Bezier curves; FindMinimum
- To: mathgroup at smc.vnet.net
- Subject: [mg22783] Bezier curves; FindMinimum
- From: dgolber at aol.com (DGolber)
- Date: Sat, 25 Mar 2000 03:58:20 -0500 (EST)
- Organization: AOL http://www.aol.com
- Sender: owner-wri-mathgroup at wolfram.com
The file below the "=====" line is a Notebook File. What I'm doing is finding the Bezier curve that best fits a bunch of points. (Begins and ends at the first and last points, and is horizontal there.) Questions: (1) Is there a package of good stuff for Bezier curves around somewhere? (2) The thing runs very slowly. Is there a way to modify what I'm doing to run faster? (I could modify the "find distance from a point to the curve" to remember the value it found the last time, since the point is the same, and the curve has changed very little. This would speed some part of it up.) (3) The "FindMinimum[bez7TargetFunc ...." converges very slowly indeed. (Thousands of evaluations of the target function!) And the parameters seem to be moving a lot, with the curve moving only a little. Seems to be some kind of bad behavior of the functions going on. Any comments? Are there any more parameters I can give it? Things to make it wiser about what's going on? Thanks Dave Golber ================== (*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. 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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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info at wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 7250, 166]*) (*NotebookOutlinePosition[ 7924, 190]*) (* CellTagsIndexPosition[ 7880, 186]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ \(\(\(edgedefpts\ = {{0, 5.4`}, {9.9480439`, 5.54248327`}, {19.88410958`, 6.04387166`}, {29.7827598`, 7.03453726`}, {39.59462195`, 8.66975694`}, {49.23291011`, 11.12350173`}, {58.56009935`, 14.57081366`}, {67.37999284`, 19.15811361`}, {75.45165162`, 24.95900645`}, {82.52512523`, 31.94126306`}, {88.39785909`, 39.96269368`}, {92.94202527`, 48.80016403`}, {96.1192025`, 58.21958287`}, {97.93646567`, 67.99360384`}, {98.42285575`, 77.9238168`}, {97.59998779`, 87.83212547`}, {95.49371832`, 97.54891289`}, {92.25476356`, 106.9519202`}, {88.47627937`, 116.1558295`}, {63.88103746`, 150.6650253`}, {58.18321955`, 157.9593264`}, {54.26254516`, 166.3593006`}, {51.7089787`, 175.277851`}, {50.1625087`, 184.4283674`}, {49.3690132`, 193.6760397`}, {49.22616918`, 202.9566534`}, {49.8734703`, 212.2133016`}, {51.91982826`, 221.2508838`}, {56.48472277`, 229.2674844`}, {75.71937801`, 269.6054168`}, {77.43072613`, 278.7797642`}, {78.16370608`, 288.0810706`}, {77.6803686`, 297.3964801`}, {75.84065874`, 306.5397911`}, {72.60073605`, 315.2848174`}, {68.00229656`, 323.3975739`}, {62.15589737`, 330.6627379`}, {55.2124192`, 336.8877347`}, {47.34722856`, 341.8975199`}, {38.77375423`, 345.5669841`}, {29.75035074`, 347.9340238`}, {20.51301702`, 349.2618733`}, {11.20010479`, 349.8933437`}, {1.86712191`, 350.0951868`}};\)\(\[IndentingNewLine]\)\)\)], "Input"], Cell[BoxData[""], "Input"], Cell[BoxData[ \(ptDistSq[p_, q_]\ := \ \ \[Sum]\+\(i = 1\)\%\(Min[Length[p], Length[q]]\)\((\((p[\ \([i]\)] - q[\([i]\)])\)^2)\)\)], "Input"], Cell[BoxData[ \(ptCurveDistSq[p_, f_, lims_]\ := \ \(FindMinimum[ ptDistSq[p, f], {t, lims}]\)[\([1]\)]\)], "Input"], Cell[BoxData[ \(devSqPtSetFromCurve[ptset_, curve_, lims_] := \[IndentingNewLine]\ \[Sum]\+\(i = \ 1\)\%\(Length[ptset]\)ptCurveDistSq[ptset[\([i]\)], curve, lims]\)], "Input"], Cell[BoxData[ \(combs[n_, i_] := \(n!\)/\((\(\((n - i)\)!\)\ \(i!\))\)\)], "Input"], Cell[BoxData[ \(bez[t_, \ n_, i_]\ := \ combs[n, i]\ t\^i\ \((1 - t)\)\^\((n - i)\)\)], "Input"], Cell[BoxData[ \(bezcurve[t_, n_, ptvec_, wtvec_]\ := \[IndentingNewLine]\[IndentingNewLine]\((\[Sum]\+\(i = 0\ \)\%n bez[t, n, i]\[Times]wtvec[\([i + 1]\)]\[Times] ptvec[\([i + 1]\)])\)\ \ /\[IndentingNewLine]\[IndentingNewLine]\((\[Sum]\+\(i = 0\)\%n bez[t, n, i]\[Times]wtvec[\([i + 1]\)])\)\)], "Input"], Cell[BoxData[ \(bezedge7[t_, p1x_, p2x_, p2y_, p3x_, p3y_, p4x_, p4y_, p5x_, p5y_, \[IndentingNewLine]p6x_, w1_, w2_, w3_, w4_, w5_, w6_]\ := \[IndentingNewLine]\ bezcurve[t, 7, \[IndentingNewLine]{\ edgedefpts[\([1]\)], \[IndentingNewLine]{p1x, \(edgedefpts[\([1]\)]\ \)[\([2]\)]}, \[IndentingNewLine]{p2x, p2y}, {p3x, p3y}, {p4x, p4y}, {p5x, p5y}, \[IndentingNewLine]{p6x, \(edgedefpts[\([\(-1\)]\)]\)[\([2]\ \)]}, \[IndentingNewLine]\ edgedefpts[\([\(-1\)]\)]}, \[IndentingNewLine]{1, w1, w2, w3, w4, w5, w6, 1}\ ]\)], "Input"], Cell[BoxData[ \(bez7TargetFunc[{p1x_, \[IndentingNewLine]p2x_, p2y_, p3x_, p3y_, p4x_, p4y_, p5x_, p5y_, \[IndentingNewLine]p6x_, \[IndentingNewLine]w1_, w2_, w3_, w4_, w5_, w6_}]\ := \ Module\[IndentingNewLine][{val}, \[IndentingNewLine]\((\(count++\); 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