(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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See the ", StyleBox["Mathematica ", FontSlant->"Italic"], "Help browser or wolfram.com for more information on using ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text"], Cell["\<\ This notebook and those for the other seven chapters use a common stylesheet \ that has been imported into the notebook. This style sheet supports \ chapters, section, subsections, numbered equations, numbered figures and \ citations. A separate copy of the stylesheet can be download from the same \ page where this notebook was downloaded. \ \>", "Text"], Cell[TextData[{ StyleBox["COPYRIGHT ISSUES:", FontWeight->"Bold"], " The authors reserve all copyrights associated with this work. Any of \ the material appearing in these notebooks (such as the polyhedral meshes in \ chapter 7) can be used and modified without restriction as long as the use is \ non\[Hyphen]commercial. We simply ask that you acknowledge the authors when \ using material from these notebooks. For those readers interested in \ commercial use of the material in these notebooks, please contact \ jwarren@cs.rice.edu. 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\[LeftDoubleBracket]i, 1\[RightDoubleBracket]\ y\^dir\[LeftDoubleBracket]i, 2\ \[RightDoubleBracket]\)\/2\)\)], "Equation"], Cell[BoxData[ \(s[dir_, x_, y_] := Simplify[\(2 d[k - 1, dir, x\^2, y\^2]\)\/d[k, dir, x, y]]\)], "Input",\ InitializationCell->True, CellTags->"EQN box spline subdivision"], Cell["Example for linear box spline", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(s[{{1, 0}, {0, 1}, {1, 1}}, x, y]\)], "Input"], Cell[BoxData[ \(1\/4\ \((1 + x)\)\ \((1 + y)\)\ \((1 + x\ y)\)\)], "Output"] }, Open ]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Subdivision for exponential B\[Hyphen]splines", "Section"], Cell[CellGroupData[{ Cell["Discretization of the differential equation", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\[Eta][ k_, \[Alpha]_] = \[ExponentialE]\^\(\(2\^\(-k\)\) \[Alpha]\)\)], \ "Input", InitializationCell->True], Cell[BoxData[ \(\[ExponentialE]\^\(2\^\(-k\)\ \[Alpha]\)\)], "Output"] }, Open ]], Cell[TextData[{ "Observe that ", Cell[BoxData[ \(\[Alpha]\/\(\[Eta][k, \[Alpha]] - 1\)\)]], " converges to ", Cell[BoxData[ \(2\^k\)]], " as ", Cell[BoxData[ \(\[Alpha] \[Rule] 0\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Limit[\[Alpha]\/\(\[Eta][k, \[Alpha]] - 1\), \[Alpha] \[Rule] 0]\)], "Input"], Cell[BoxData[ \(2\^k\)], "Output"] }, Open ]], Cell[TextData[{ "Difference masks for given set of roots ", Cell[BoxData[ \(\[Alpha]\)]], " on the grid ", Cell[BoxData[ \(\(1\/2\^k\) \[DoubleStruckCapitalZ]\)]], ". Add special case for ", Cell[BoxData[ \(\[Alpha]\[LeftDoubleBracket]i\[RightDoubleBracket] \[Equal] 0\)]], "." }], "Text"], Cell[BoxData[ \(d[k_, \[Alpha]_, x_] := \[Product]\+\(i = 1\)\%\(Length[\[Alpha]]\)If[\[Alpha]\ \[LeftDoubleBracket]i\[RightDoubleBracket] \[Equal] 0, 2\^k, \[Alpha]\[LeftDoubleBracket]i\[RightDoubleBracket]\/\(\[Eta]\ [k, \[Alpha]\[LeftDoubleBracket]i\[RightDoubleBracket]] - 1\)] \((1 - \[Eta][ k, \[Alpha]\[LeftDoubleBracket]i\[RightDoubleBracket]] x)\)\)], "Input", InitializationCell->True, CellTags->"EQN exponential difference mask"], Cell[TextData[{ "Example of difference mask for differential operator ", Cell[BoxData[ \(\[ScriptCapitalD][\[ScriptX]]\^4 - \ \[ScriptCapitalD][\[ScriptX]]\^2\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CoefficientList[d[0, {1, \(-1\), 0, 0}, x], x] // Simplify\)], "Input"], Cell[BoxData[ \({\[ExponentialE]\/\((\(-1\) + \[ExponentialE])\)\^2, \(-\(\((1 + \ \[ExponentialE])\)\^2\/\((\(-1\) + \[ExponentialE])\)\^2\)\), \(2\ \((1 + \ \[ExponentialE] + \[ExponentialE]\^2)\)\)\/\((\(-1\) + \[ExponentialE])\)\^2, \ \(-\(\((1 + \[ExponentialE])\)\^2\/\((\(-1\) + \[ExponentialE])\)\^2\)\), \ \[ExponentialE]\/\((\(-1\) + \[ExponentialE])\)\^2}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["A subdivision scheme for exponential splines", "Subsection", CellTags->"SUBSEC subdiv exponential spline"], Cell[TextData[{ "Subdivision mask for exponential splines as function of level ", Cell[BoxData[ \(k\)]], " and given set of roots ", Cell[BoxData[ \(\[Alpha]\)]], ", use symbolic simplification to cancel out factors in denominator" }], "Text"], Cell[BoxData[ \(s[k_, \[Alpha]_, x_] := Simplify[\(2 d[k, \[Alpha], x\^2]\)\/d[k + 1, \[Alpha], x]]\)], "Input"], Cell["An alternative function without need for simplification", "Text"], Cell[BoxData[ \(s[k_, \[Alpha]_, x_] := 2 \(\[Product]\+\(i = 1\)\%\(Length[\[Alpha]]\)\((\(1 + \[ExponentialE]\ \^\(\[Alpha]\[LeftDoubleBracket]i\[RightDoubleBracket] 2\^\(-\((k + 1)\)\)\)\ \ x\)\/\(1 + \[ExponentialE]\^\(\[Alpha]\[LeftDoubleBracket]i\ \[RightDoubleBracket] 2\^\(-\((k + 1)\)\)\)\))\)\)\)], "Input", InitializationCell->True, CellTags->"EQN exponential subdivision mask"], Cell[TextData[{ "An example for the differential operator ", Cell[BoxData[ \(\[ScriptCapitalD][\[ScriptX]]\^4 - \ \[ScriptCapitalD][\[ScriptX]]\^2\)]], " on the grid ", Cell[BoxData[ \(\[DoubleStruckCapitalZ]\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(CoefficientList[s[0, {1, \(-1\), 0, 0}, x], x] // Simplify\)], "Input"], Cell[BoxData[ \({\@\[ExponentialE]\/\(2\ \((1 + \@\[ExponentialE])\)\^2\), 1\/2, \(1 + \@\[ExponentialE] + \[ExponentialE]\)\/\((1 + \@\ \[ExponentialE])\)\^2, 1\/2, \@\[ExponentialE]\/\(2\ \((1 + \@\[ExponentialE])\)\^2\)}\)], \ "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Exponential B\[Hyphen]splines as piecewise analytic functions", \ "Subsection"], Cell[CellGroupData[{ Cell["Recurrence for exponential analog of truncated powers", "Subsubsection"], Cell[TextData[{ "Again, for ", StyleBox["Mathematica", FontSlant->"Italic"], " to reduce ", Cell[BoxData[ \(\[ScriptX] > \(-\[Infinity]\)\)]], " to be ", Cell[BoxData[ \(True\)]], "." }], "Text"], Cell[BoxData[{ \(\(Unprotect[Greater];\)\), "\[IndentingNewLine]", \(\(Greater[_, \(-\[Infinity]\)] = True;\)\), "\[IndentingNewLine]", \(\(Protect[Greater];\)\)}], "Input", InitializationCell->True], Cell[TextData[{ "Define an integral operator for the exponential analog of truncated \ powers. In the text, the operator has the form ", Cell[BoxData[ \(\[ScriptCapitalI][\[ScriptX], \[Alpha]] \[ScriptF][\[ScriptX]]\)]], "." }], "Text"], Cell[BoxData[ \(\[ScriptCapitalI][\[ScriptF]_, \[Alpha]_] := Module[{\[ScriptG]}, \[IndentingNewLine]\[ScriptG][\[ScriptX]_] = \ \[Integral]\_0\%\[Infinity]\( \[ExponentialE]\^\(\[Alpha]\ \[ScriptT]\)\) \ \[ScriptF][\[ScriptX] - \[ScriptT]] \[DifferentialD]\[ScriptT]; \ \[ScriptG]]\)], "Input", InitializationCell->True, CellTags->"def exp integral"], Cell[TextData[{ "Observe that ", Cell[BoxData[ \(\[ScriptCapitalD][\[ScriptX]] - \[Alpha]\)]], " is the left inverse of ", Cell[BoxData[ \(\[ScriptCapitalI][\[ScriptX], \[Alpha]]\)]], " for the ", Cell[BoxData[ \(DiracDelta\)]], " function." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(With[{\[ScriptF] = \[ScriptCapitalI][ DiracDelta, \[Alpha]]}, \ \[IndentingNewLine]D[\[ScriptF][\[ScriptX]], \[ScriptX]] - \[Alpha]\ \ \[ScriptF][\[ScriptX]]]\)], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\(\[ScriptX]\ \[Alpha]\)\ DiracDelta[\[ScriptX]]\)], \ "Output"] }, Open ]], Cell[TextData[{ "Given a list of roots ", Cell[BoxData[ \(\[Alpha]\)]], ", we construct the exponential truncated powers as follows:" }], "Text"], Cell[BoxData[ \(makeTruncPower[\[Alpha]_] := Fold[\[ScriptCapitalI], DiracDelta, \[Alpha]]\)], "Input", InitializationCell->True, CellTags->"def scc"], Cell[TextData[{ "Example for ", Cell[BoxData[ \(\[Alpha] = {1, \(-1\), 0, 0}\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(\[ScriptC] = makeTruncPower[{1}];\)\), "\[IndentingNewLine]", \(\[ScriptC][\[ScriptX]]\), "\[IndentingNewLine]", \(Plot[\[ScriptC][\[ScriptX]], {\[ScriptX], \(-1\), 1}, PlotRange \[Rule] All]\)}], "Input"], Cell[BoxData[ \(\[ExponentialE]\^\[ScriptX]\ UnitStep[\[ScriptX]]\)], "Output"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 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The factor of ", Cell[BoxData[ \(\[Alpha]\/\(\(\[ExponentialE]\^\[Alpha]\) 1\)\)]], " normalizes exponential step function to have unit integral. The function \ ", Cell[BoxData[ \(\[ScriptCapitalS][\[ScriptF], \[Alpha]]\)]], " returns the continuous convolution of ", Cell[BoxData[ \(\[ScriptF][\[ScriptX]]\)]], " and the exponential step function ", Cell[BoxData[ \(\[ScriptU][\[ScriptX]]\)]], " defined above. " }], "Text"], Cell[BoxData[ \(\[ScriptCapitalS][\[ScriptF]_, \[Alpha]_] := Module[{\[ScriptG]}, \[IndentingNewLine]\[ScriptG][\[ScriptX]_] = If[\[Alpha] \[Equal] 0, 1, \[Alpha]\/\(\[ExponentialE]\^\[Alpha] - 1\)] \ \(\[Integral]\_0\%1\( \[ExponentialE]\^\(\[Alpha]\ \[ScriptT]\)\) \[ScriptF][\ \[ScriptX] - \[ScriptT]] \[DifferentialD]\[ScriptT]\); \[ScriptG]]\)], "Input",\ InitializationCell->True, CellTags->"EQN integral recurrence for exp B-splines"], Cell[TextData[{ Cell[BoxData[ \(makeExpSpline[\[Alpha]]\)]], " takes a list of roots ", Cell[BoxData[ \(\[Alpha]\)]], " and recursively construct the exponential basis function by applying ", Cell[BoxData[ \(\[ScriptCapitalS]\)]], ". The base case is when ", Cell[BoxData[ \(\[ScriptN][\[ScriptX]]\)]], " is the function ", Cell[BoxData[ \(DiracDelta[\[ScriptX]]\)]], "." }], "Text"], Cell[BoxData[ \(makeExpSpline[\[Alpha]_] := Fold[\[ScriptCapitalS], DiracDelta, \[Alpha]]\)], "Input", InitializationCell->True], Cell[TextData[{ "Example for the differential operator ", Cell[BoxData[ \(\[ScriptCapitalD][\[ScriptX]]\^4 - \ \[ScriptCapitalD][\[ScriptX]]\^2\)]], ", i.e. ", Cell[BoxData[ \(\[Alpha] = {1, \(-1\), 0, 0}\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(\[ScriptN] = makeExpSpline[{1, \(-1\), 0, 0}] // FullSimplify;\)\), "\[IndentingNewLine]", \(Plot[\ \[ScriptN][\[ScriptX]], {\[ScriptX], \(-1\), 5}, PlotRange \[Rule] All]\)}], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier 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in tension", "Subsection"], Cell["See examples above", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Mixed trigonometric splines", "Subsection"], Cell[TextData[{ "Show that the truncated powers for the differential operator ", Cell[BoxData[ \(\[ScriptCapitalD][\[ScriptX]]\^4 + \ \[ScriptCapitalD][\[ScriptX]]\^2\)]], " are real valued." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(\[ScriptC] = makeTruncPower[{\[ImaginaryI], \(-\[ImaginaryI]\), 0, 0}];\)\), "\[IndentingNewLine]", \(FullSimplify[\[ScriptC][\[ScriptX]]]\)}], "Input"], Cell[BoxData[ \(\((\[ScriptX] - Sin[\[ScriptX]])\)\ UnitStep[\[ScriptX]]\)], "Output"] }, Open ]], Cell["\<\ Observe that difference coefficients used in construct exponential \ B\[Hyphen]spline basis functions are also real.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FullSimplify[ CoefficientList[d[0, {\[ImaginaryI], \(-\[ImaginaryI]\), 0, 0}, x], x]]\)], "Input"], Cell[BoxData[ \({1\/4\ Csc[1\/2]\^2, \(-Cot[1\/2]\^2\), \(-2\) + 3\/2\ Csc[1\/2]\^2, \(-Cot[1\/2]\^2\), 1\/4\ Csc[1\/2]\^2}\)], "Output"] }, Open ]], Cell["See next subsection for examples", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["The unified subdivision scheme", "Subsection"], Cell["\<\ In the next two examples, we implement univariate and bivariate (tensor \ product) instances of this scheme and plot some examples.\ \>", "Text"], Cell[CellGroupData[{ Cell["Code for curves", "Subsubsection"], Cell[TextData[{ "Based on these two examples, we can build a general, order four \ subdivision scheme that has ", Cell[BoxData[ \(\[ScriptCapitalC]\^2\)]], " continuity and linear precision. Depending on the choice of an initial \ parameter ", Cell[BoxData[ \(\[Sigma]\_0\)]], ", the scheme also reproduces hyperbolic functions if ", Cell[BoxData[ \(\[Sigma]\_0 > 1\)]], ", trigonometric functions if ", Cell[BoxData[ \(\[Sigma]\_0 < 1\)]], " or higher order polynomials if ", Cell[BoxData[ \(\[Sigma]\_0 \[Equal] 1\)]], ".", " ", "Given initial ", Cell[BoxData[ \(\[Sigma]\_0\)]], ", we suggest applying subdivision mask of the form" }], "Text"], Cell[BoxData[ \(\(\(s\_\(k - 1\)[ x]\)\(=\)\(\(\(\((1 + x)\)\^2\) \((1 + 2 \[Sigma]\_k\ x + x\^2)\)\)\/\(4 \( x\^2\) \((1 + \[Sigma]\_k)\)\)\)\(\ \)\)\)], "Equation"], Cell[TextData[{ "where ", Cell[BoxData[ \(\[Sigma]\)]], " is updated via ", Cell[BoxData[ \(\[Sigma]\_k = \@\(\(1 + \[Sigma]\_\(k - 1\)\)\/2\)\)]], ". Given an initial regular ", Cell[BoxData[ \(n\)]], "\[Hyphen]", "gon, choosing ", Cell[BoxData[ \(\[Sigma]\_0 = Cos[\(2 \[Pi]\)\/n]\)]], " forces convergence to a circle." }], "Text"], Cell[TextData[{ Cell[BoxData[ \(subdivide\)]], " is specialized subdivision procedure for the formula above. ", Cell[BoxData[ \(p\)]], " is generating function whose coefficients are points, ", Cell[BoxData[ \(\[Sigma]\)]], " is initial tension for subdivision mask, ", Cell[BoxData[ \(n\)]], " is number of points, ", Cell[BoxData[ \(k\)]], " is number of rounds of subdivision" }], "Text"], Cell[BoxData[ \(subdivide[p_, \[Sigma]_, n_, k_] := If[k \[Equal] 0, p, With[{n\[Sigma] = \@\(\(1 + \[Sigma]\)\/2\)}, \ \[IndentingNewLine]subdivide[ Expand[\(\(\((1 + x)\)\^2\) \((1 + 2 n\[Sigma]\ x + \ x\^2)\)\)\/\(4 \( x\^2\) \((1 + n\[Sigma])\)\)*\((p /. {x \[Rule] x\^2})\)] /. \ {x\^a_ \[Rule] x\^Mod[a, 2 n]}, n\[Sigma], 2 n, k - 1]]]\)], "Input", InitializationCell->True, CellTags->"EQN unified scheme"], Cell[TextData[{ "Helper function that closes open polygon ", Cell[BoxData[ \(p\)]], "." }], "Text"], Cell[BoxData[ \(periodize[p_] := Append[p, First[p]]\)], "Input", InitializationCell->True], Cell["Create example of various tensions for an initial diamond", "Text"], Cell[CellGroupData[{ 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