "Text", CellChangeTimes->{{3.36416860476375*^9, 3.36416866091606*^9}, { 3.36416957487991*^9, 3.36416958863064*^9}, {3.3641696952841*^9, 3.36416984973525*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->"Previous Slide"], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->"Next Slide"] }], "PreviousNext", CellChangeTimes->{{3.36415288822036*^9, 3.36415289144661*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}]& ), ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}]& ), ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageLast"]}]& ), ButtonNote->"Last Slide"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}, ButtonFunction:>Null]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Numbers of real cylinders that arise", "Section", CellChangeTimes->{{3.36413448300955*^9, 3.36413448509781*^9}, { 3.36413512543128*^9, 3.36413512860691*^9}, {3.36413580231783*^9, 3.36413580624819*^9}, {3.36413590720543*^9, 3.36413591019657*^9}, { 3.36414886187831*^9, 3.36414887751073*^9}, {3.36415147336487*^9, 3.36415147802102*^9}, {3.36415343256285*^9, 3.36415343931916*^9}, { 3.36416351532414*^9, 3.36416352304842*^9}, {3.36416396097275*^9, 3.36416396873205*^9}, {3.36416787899981*^9, 3.36416788029368*^9}, { 3.36416967723973*^9, 3.3641696910012*^9}, 3.36416985702353*^9, { 3.3641699002186*^9, 3.36416990418562*^9}, {3.3641707112057*^9, 3.3641707130673*^9}}], Cell[CellGroupData[{ Cell["Basics", "Subsection", CellChangeTimes->{{3.36417019404984*^9, 3.36417020712061*^9}, 3.36417075329776*^9}], Cell["Already saw number is even.", "Text", CellDingbat->"\[ClubSuit]", CellChangeTimes->{{3.36416860476375*^9, 3.36416866091606*^9}, { 3.36416957487991*^9, 3.36416958863064*^9}, {3.3641696952841*^9, 3.36416984973525*^9}, {3.36416990844802*^9, 3.36416999249911*^9}, 3.36417075329797*^9}], Cell["Already saw it can be as large as 6.", "Text", CellDingbat->"\[DiamondSuit]", CellChangeTimes->{{3.36416860476375*^9, 3.36416866091606*^9}, { 3.36416957487991*^9, 3.36416958863064*^9}, {3.3641696952841*^9, 3.36416984973525*^9}, {3.36416990844802*^9, 3.3641699965146*^9}, 3.3641707532981*^9}], Cell["\<\ Using pseudorandom sets of five points in unit cube will show the other \ possibilities all arise.\ \>", "Text", CellDingbat->"\[HeartSuit]", CellChangeTimes->{{3.36416860476375*^9, 3.36416866091606*^9}, { 3.36416957487991*^9, 3.36416958863064*^9}, {3.3641696952841*^9, 3.36416984973525*^9}, {3.36416990844802*^9, 3.36416999556182*^9}, 3.36417075329822*^9}], Cell["\<\ Obvious on reflection: We get no real cylinders whenever one point is in the \ (tetrahedral) hull of the other four. Reason: All projections onto planes \ keep it inside the hull of the projected quadrilateral, hence planar \ quadratic through the five projected points cannot be elliptical, hence not a \ circle.\ \>", "Text", CellDingbat->"\[SpadeSuit]", CellChangeTimes->{{3.36416860476375*^9, 3.36416866091606*^9}, { 3.36416957487991*^9, 3.36416958863064*^9}, {3.3641696952841*^9, 3.36416984973525*^9}, {3.36416990844802*^9, 3.36416999556182*^9}, { 3.36417007144209*^9, 3.36417017527734*^9}, 3.36417075329835*^9}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->"Previous Slide"], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->"Next Slide"] }], "PreviousNext", CellChangeTimes->{{3.36415288822036*^9, 3.36415289144661*^9}}] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}]& ), ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}]& ), ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageLast"]}]& ), ButtonNote->"Last Slide"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}, ButtonFunction:>Null]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Numbers of real cylinders...", "Section", CellChangeTimes->{{3.36413448300955*^9, 3.36413448509781*^9}, { 3.36413512543128*^9, 3.36413512860691*^9}, {3.36413580231783*^9, 3.36413580624819*^9}, {3.36413590720543*^9, 3.36413591019657*^9}, { 3.36414886187831*^9, 3.36414887751073*^9}, {3.36415147336487*^9, 3.36415147802102*^9}, {3.36415343256285*^9, 3.36415343931916*^9}, { 3.36416351532414*^9, 3.36416352304842*^9}, {3.36416396097275*^9, 3.36416396873205*^9}, {3.36416787899981*^9, 3.36416788029368*^9}, { 3.36416967723973*^9, 3.3641696910012*^9}, 3.36416985702353*^9, { 3.3641699002186*^9, 3.36416990418562*^9}, {3.3641707112057*^9, 3.36417073746307*^9}}], Cell[CellGroupData[{ Cell["Interesting to study", "Subsection", CellChangeTimes->{{3.36417021619877*^9, 3.36417022006138*^9}}], Cell[TextData[{ "How often do such random examples have one point enclosed by the other \ four? Related to expected volume of a tetrahedron with points picked randomly \ in unit cube (generalization of a problem of Sylvester). Only recently \ solved, using symbolic calculus among other things. It is ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"3977", "/", "216000"}], "-", RowBox[{ SuperscriptBox["\[Pi]", "2"], "/", "2160"}]}], TraditionalForm]]], "." }], "Text", CellDingbat->"\[FilledSmallCircle]", CellChangeTimes->{{3.36417022636844*^9, 3.36417037219723*^9}}], Cell["\<\ How do other cases of no real cylinders arise. This is discussed in the \ companion to this talk.