Re: Interpolation of data to form a parametric curve

*To*: mathgroup at smc.vnet.net*Subject*: [mg80407] Re: [mg80382] Interpolation of data to form a parametric curve*From*: Carl Woll <carlw at wolfram.com>*Date*: Tue, 21 Aug 2007 05:09:24 -0400 (EDT)*References*: <200708201006.GAA04997@smc.vnet.net>

Hugh wrote: >Below I give some example data for a 2D curve. I then interpolate the >x and y data to give a parametric version of the curve. This works >well as the plot shows, and I could also use the Spline package. >However, this data is parameterized with respect to point number while >I need the data parameterized with respect to distance along the curve >or alternatively as a distance going from 0 to 1. I can see how to get >distance in terms of point number, by using NDSolve, but how do I get >the inverse -point number in terms of distance? If I have point number >in terms of distance then presumably I can rework the interpolation as >a new function. Any suggestions? >Thanks >Hugh Goyder > >d = {{0., 1.2}, {0.180347569987808, > 1.1598301273032612}, {0.31554453682333494, > 1.0539181001894673}, {0.37759261784475534, 0.9204838518536992}, > {0.3662469376233495, 0.8067797622536416}, {0.3090169943749474, > 0.7510565162951535}, {0.2505675022261833, 0.767973087013262}, > {0.23556798830604195, > 0.8430236535910302}, {0.2915423708426846, > 0.938110078918853}, {0.418269744520502, 1.0061313243770045}, > {0.5877852522924731, 1.0090169943749474}, {0.7549810402071845, > 0.9323166416507785}, {0.8747584091877195, 0.7907720262964009}, > {0.9191799306804422, 0.6227437070536992}, {0.8880702932342837, > 0.4756205908737}, {0.8090169943749471, 0.387785252292473}, > {0.7267708750435204, 0.37402339610400703}}; > >nn = Length[d]; > >fx = Interpolation[d[[All, 1]]]; >fy = Interpolation[d[[All, 2]]]; > >ParametricPlot[{fx[n], fy[n]}, {n, 1, nn}, > Epilog -> {Point[#] & /@ d}, AspectRatio -> Automatic] > >(* Get distance in terms of point number *) > >dfx = Derivative[1][fx]; dfy = Derivative[1][fy]; > >sol = NDSolve[{Derivative[1][n][t] == Sqrt[dfx[t]^2 + dfy[t]^2], > n[1] == 0}, {n}, {t, 1, nn}]; > > > sol gives you a function of distance per node, n[t]. You are interested in the function t[n]. Now, for a given distance s, we have: n[t[s]] == s so, n'[t[s]] t'[s] == 1 So, plug this equation into NDSolve, using n'[t[s]] == Sqrt[dfx[t[s]]^2+dfy[t[s]]^2]: NDSolve[{Sqrt[dfx[t[s]]^2 + dfy[t[s]]^2] t'[s] == 1, t[0] == 1}, t, {s, 0, 2.18121}] Carl Woll Wolfram Research

**References**:**Interpolation of data to form a parametric curve***From:*Hugh <h.g.d.goyder@cranfield.ac.uk>