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The supported coordinates are polar, equidistant and horocyclic. Their \ coordinate lines are circles and lines; equidstant and lines; horocycles and \ lines, respectively. It is well known that there doesn't exist a coordinate \ system with only lines being a coordinate lines. Hence, the supported \ coordinate systems are among the simplest in the hyperbolic plane.\ \>", "Address", TextAlignment->Center, TextJustification->0], Cell["The commands in brief", "Section", TextAlignment->Center, TextJustification->0], Cell["\<\ First we breifly list the commands for the three kinds of coordinates, and \ command for drawing the coordinate net.\ \>", "Text", CellMargins->{{18, Inherited}, {Inherited, Inherited}}], Cell[BoxData[ FormBox[GridBox[{ { StyleBox[\(CirclePt[{r, \ phi}, \ options]\), FontFamily->"Courier New"], \(returns\ the\ LPoint\ \n object\ with\ hyperbolic\ polar\ coordinates\ \((r, \ phi)\) . \ The\ origin\ and\n phi = 0\ line\ direction\ are\ given\ by\ the\ Origin\ and\ \ XAxesPt\ options .. \)}, { StyleBox[\(EquiPt[{d, \ h}, \ options]\), "MR", FontFamily->"Courier New", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontColor->GrayLevel[0], FontVariations->{"Underline"->False, "StrikeThrough"->False}], \(EquiPt[{d, \ h}, \ options]\ returns\ the\ LPoint\ object\ \n with\ hyperbolic\ equidistant\ coordinates\ \((d, \ h)\) . \ The\ origin\ and\ \n d - axes\ direction\ are\ given\ by\ the\ Origin\ and\ XAxesPt\ \ \(\(options\)\(.\)\)\)}, { StyleBox[\(HoroPt[{d, \ h}, \ options]\), "MR", FontFamily->"Courier New", FontSize->12, FontWeight->"Plain", FontSlant->"Plain", FontColor->GrayLevel[0], FontVariations->{"Underline"->False, "StrikeThrough"->False}], \(returns\ the\ LPoint\ object\n with\ hyperbolic\ horocyclic\ coordinates\ \((d, h)\) . The\ origin\ and\n the\ infinite\ point\ of\ h - axes\ is\ given\ by\ the\ option\ \(\(InfPoint\)\(.\)\)\)}, { StyleBox[\(DrawNet[ t, {x1, x2, xstep, xdense}, {y1, y2, ystep, ydense}, opts]\), FontFamily-> "Courier New"], \(returns\ a\ coordinate\ net\ on\ the\ \n interval\ \((x1, \ x2)\) x \((y1, y2)\)\ in\ the\ coordinates\ which\ are\ specified\ by\n the\ parameter\ t\ being\ one\ of\ functions\ CirclePt, \ EquiPt, \ HoroPt . \ \nThe\ parameters\ x \((y)\) step\ specify\ the\ density\ of\ the\ net\ while\n x \((y)\) dense\ specify\ the\ smoothness\ of\ the\ curves\ \ \(\(drawn\)\(.\)\)\)} }], NotebookDefault]], "DefinitionBox", CellMargins->{{19.5625, Inherited}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->0, GridBoxOptions->{ColumnAlignments->{Right, Left}}, CellTags->{"Notation:S1", "Notation:S1.1", "Notation:Definition:Notation"}], Cell["The supported commands of the package.", "Text", CellTags->{"Notation:S1", "Notation:S1.1"}], Cell["Installing the package", "Section", TextAlignment->Center, TextJustification->0], Cell[TextData[{ "To use a package L2Coordinates.m you will need also a package \ L2Primitives.m. We highly recommend reading the userguide.nb for \ familiarizing with hyperbolic geometry and its basic objects supported by the \ package L2Primitives. Copy them both to any directory that can be found in \ the ", StyleBox["Mathematica", FontSlant->"Italic"], " path. A good choice is C:\\...