\ \>", "Text", CellDingbat->"\[FilledSmallCircle]", CellChangeTimes->{{3.36417022636844*^9, 3.36417037219723*^9}, { 3.36417041105955*^9, 3.36417042915726*^9}}], Cell["\<\ More generally, can we (either algebraically or geometrically) classify the \ cases of 0, 2, 4, or 6 real solutions? Little seems to be known about this. Suggested by referee: look at discriminant varieties. Me: Good idea. What's a discriminat variety? Okay, that wasn't quite my response. But I will say the computations appear \ to be daunting. One problem is that it takes extra polynomials and variables \ just to enforce the condition that no pair of points coincides.\ \>", "Text", CellDingbat->"\[FilledSmallCircle]", CellChangeTimes->{{3.36417022636844*^9, 3.36417037219723*^9}, { 3.36417041105955*^9, 3.36417042915726*^9}, {3.36417046756441*^9, 3.36417065499803*^9}, {3.36417102478394*^9, 3.36417103032472*^9}, { 3.36432394987536*^9, 3.36432395057322*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->"Previous Slide"], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->"Next Slide"] }], "PreviousNext", CellChangeTimes->{{3.36415288822036*^9, 3.36415289144661*^9}}] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}]& ), ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}]& ), ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageLast"]}]& ), ButtonNote->"Last Slide"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}, ButtonFunction:>Null]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Proving the generic number of cylinders is six", "Section", CellChangeTimes->{{3.36413448300955*^9, 3.36413448509781*^9}, { 3.36413512543128*^9, 3.36413512860691*^9}, {3.36413580231783*^9, 3.36413580624819*^9}, {3.36413590720543*^9, 3.36413591019657*^9}, { 3.36414886187831*^9, 3.36414887751073*^9}, {3.36415147336487*^9, 3.36415147802102*^9}, {3.36415343256285*^9, 3.36415343931916*^9}, { 3.36416351532414*^9, 3.36416352304842*^9}, {3.36416396097275*^9, 3.36416396873205*^9}, {3.36416787899981*^9, 3.36416788029368*^9}, { 3.36416967723973*^9, 3.3641696910012*^9}, 3.36416985702353*^9, { 3.3641699002186*^9, 3.36416990418562*^9}, {3.3641707112057*^9, 3.36417073746307*^9}, {3.36417116679818*^9, 3.36417118809625*^9}}], Cell["\<\ To simplify the computations we reformulate so as to have two equations in \ two variables. Idea:\ \>", "Text", CellChangeTimes->{{3.36417123110423*^9, 3.36417134562939*^9}}], Cell[TextData[{ "Without loss of generality we have one point at the origin, another at ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"1", ",", "0", ",", "0"}], ")"}], TraditionalForm]]], ", and a third in the ", Cell[BoxData[ FormBox[ RowBox[{"x", " ", "y"}], TraditionalForm]]], " coordinate plane." }], "Text", CellChangeTimes->{{3.36417123110423*^9, 3.36417135370999*^9}}], Cell[TextData[{ "Project these onto the set of planes through the origin, parametrized \ generically by a normal vector ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b", ",", "1"}], ")"}], TraditionalForm]]], "." }], "Text", CellChangeTimes->{{3.36417123110423*^9, 3.36417135814658*^9}}], Cell["\<\ In each such plane these three points determine a circle, and we get one \ equation for each of the remaining two points in order that they project onto \ the same circle (which is the condition that the five be cocylindrical).\ \>", "Text", CellChangeTimes->{{3.36417123110423*^9, 3.36417136805966*^9}}], Cell[TextData[{ "So our points are ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"0", ",", "0", ",", "0"}], ")"}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"1", ",", "0", ",", "0"}], ")"}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "2"], ",", SubscriptBox["y", "2"], ",", "0"}], ")"}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "3"], ",", SubscriptBox["y", "3"], ",", SubscriptBox["z", "3"]}], ")"}], TraditionalForm]]], ", and ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "4"], ",", SubscriptBox["y", "4"], ",", SubscriptBox["z", "4"]}], ")"}], TraditionalForm]]], " and our variables are ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], TraditionalForm]]], " where direction vector is ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b", ",", "1"}], ")"}], TraditionalForm]]], ". 