\\Wolfram \ Research\\Mathematica\\4.0\\AddOns\\ExtraPackages, in the directory where ", StyleBox["Mathematica", FontSlant->"Italic"], " files can be found. " }], "Text"], Cell["This loads the package. ", "Text"], Cell[BoxData[ \(<< L2Coordinates.m\)], "Input", CellLabel->"In[1]:="], Cell["\<\ The important options for setting up the coordinate system are origin \ (Origin) and the direction of x-axes (XAxesPt). In the case of horocyclic \ cooordinates we rather choose the direction of y-axes (InfPoint). The \ InfPoint has to lie on the unit circle. The Origin and XAxesPt are given in \ KleinDisk model and has to lie inside of the unit circle (see userguide.nb \ for details). Here are the default values for those parameters.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Options[L2Plots]\)], "Input", CellLabel->"In[2]:="], Cell[BoxData[ \({Origin \[Rule] LPoint[{0, 0}], XAxesPt \[Rule] LPoint[{2\/3, 0}], InfPoint \[Rule] LPoint[{\(-\(1\/\@2\)\), \(-\(1\/\@2\)\)}]}\)], "Output", CellLabel->"Out[2]="] }, Open ]], Cell["\<\ We can choose them if we decide to change the default coordinate system. For \ instance:\ \>", "Text"], Cell[BoxData[{ RowBox[{" ", RowBox[{ RowBox[{"SetOptions", "[", RowBox[{"L2Plots", StyleBox[",", "MR"], \(Origin \[Rule] LPoint[{1/2, 1/6}]\)}], "]"}], ";"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"SetOptions", "[", RowBox[{"L2Plots", StyleBox[",", "MR"], \(XAxesPt -> LPoint[{2/3, 0}]\)}], StyleBox["]", "TI"]}], StyleBox[";", "TI"]}]}], "Input", CellLabel->"In[7]:="], Cell["Now we see that Origin and XAxesPt have been changed.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Options[L2Plots]\)], "Input", CellLabel->"In[9]:="], Cell[BoxData[ \({Origin \[Rule] LPoint[{1\/2, 1\/6}], XAxesPt \[Rule] LPoint[{2\/3, 0}], InfPoint \[Rule] LPoint[{\(-\(1\/\@2\)\), \(-\(1\/\@2\)\)}]}\)], "Output", CellLabel->"Out[9]="] }, Open ]], Cell["Equidistant coordinates", "Section", TextAlignment->Center, TextJustification->0], Cell["\<\ Before explaining the coordinates lets draw the (-2, 2)x(-1,1) coordinate net \ and a particular point with coordinates (1, 1/2) in PoincareDisk mode.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(net\ = \ DrawNet[EquiPt, \ \ {\(-2\), 2, \ 20, \ 0.1}, {\(-1\), 1, \ 40, \ 0.1}, \ \ Model \[Rule] PoincareDisk];\)\), "\[IndentingNewLine]", \(\(point\ = \ \ LToGraphics[ EquiPt[{1, \ 1/2}], \ \ Model \[Rule] PoincareDisk]\ ;\)\), "\[IndentingNewLine]", \(peq\ = \ Show[Graphics[\ {net, \ {PointSize[0.02], \ point}}]\ , Graphics[{Thickness[0.01], Circle[{0, 0}, 1]}], \ AspectRatio \[Rule] Automatic]\)}], "Input", CellLabel->"In[61]:="], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output", CellLabel->"Out[63]="] }, Open ]], Cell["\<\ The green lines are coordinate