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{3.36413590720543*^9, 3.36413591019657*^9}, { 3.36414886187831*^9, 3.36414887751073*^9}, {3.36415147336487*^9, 3.36415147802102*^9}, {3.36415343256285*^9, 3.36415343931916*^9}, { 3.36416351532414*^9, 3.36416352304842*^9}, {3.36416396097275*^9, 3.36416396873205*^9}, {3.36416787899981*^9, 3.36416788029368*^9}, { 3.36416967723973*^9, 3.3641696910012*^9}, 3.36416985702353*^9, { 3.3641699002186*^9, 3.36416990418562*^9}, {3.3641707112057*^9, 3.36417073746307*^9}, {3.36417116679818*^9, 3.36417118809625*^9}, { 3.36417160522815*^9, 3.36417160794393*^9}}], Cell["Now we must count solutions to this system.", "Text", CellChangeTimes->{{3.36417164681457*^9, 3.36417165475381*^9}}], Cell[TextData[{ "PROOF 1. We form a Gr\[ODoubleDot]bner basis with respect to a degree based \ term ordering for the polynomials. Looking at the head terms we find that \ there are ", Cell[BoxData[ FormBox["6", TraditionalForm]]], " monomials in ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"a", ",", "b"}], ")"}], TraditionalForm]]], " that are not reducible with respect to this basis and hence ", Cell[BoxData[ FormBox["6", TraditionalForm]]], " solutions to the system." }], "Text", CellChangeTimes->{ 3.36067647060585*^9, {3.3641716684107*^9, 3.36417168009474*^9}}], Cell[TextData[{ "PROOF 2. We compute the resultant of the pair of polynomials with respect \ to one of the two variables. We obtain a polynomial of degree ", Cell[BoxData[ FormBox["6", TraditionalForm]]], " in the other (with large symbolic coefficients). This means there are at \ most ", Cell[BoxData[ FormBox["6", TraditionalForm]]], " solutions. As we already know there are at least that many, this suffices \ to show that there are generically six solutions." }], "Text", CellChangeTimes->{{3.36062165629936*^9, 3.36062167337597*^9}, { 3.3641716999706*^9, 3.36417170127962*^9}}], Cell[TextData[{ ButtonBox["\[FilledLeftTriangle]\[ThickSpace]\[ThickSpace]\[ThickSpace]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}], ButtonNote->"Previous Slide"], "\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]", ButtonBox["\[ThickSpace]\[ThickSpace]\[ThickSpace]\[FilledRightTriangle]", BaseStyle->"SlidePreviousNextLink", Appearance->{Automatic, None}, ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}], ButtonNote->"Next Slide"] }], "PreviousNext", CellChangeTimes->{{3.36415288822036*^9, 3.36415289144661*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPagePrevious"]}]& ), ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageNext"]}]& ), ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], BaseStyle->"SlideHyperlink", ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`ButtonNotebook[], "ScrollPageLast"]}]& ), ButtonNote->"Last Slide"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}, ButtonFunction:>Null]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["The generic number...", "Section", CellChangeTimes->{{3.36413448300955*^9, 3.36413448509781*^9}, { 3.36413512543128*^9, 3.36413512860691*^9}, {3.36413580231783*^9, 3.36413580624819*^9}, {3.36413590720543*^9, 3.36413591019657*^9}, { 3.36414886187831*^9, 3.36414887751073*^9}, {3.36415147336487*^9, 3.36415147802102*^9}, {3.36415343256285*^9, 3.36415343931916*^9}, { 3.36416351532414*^9, 3.36416352304842*^9}, {3.36416396097275*^9, 3.36416396873205*^9}, {3.36416787899981*^9, 3.36416788029368*^9}, { 3.36416967723973*^9, 3.3641696910012*^9}, 3.36416985702353*^9, { 3.3641699002186*^9, 3.36416990418562*^9}, {3.3641707112057*^9, 3.36417073746307*^9}, {3.36417116679818*^9, 3.36417118809625*^9}, { 3.36417160522815*^9, 3.36417160794393*^9}}], Cell["Remarks about these proofs.", "Text", CellChangeTimes->{{3.36417186240012*^9, 3.36417187516362*^9}}], Cell["\<\ Both rely on having a fairly simple formulation of the problem.\ \>", "Text", CellDingbat->"\[EmptyUpTriangle]", CellChangeTimes->{{3.36417186240012*^9, 3.36417190574627*^9}}], Cell["\<\ With more than two equations, a \"standard\" resultant would not likely \ suffice unless used in iterated fashion.