axes. Their intersection is the Origin \ specified as an Option of L2Plots. The point with coordinates (d,h) is \ obtained by measuring d (in this case 1) along the d-axes (starting form the \ Origin and going in the direction of XAxesPt) and then going orthogonaly \ h-units(1/2 in this case). Note that these coordinates does not commute: \ going first h units vertically and then d units horizontaly would result in \ a different point. The coordinate lines are equidistants (h = const) colored \ red and lines (d= const) colored blue. Only the equidistant h=0 is a line. The metric of the hyperbolic plane in this coordinates takes form (don't \ confuse with d denoting both the coordinate and the differential):\ \>", "Text"], Cell[BoxData[ \(ds\^2 = \ dd\^2 + \ \(\(ch\^2\) \(d\)\(\ \)\(dh\^2\)\(\ \)\)\)], \ "NumberedEquation"], Cell["\<\ We can draw the same coordinate net in the HalfPlane and KleinDisk model of \ the hyperbolic plane (the are all isometric to each other).\ \>", "Text"], Cell[BoxData[{ \(\(net\ = \ DrawNet[EquiPt, \ \ {\(-2\), 2, \ 20, \ 0.1}, {\(-1\), 1, \ 40, \ 0.1}, \ \ Model \[Rule] KleinDisk];\)\), "\[IndentingNewLine]", \(\(point\ = \ \ LToGraphics[ EquiPt[{1, \ 1/2}], \ \ Model \[Rule] KleinDisk]\ ;\)\), "\[IndentingNewLine]", \(keq\ = \ Show[Graphics[\ {net, \ {PointSize[0.02], \ point}}]\ , Graphics[{Thickness[0.01], Circle[{0, 0}, 1]}], \ AspectRatio \[Rule] Automatic]\)}], "Input", CellLabel->"In[13]:="], Cell[BoxData[{ \(\(net\ = \ DrawNet[EquiPt, \ \ {\(-2\), 2, \ 20, \ 0.1}, {\(-1\), 1, \ 40, \ 0.1}, \ \ Model \[Rule] HalfPlane];\)\), "\[IndentingNewLine]", \(\(point\ = \ \ LToGraphics[ EquiPt[{1, \ 1/2}], \ \ Model \[Rule] HalfPlane]\ ;\)\), "\[IndentingNewLine]", \(heq\ = \ Show[Graphics[\ {net, \ {PointSize[0.02], \ point}}]\ , Graphics[{Thickness[0.01], Line[{{\(-1\), 0}, \ {2, 0}}]}], \ AspectRatio \[Rule] Automatic]\)}], "Input", CellLabel->"In[16]:="], Cell[BoxData[ \(GraphicsArray[{peq, \ keq, \ heq}] // Show\)], "Input", CellLabel->"In[19]:="], Cell["Horocyclic coordinates", "Section", TextAlignment->Center, TextJustification->0], Cell["\<\ Lets draw the (-2, 2)x(-1,1) coordinate net and a particular point in \ PoincareDisk mode.\ \>", "Text"], Cell[BoxData[{ \(\(net\ = \ DrawNet[HoroPt, \ \ {\(-2\), 2, \ 20, \ 0.05}, {\(-1\), 1, \ 40, \ 0.05}, \ \ Model \[Rule] PoincareDisk];\)\), "\[IndentingNewLine]", \(\(point = LToGraphics[HoroPt[{1, 1/2}], Model \[Rule] PoincareDisk];\)\), "\[IndentingNewLine]", \(hho\ = \ Show[Graphics[\ {net, \ {PointSize[0.02], \ point}}]\ , Graphics[{Thickness[0.01], Circle[{0, 0}, 1]}], \ AspectRatio \[Rule] Automatic]\)}], "Input", CellLabel->"In[20]:="], Cell["\<\ The green lines are coordinate axes. Their intersection is the Origin \ specified as an Option of L2Plots. The infinite point of both coordinate \ lines is specified by the InfPoint option of L2Plots. The point with \ coordinates (d,h) is obtained starting from the Origin and measuring d along \ the d-axes (horocycle) and then going orthogonaly h-units along a line. These \ coordinates also does not commute. The coordinate lines are mutually \ congruent horocycles (h = const) colored red and lines (d= const) colored \ blue. The metric of the hyperbolic plane in this coordinates takes form:\ \>", "Text"], Cell[BoxData[ \(ds\^2 = \ dd\^2 + \ \(\(e\^\(2 d\)\) \(dh\^2\)\(\ \)\)\)], "NumberedEquation"], Cell["\<\ Similar pictures can be drawn in other models of the hyperbolic plane as with \ equidistant coordinates.\ \>", "Text"], Cell["Polar coordinates ", "Section", TextAlignment->Center, TextJustification->0], Cell["\<\ These coordinates are counterpart of the polar coordinates in the Euclidean \ plane. They are defined on the region (0, \[Infinity])x(0, 2\[Pi]). Lets draw \ \ \>", "Text"], Cell[BoxData[{ \(\(net\ = \ DrawNet[CirclePt, \ \ {0, 2.5, \ 20, \ 0.03}, {0, 3\ Pi/2, \ 10, \ Pi/80}, \ \ Model \[Rule] PoincareDisk];\)\), "\[IndentingNewLine]", \(pci = \ Show[Graphics[\ net]\ , Graphics[{Thickness[0.01], Circle[{0, 0}, 1]}], \ \ AspectRatio \[Rule] Automatic]\)}], "Input", CellLabel->"In[23]:="], Cell["\<\ The green line is the x-axis (phi =0). The coordinate lines are circles (r= \ const) and lines (phi = const).\ \>", "Text"], Cell["Some applications ", "Section", TextAlignment->Center, TextJustification->0], Cell["Example 1", "Subsection"], Cell["\<\ In this example we combine functions from both L2Primitives and L2Coordinates \ package. As first, we create two points.\ \>", "Text"], Cell[BoxData[{ \(\(A\ = \ LPoint[{1/2, \ 1/2}];\)\), "\[IndentingNewLine]", \(\(B\ = \ LPoint[{\(-1\)/3, \ 1/4}];\)\)}], "Input", CellLabel->"In[25]:="], Cell[BoxData[ \(\(height\ = \ 3;\)\)], "Input", CellLabel->"In[27]:="], Cell["\<\ We would like to construc a Sacceri quadrilateral with the base AB and height \ equals 3. Using the coordinates we easily find the remaining vertices of the \ quadrilateral.\ \>", "Text"], Cell[BoxData[{ \(\(DD\ = \ EquiPt[{0, \ height}, \ Origin \[Rule] A, \ XAxesPt \[Rule] B];\)\), "\[IndentingNewLine]", \(\(CC\ = \ EquiPt[{LDistance[A, B], \ height}, \ Origin \[Rule] A, \ XAxesPt \[Rule] B];\)\)}], "Input", CellLabel->"In[28]:="], Cell[BoxData[ \(\(sakeri = LLine[{A, \ B, \ CC, \ DD, A}];\)\)], "Input", CellLabel->"In[30]:="], Cell[BoxData[ \(Having\ the\ quadrilateral, \ we\ would\ like\ to\ "\"\ if\ it\ is\ symmetric\ with\ respect\ \ to\ the\ lines\ joining\ midpoints\ of\ the\ \ opposite\ \ \(\(sides\)\(.\)\)\)], "Input"], Cell[BoxData[{ \(\(EE\ = \ Mid[A, \ B];\)\), "\[IndentingNewLine]", \(\(FF\ \ = \ Mid[CC, \ DD];\)\), "\[IndentingNewLine]", \(\(GG\ = \ Mid[A, \ DD];\)\), "\[IndentingNewLine]", \(\(HH\ = \ Mid[B, \ CC];\)\), "\[IndentingNewLine]", \(\(sakeri1\ = \ \(L2Reflection[LLine[{EE, FF}]]\)[ sakeri];\)\), "\[IndentingNewLine]", \(\(sakeri2\ = \ \(L2Reflection[LLine[{GG, HH}]]\)[ sakeri];\)\)}], "Input", CellLabel->"In[31]:="], Cell[BoxData[ \(\(labels\ = \ {Text["\", \ A[\([1]\)], \ {7, \ 3}], \ Text["\", \ B[\([1]\)], \ {0, \ 6}], \ Text["\", \ CC[\([1]\)], \ {\(-1\), \ \(-1\)}], \ Text["\", \ DD[\([1]\)], \ {0, \ \(-3\)}]};\)\)], "Input", CellLabel->"In[56]:="], Cell["We draw the picture.", "Text"], Cell[BoxData[ \(Show[ Graphics[LToGraphics[{sakeri, \ sakeri1, \ sakeri2, \ EE, \ FF, \ GG, \ HH}, \ Model \[Rule] PoincareDisk]]\ , \ Graphics[labels], Graphics[{Thickness[0.01], Circle[{0, 0}, 1]}], \[IndentingNewLine]\ AspectRatio \[Rule] Automatic]\)], "Input", CellLabel->"In[57]:="], Cell["\<\ We see only two, rather than three quadrilaterals. That is since sakeri and \ sakeri1 coincide what means that the quadrilateral ABCD is symmetric with \ respect to line joining the midpoints of AB and CD. That is not the case with \ the sakeri2.\ \>", "Text"], Cell["Example 2", "Subsection"], Cell["\<\ Although it doesn't make much sense, we can choose our favourite coordinates, \ and draw the graph of the function Sin[6 x]. You will notice that there is \ some hidden simmetry in the curve, although that symmetry is not an Euclidean \ one.\ \>", "Text"], Cell[BoxData[{ \(\(net\ = \ DrawNet[EquiPt, \ \ {\(-Pi\), Pi, \ 20, \ 0.2}, {\(-1\), 1, \ 35, \ 0.2}, \ \ Model \[Rule] PoincareDisk];\)\), "\[IndentingNewLine]", \(\(sin\ = \ Graphics[{Thickness[0.009], Line[Table[\(LToGraphics[EquiPt[{t, \ Sin[6\ t]}], \ Model \[Rule] PoincareDisk]\)[\([1]\)], \ {t, \ \(-Pi\), \ Pi, \ 1/40}]]}];\)\), "\[IndentingNewLine]", \(\(Show[Graphics[\ net], \ sin\ , Graphics[{Thickness[0.01], Circle[{0, 0}, 1]}], \ AspectRatio \[Rule] Automatic];\)\)}], "Input", CellLabel->"In[82]:="], Cell["\<\ Finally, using several Sin-like graphs we get an interesting pictures. It \ has some unusuall symmetries (actually, it has both hyperbolic translational \ symmetries, and a miror symmetry). The sky is the limit :)\ \>", "Text"], Cell[BoxData[ \(\(msin\ = \ Graphics[{Thickness[0.009], Line[Table[\(LToGraphics[EquiPt[{t, \ \(-Sin[6\ t]\)}], \ Model \[Rule] PoincareDisk]\)[\([1]\)], \ {t, \ \(-Pi\), \ Pi, \ 1/40}]]}];\)\)], "Input", CellLabel->"In[80]:="], Cell[BoxData[ \(\(curve\ = \ Graphics[{Thickness[0.009]\ , \ RGBColor[1, 0.2, 1], Line[Table[\(LToGraphics[EquiPt[{t, \ Sin[6\ t] + 0.5}], \ Model \[Rule] PoincareDisk]\)[\([1]\)], \ {t, \(-Pi\)\ , \ Pi, \ 1/40}]]}];\)\)], "Input", CellLabel->"In[85]:="], Cell[BoxData[ \(\(mcurve\ = \ \ Graphics[{Thickness[0.009], \ RGBColor[1, 0.2, 1], Line[Table[\(LToGraphics[EquiPt[{t, \ \(-Sin[\ 6\ t]\) - 0.5}], \ Model \[Rule] PoincareDisk]\)[\([1]\)], \ {t, \ \(-Pi\), \ Pi, \ 1/40}]]}];\)\)], "Input", CellLabel->"In[87]:="], Cell[BoxData[ \(\(Show[sin, \ msin, \ curve, \ mcurve, \ AspectRatio \[Rule] Automatic];\)\)], "Input", CellLabel->"In[89]:="] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 715}}, ScreenStyleEnvironment->"Brackets", WindowToolbars->{"RulerBar", "EditBar"}, WindowSize->{1014, 648}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, StyleDefinitions -> "HelpBrowser.nb" ] (*********************************************************************** Cached data follows. 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