\ \>", "Text", CellDingbat->"\[EmptyDownTriangle]", CellChangeTimes->{{3.36417186240012*^9, 3.36417195913982*^9}}], Cell["\<\ For even slightly more complicated problems, a Gr\[ODoubleDot]bner basis \ computation involving symbolic coefficients is likely to bog down.\ \>", "Text", CellDingbat->"\[EmptyUpTriangle]", CellChangeTimes->{{3.36417186240012*^9, 3.36417190574627*^9}, { 3.36417198414467*^9, 3.36417201224852*^9}}], Cell["\<\ A related Gr\[ODoubleDot]bner basis approach is to use the Bezout bound and \ subtract off the count of solutions at infinity. Works for this problem but \ often fails because of presence of dimensional component(s) at infinity.\ \>", "Text", CellDingbat->"\[EmptyDownTriangle]", CellChangeTimes->{{3.36417186240012*^9, 3.36417190574627*^9}, { 3.36417198414467*^9, 3.36417201224852*^9}, {3.36417204364082*^9, 3.36417209777221*^9}}], Cell["\<\ Another Gr\[ODoubleDot]bner basis approach that seems promising is to work in \ a mixed algebra with symbolic perturbation variables. Idea is to show we have \ six solutions not just at a given configuration, but at all configurations in \ a neighborhood thereof. 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FormBox[ RowBox[{"grid", AdjustmentBox[ StyleBox["Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], "webMathematica" -> FormBox[ RowBox[{"web", AdjustmentBox[ StyleBox["Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], Inherited}, CounterIncrements -> "Subsubsection", FontFamily -> "Times", FontSize -> 12, FontWeight -> "Bold"], Cell[ StyleData["Subsubsection", "Presentation"], CellMargins -> {{80, 30}, {8, 12}}, FontSize -> 18], Cell[ StyleData["Subsubsection", "Printout"], CellMargins -> {{18, 30}, {0, 12}}, FontSize -> 10]}, Closed]]}, Open]], Cell[ CellGroupData[{ Cell["Styles for Body Text", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["Text"], CellMargins -> {{55, 10}, {6, 6}}, InputAutoReplacements -> {"TeX" -> StyleBox[ RowBox[{"T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "LaTeX" -> StyleBox[ RowBox[{"L", StyleBox[ AdjustmentBox[ "A", BoxMargins -> {{-0.36, -0.1}, {0, 0}}, BoxBaselineShift -> -0.2], FontSize -> Smaller], "T", AdjustmentBox[ "E", BoxMargins -> {{-0.075, -0.085}, {0, 0}}, BoxBaselineShift -> 0.5], "X"}]], "mma" -> "Mathematica", "Mma" -> "Mathematica", "MMA" -> "Mathematica", "gridMathematica" -> FormBox[ RowBox[{"grid", AdjustmentBox[ StyleBox["Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], "webMathematica" -> FormBox[ RowBox[{"web", AdjustmentBox[ StyleBox["Mathematica", FontSlant -> "Italic"], BoxMargins -> {{-0.175, 0}, {0, 0}}]}], TextForm], Inherited}, TextJustification -> 1, LineSpacing -> {1, 2}, FontFamily -> "Times"], Cell[ StyleData["Text", "Presentation"], CellMargins -> {{80, 10}, {10, 8}}, FontSize -> 18], Cell[ StyleData["Text", "SlideShow"], CellMargins -> {{80, 10}, {10, 8}}, FontSize -> 18], Cell[ StyleData["Text", "Printout"], CellMargins -> {{18, 4}, {4, 4}}, Hyphenation -> True, LineSpacing -> {1, 3}, FontSize -> 18]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Commentary"], CellMargins -> {{55, 10}, {2, 6}}, TextJustification -> 1, LineSpacing -> {1, 2}, FontFamily -> "Helvetica", FontSize -> 10, FontColor -> RGBColor[0, 0, 0.4]], Cell[ StyleData["Commentary", "Presentation"], CellMargins -> {{80, 10}, {2, 10}}, FontSize -> 15], Cell[ StyleData["Commentary", "Printout"], CellMargins -> {{18, 30}, {3, 0}}, Hyphenation -> True, LineSpacing -> {1, 3}, FontFamily -> "Times", FontSize -> 10]}, Closed]]}, Open]], Cell[ CellGroupData[{ Cell["Styles for Input/Output", "Section"], Cell[ "The cells in this section define styles used for input and output \ to the kernel. Be careful when modifying, renaming, or removing these \ styles, because the front end associates special meanings with these style \ names.", "Text"], Cell[ CellGroupData[{ Cell[ StyleData["Input"], CellFrame -> {{1, 1}, {0, 1}}, CellMargins -> {{55, 10}, {0, 0}}, Evaluatable -> True, CellGroupingRules -> "InputGrouping", CellHorizontalScrolling -> True, PageBreakWithin -> False, GroupPageBreakWithin -> False, CellLabelPositioning -> Automatic, CellLabelMargins -> {{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType -> DefaultInputFormatType, ShowAutoStyles -> True, "TwoByteSyntaxCharacterAutoReplacement" -> True, HyphenationOptions -> { "HyphenationCharacter" -> "\[Continuation]"}, LanguageCategory -> "Formula", FormatType -> InputForm, ShowStringCharacters -> True, NumberMarks -> True, LinebreakAdjustments -> {0.85, 2, 10, 0, 1}, FontSize -> 12, FontWeight -> "Bold", Background -> GrayLevel[0.966674]], Cell[ StyleData["Input", "Presentation"], CellFrame -> {{2, 2}, {0, 2}}, CellMargins -> {{80, 18}, {0, 2}}, CellFrameMargins -> {{10, 10}, {12, 12}}, FontSize -> 18], Cell[ StyleData["Input", "SlideShow"], CellFrame -> {{2, 2}, {0, 2}}, CellMargins -> {{80, 18}, {0, 2}}, CellFrameMargins -> {{10, 10}, {12, 12}}, FontSize -> 18], Cell[ StyleData["Input", "Printout"], CellMargins -> {{55, 10}, {0, 10}}, LinebreakAdjustments -> {0.85, 2, 10, 1, 1}, FontSize -> 18, Background -> GrayLevel[0.850004]]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["InlineInput"], Evaluatable -> True, CellGroupingRules -> "InputGrouping", CellHorizontalScrolling -> True, PageBreakWithin -> False, GroupPageBreakWithin -> False, DefaultFormatType -> DefaultInputFormatType, ShowAutoStyles -> True, "TwoByteSyntaxCharacterAutoReplacement" -> True, HyphenationOptions -> { "HyphenationCharacter" -> "\[Continuation]"}, AutoItalicWords -> {}, FormatType -> InputForm, ShowStringCharacters -> True, NumberMarks -> True, CounterIncrements -> "Input", FontWeight -> "Bold"], Cell[ StyleData["InlineInput", "Presentation"], CellMargins -> {{10, 18}, {6, 4}}, FontSize -> 18], Cell[ StyleData["InlineInput", "SlideShow"], CellMargins -> {{10, 18}, {6, 4}}, FontSize -> 18], Cell[ StyleData["InlineInput", "Printout"], CellMargins -> {{10, 18}, {6, 4}}, FontSize -> 18]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Output"], CellFrame -> {{1, 1}, {1, 0}}, CellMargins -> {{55, 10}, {15, 0}}, CellEditDuplicate -> True, CellGroupingRules -> "OutputGrouping", CellHorizontalScrolling -> True, PageBreakWithin -> False, GroupPageBreakWithin -> False, GeneratedCell -> True, CellAutoOverwrite -> True, CellLabelMargins -> {{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType -> DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement" -> True, HyphenationOptions -> { "HyphenationCharacter" -> "\[Continuation]"}, LanguageCategory -> "Formula", FormatType -> InputForm, FontSize -> 12, Background -> GrayLevel[0.850004]], Cell[ StyleData["Output", "Presentation"], CellFrame -> {{2, 2}, {2, 0}}, CellMargins -> {{80, 18}, {16, 0}}, CellFrameMargins -> {{12, 10}, {12, 12}}, FontSize -> 18], Cell[ StyleData["Output", "SlideShow"], CellFrame -> {{2, 2}, {2, 0}}, CellMargins -> {{80, 18}, {16, 0}}, CellFrameMargins -> {{12, 10}, {12, 12}}, FontSize -> 18], Cell[ StyleData["Output", "Printout"], CellMargins -> {{55, Inherited}, {10, 0}}, FontSize -> 18]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["InputOnly"], CellFrame -> 1, CellMargins -> {{55, 10}, {15, 0}}, Evaluatable -> True, CellGroupingRules -> "InputGrouping", CellHorizontalScrolling -> True, PageBreakWithin -> False, GroupPageBreakWithin -> False, CellLabelPositioning -> Automatic, CellLabelMargins -> {{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType -> DefaultInputFormatType, ShowAutoStyles -> True, "TwoByteSyntaxCharacterAutoReplacement" -> True, HyphenationOptions -> { "HyphenationCharacter" -> "\[Continuation]"}, LanguageCategory -> "Formula", FormatType -> InputForm, ShowStringCharacters -> True, NumberMarks -> True, LinebreakAdjustments -> {0.85, 2, 10, 0, 1}, FontSize -> 12, FontWeight -> "Bold", Background -> GrayLevel[0.966674]], Cell[ StyleData["InputOnly", "Presentation"], CellFrame -> 2, CellMargins -> {{80, 18}, {24, 4}}, CellFrameMargins -> {{10, 10}, {12, 12}}, FontSize -> 18], Cell[ StyleData["InputOnly", "Printout"], CellMargins -> {{55, Inherited}, {10, 10}}, LinebreakAdjustments -> {0.85, 2, 10, 1, 1}, FontSize -> 10]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Message"], CellFrame -> {{1, 1}, {0, 0}}, CellDingbat -> "\[LongDash]", CellMargins -> {{55, 10}, {0, 0}}, CellGroupingRules -> "OutputGrouping", PageBreakWithin -> False, GroupPageBreakWithin -> False, GeneratedCell -> True, CellAutoOverwrite -> True, ShowCellLabel -> False, CellLabelMargins -> {{23, Inherited}, {Inherited, Inherited}}, DefaultFormatType -> DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement" -> True, AutoStyleOptions -> {"SyntaxErrorStyle" -> None}, HyphenationOptions -> { "HyphenationCharacter" -> "\[Continuation]"}, FormatType -> InputForm, StyleMenuListing -> None, FontFamily -> "Helvetica", FontSize -> 10, FontSlant -> "Oblique", FontColor -> GrayLevel[0], Background -> GrayLevel[1]], Cell[ StyleData["Message", "Presentation"], CellFrame -> {{2, 2}, {0, 0}}, CellMargins -> {{80, 18}, {0, 0}}, CellFrameMargins -> 12, FontSize -> 15], Cell[ StyleData["Message", "Printout"], CellMargins -> {{55, Inherited}, {0, 0}}, FontSize -> 8]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Print"], CellMargins -> {{55, 26}, {1, 6}}, CellGroupingRules -> "OutputGrouping", CellHorizontalScrolling -> True, PageBreakWithin -> False, GroupPageBreakWithin -> False, GeneratedCell -> True, CellAutoOverwrite -> True, ShowCellLabel -> False, CellLabelMargins -> {{23, Inherited}, { Inherited, Inherited}}, DefaultFormatType -> DefaultOutputFormatType, "TwoByteSyntaxCharacterAutoReplacement" -> True, TextAlignment -> Left, HyphenationOptions -> { "HyphenationCharacter" -> "\[Continuation]"}, FormatType -> InputForm, StyleMenuListing -> None, Background -> GrayLevel[0.700008]], Cell[ StyleData["Print", "Presentation"], CellMargins -> {{80, 42}, {1, 10}}, CellFrameMargins -> 12, FontSize -> 18], Cell[ StyleData["Print", "Printout"], CellMargins -> {{54, Inherited}, {2, 6}}, FontSize -> 10]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Graphics"], CellFrame -> {{1, 1}, {0, 0}}, CellMargins -> {{55, 10}, {0, 0}}, CellGroupingRules -> "GraphicsGrouping", CellHorizontalScrolling -> True, PageBreakWithin -> False, GeneratedCell -> True, CellAutoOverwrite -> True, ShowCellLabel -> False, DefaultFormatType -> DefaultOutputFormatType, FormatType -> InputForm, ImageMargins -> {{35, Inherited}, {Inherited, 0}}, StyleMenuListing -> None, Background -> GrayLevel[0.850004]], Cell[ StyleData["Graphics", "Presentation"], CellFrame -> {{2, 2}, {0, 0}}, CellMargins -> {{80, 18}, {0, 0}}, CellFrameMargins -> 12, FontSize -> 18], Cell[ StyleData["Graphics", "SlideShow"], CellFrame -> {{2, 2}, {0, 0}}, CellMargins -> {{80, 18}, {0, 0}}, CellFrameMargins -> 12, FontSize -> 18], Cell[ StyleData["Graphics", "Printout"], CellMargins -> {{55, Inherited}, {0, 0}}, ImageSize -> {0.0625, 0.0625}]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["CellLabel"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontSize -> 10, FontSlant -> "Oblique", FontColor -> RGBColor[0.6, 0, 0.6]], Cell[ StyleData["CellLabel", "Presentation"], FontSize -> 15], Cell[ StyleData["CellLabel", "Printout"], CellMargins -> {{0, Inherited}, {Inherited, Inherited}}, FontSize -> 8]}, Closed]]}, Open]], Cell[ CellGroupData[{ Cell["Inline Formatting", "Section"], Cell[ "These styles are for modifying individual words or letters in a \ cell exclusive of the cell tag.", "Text"], Cell[ StyleData["RM"], StyleMenuListing -> None, FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["BF"], StyleMenuListing -> None, FontWeight -> "Bold"], Cell[ StyleData["IT"], StyleMenuListing -> None, FontSlant -> "Italic"], Cell[ StyleData["TR"], StyleMenuListing -> None, FontFamily -> "Times", FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["TI"], StyleMenuListing -> None, FontFamily -> "Times", FontWeight -> "Plain", FontSlant -> "Italic"], Cell[ StyleData["TB"], StyleMenuListing -> None, FontFamily -> "Times", FontWeight -> "Bold", FontSlant -> "Plain"], Cell[ StyleData["TBI"], StyleMenuListing -> None, FontFamily -> "Times", FontWeight -> "Bold", FontSlant -> "Italic"], Cell[ StyleData["MR"], "TwoByteSyntaxCharacterAutoReplacement" -> True, HyphenationOptions -> {"HyphenationCharacter" -> "\[Continuation]"}, StyleMenuListing -> None, FontFamily -> "Courier", FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["MO"], "TwoByteSyntaxCharacterAutoReplacement" -> True, HyphenationOptions -> {"HyphenationCharacter" -> "\[Continuation]"}, StyleMenuListing -> None, FontFamily -> "Courier", FontWeight -> "Plain", FontSlant -> "Italic"], Cell[ StyleData["MB"], "TwoByteSyntaxCharacterAutoReplacement" -> True, HyphenationOptions -> {"HyphenationCharacter" -> "\[Continuation]"}, StyleMenuListing -> None, FontFamily -> "Courier", FontWeight -> "Bold", FontSlant -> "Plain"], Cell[ StyleData["MBO"], "TwoByteSyntaxCharacterAutoReplacement" -> True, HyphenationOptions -> {"HyphenationCharacter" -> "\[Continuation]"}, StyleMenuListing -> None, FontFamily -> "Courier", FontWeight -> "Bold", FontSlant -> "Italic"], Cell[ StyleData["SR"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["SO"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontWeight -> "Plain", FontSlant -> "Italic"], Cell[ StyleData["SB"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontWeight -> "Bold", FontSlant -> "Plain"], Cell[ StyleData["SBO"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontWeight -> "Bold", FontSlant -> "Italic"], Cell[ CellGroupData[{ Cell[ StyleData["SO10"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontSize -> 10, FontWeight -> "Plain", FontSlant -> "Italic"], Cell[ StyleData["SO10", "Printout"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontSize -> 7, FontWeight -> "Plain", FontSlant -> "Italic"]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Inert"], StyleMenuListing -> None, Background -> RGBColor[ 0.870588235294118, 0.905882352941176, 0.972549019607843]], Cell[ StyleData["Inert", "Printout"], StyleMenuListing -> None, Background -> GrayLevel[1]]}, Closed]], Cell[ StyleData["Small"], StyleMenuListing -> None, FontSize -> 9], Cell[ StyleData["Medium"], StyleMenuListing -> None, FontSize -> 12], Cell[ StyleData["Large"], StyleMenuListing -> None, FontSize -> 24]}, Open]], Cell[ CellGroupData[{ Cell["Unique Styles", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["Author"], CellMargins -> {{20, 30}, {45, 5}}, CellGroupingRules -> {"TitleGrouping", 20}, PageBreakBelow -> False, CellFrameMargins -> {{0, 4}, {8, 4}}, LineSpacing -> {1, 0}, CounterAssignments -> {{"Section", 0}, {"Equation", 0}, { "Figure", 0}}, FontFamily -> "Helvetica", FontSize -> 14, FontSlant -> "Italic"], Cell[ StyleData["Author", "Presentation"], CellMargins -> {{28, 40}, {68, 8}}, FontSize -> 21], Cell[ StyleData["Author", "SlideShow"], CellMargins -> {{28, 40}, {68, 8}}, FontSize -> 21], Cell[ StyleData["Author", "Printout"], CellMargins -> {{18, 30}, {45, 5}}, FontSize -> 21]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Abstract"], CellFrame -> False, CellMargins -> {{18, 140}, {4, 30}}, LineSpacing -> {0.9, 0}, FontFamily -> "Times", FontSize -> 12], Cell[ StyleData["Abstract", "Presentation"], CellMargins -> {{28, 180}, {6, 42}}, FontSize -> 18], Cell[ StyleData["Abstract", "SlideShow"], CellMargins -> {{28, 180}, {6, 42}}, FontSize -> 18], Cell[ StyleData["Abstract", "Printout"], Hyphenation -> True, LineSpacing -> {1, 2}, FontSize -> 20]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Caption"], CellMargins -> {{55, 10}, {5, 3}}, PageBreakAbove -> False, FontFamily -> "Helvetica", FontSize -> 9], Cell[ StyleData["Caption", "Presentation"], CellMargins -> {{80, 18}, {6, 6}}, FontSize -> 14], Cell[ StyleData["Caption", "Printout"], CellMargins -> {{55, 55}, {5, 4}}, Hyphenation -> True, LineSpacing -> {1, 2}, FontSize -> 8]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Reference"], CellMargins -> {{24, 40}, {6, 6}}, TextJustification -> 1, LineSpacing -> {1, 0}, FontFamily -> "Times"], Cell[ StyleData["Reference", "Presentation"], CellMargins -> {{38, 40}, {10, 10}}, FontSize -> 18], Cell[ StyleData["Reference", "Printout"], CellMargins -> {{18, 4}, {4, 4}}, Hyphenation -> True, FontSize -> 9]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["PictureGroup"], CellFrame -> {{1, 1}, {0, 0}}, CellMargins -> {{55, Inherited}, {0, 0}}, CellGroupingRules -> "GraphicsGrouping", CellHorizontalScrolling -> True, PageBreakWithin -> False, ShowCellLabel -> False, ImageMargins -> {{35, Inherited}, {Inherited, 0}}, StyleMenuListing -> None, Background -> GrayLevel[0.850004]], Cell[ StyleData["PictureGroup", "Presentation"], CellFrame -> {{2, 2}, {0, 0}}, CellMargins -> {{80, 6}, {0, 0}}, CellFrameMargins -> 12, FontSize -> 18], Cell[ StyleData["PictureGroup", "Printout"], CellMargins -> {{55, Inherited}, {0, 0}}, ImageSize -> {0.0625, 0.0625}]}, Closed]]}, Open]], Cell[ CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell[ "The cells below define styles useful for making hypertext \ ButtonBoxes. The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.", "Text"], Cell[ CellGroupData[{ Cell[ StyleData["Hyperlink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookLocate[#2]}]& ), ButtonNote -> ButtonData}], Cell[ StyleData["Hyperlink", "Printout"], FontSize -> 18, FontColor -> GrayLevel[0]]}, Open]], Cell[ "The following styles are for linking automatically to the on-line \ help system.", "Text"], Cell[ CellGroupData[{ Cell[ StyleData["MainBookLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["MainBook", #]}]& )}], Cell[ StyleData["MainBookLink", "Printout"], FontSize -> 10, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["AddOnsLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontFamily -> "Courier", FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["AddOns", #]}]& )}], Cell[ StyleData["AddOnsLink", "Printout"], FontSize -> 10, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["RefGuideLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontFamily -> "Courier", FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["RefGuide", #]}]& )}], Cell[ StyleData["RefGuideLink", "Printout"], FontSize -> 10, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["GettingStartedLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["GettingStarted", #]}]& )}], Cell[ StyleData["GettingStartedLink", "Printout"], FontSize -> 10, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["DemosLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["Demos", #]}]& )}], Cell[ StyleData["DemosLink", "Printout"], FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["TourLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["Tour", #]}]& )}], Cell[ StyleData["TourLink", "Printout"], FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["OtherInformationLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["OtherInformation", #]}]& )}], Cell[ StyleData["OtherInformationLink", "Printout"], FontSize -> 10, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["MasterIndexLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["MasterIndex", #]}]& )}], Cell[ StyleData["MasterIndexLink", "Printout"], FontColor -> GrayLevel[0]]}, Closed]]}, Open]], Cell[ CellGroupData[{ Cell["Palette Styles", "Section"], Cell[ "The cells below define styles that define standard ButtonFunctions, \ for use in palette buttons.", "Text"], Cell[ StyleData["Paste"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, After]}]& )}], Cell[ StyleData["Evaluate"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[], All]}]& )}], Cell[ StyleData["EvaluateCell"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[], All]}]& )}], Cell[ StyleData["CopyEvaluate"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[], All]}]& )}], Cell[ StyleData["CopyEvaluateCell"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[], All]}]& )}]}, Closed]], Cell[ CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["DisplayFormula"], CellMargins -> {{55, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling -> True, DefaultFormatType -> DefaultInputFormatType, "TwoByteSyntaxCharacterAutoReplacement" -> True, HyphenationOptions -> { "HyphenationCharacter" -> "\[Continuation]"}, LanguageCategory -> "Formula", ScriptLevel -> 0, SingleLetterItalics -> True, StyleMenuListing -> None, UnderoverscriptBoxOptions -> {LimitsPositioning -> True}], Cell[ StyleData["DisplayFormula", "Presentation"], CellMargins -> {{80, 18}, {6, 6}}, FontSize -> 18], Cell[ StyleData["DisplayFormula", "Printout"], CellMargins -> {{18, 4}, {4, 4}}]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["ChemicalFormula"], CellMargins -> {{55, 10}, {Inherited, 0}}, CellHorizontalScrolling -> True, DefaultFormatType -> DefaultInputFormatType, "TwoByteSyntaxCharacterAutoReplacement" -> True, TextJustification -> 1, HyphenationOptions -> { "HyphenationCharacter" -> "\[Continuation]"}, LanguageCategory -> "Formula", AutoSpacing -> False, ScriptBaselineShifts -> {0.6, Automatic}, ScriptLevel -> 1, SingleLetterItalics -> False, ZeroWidthTimes -> True], Cell[ StyleData["ChemicalFormula", "Presentation"], CellMargins -> {{80, 18}, {6, 0}}, FontSize -> 18], Cell[ StyleData["ChemicalFormula", "Printout"], CellMargins -> {{18, 4}, {4, 4}}, LineSpacing -> {1, 3}, FontSize -> 10]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Program"], CellMargins -> {{55, 10}, {Inherited, 0}}, CellHorizontalScrolling -> True, Hyphenation -> False, LanguageCategory -> "Formula", FontFamily -> "Courier"], Cell[ StyleData["Program", "Presentation"], CellMargins -> {{80, 18}, {6, 0}}, FontSize -> 18], Cell[ StyleData["Program", "Printout"], CellMargins -> {{18, 4}, {4, 4}}, LineSpacing -> {1, 3}, FontSize -> 9.5]}, Closed]]}, Open]], Cell[ CellGroupData[{ Cell["Styles for Automatic Numbering", "Section"], Cell[ "The following styles are useful for numbered equations, figures, \ etc. They automatically give the cell a FrameLabel containing a reference to \ a particular counter, and also increment that counter.", "Text"], Cell[ CellGroupData[{ Cell[ StyleData["NumberedEquation"], CellMargins -> {{55, 85}, {Inherited, Inherited}}, CellFrameLabels -> {{None, Cell[ 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