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Re: Elasticity functions for Bezier and rectangular

  • To: mathgroup at smc.vnet.net
  • Subject: [mg132188] Re: Elasticity functions for Bezier and rectangular
  • From: Tomas Garza <tgarza10 at msn.com>
  • Date: Sun, 12 Jan 2014 02:33:03 -0500 (EST)
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Yes, Emilio, Bob's right.I copied and pasted his code from the email and it ran without a hitch.I;m on a MacBook Pro with OS Mavericks and Mathematica version 9.0.1.0
-Tomas

> From: hanlonr357 at gmail.com
> Subject: Re: Elasticity functions for Bezier and rectangular forms
> To: mathgroup at smc.vnet.net
> Date: Sat, 11 Jan 2014 02:36:11 -0500
>
>
> Either it is a version issue or you need to start with a fresh kernel.
>
>
> $Version
>
>
> "9.0 for Mac OS X x86 (64-bit) (January 24, 2013)"
>
>
> pts = {p[0], p[1], p[2], p[3]} =
>    Rationalize[{{0, .65}, {1.8, 2.6}, {4, 2.5}, {4, 5}}];
>
>
> n = Length[pts] - 1;
>
>
> bcurve = Sum[Binomial[n, k]*(1 - t)^(n - k)*t^k p[k], {k, 0, n}] //
> Simplify;
>
>
> Solve[{x == bcurve[[1]], 0 <= x <= 4}, t]
>
>
> {{t -> ConditionalExpression[
>          Root[5*x - 27*#1 -
>                6*#1^2 + 13*#1^3 & ,
>            1], 0 < x < 4]},
>    {t -> ConditionalExpression[
>          Root[5*x - 27*#1 -
>                6*#1^2 + 13*#1^3 & ,
>            2], 0 < x < 4]},
>    {t -> ConditionalExpression[
>          Root[5*x - 27*#1 -
>                6*#1^2 + 13*#1^3 & ,
>            3], 0 < x < 4]}}
>
>
> First /@ (bcurve[[2]] /. %)
>
>
> {(1/20)*(13 + 117*
>           Root[5*x - 27*#1 -
>                 6*#1^2 + 13*#1^3 & ,
>             1] - 123*
>           Root[5*x - 27*#1 -
>                   6*#1^2 + 13*#1^3 & ,
>               1]^2 +
>         93*Root[5*x - 27*#1 -
>                   6*#1^2 + 13*#1^3 & ,
>               1]^3), (1/20)*
>      (13 + 117*Root[
>             5*x - 27*#1 - 6*#1^2 +
>                 13*#1^3 & , 2] -
>         123*Root[5*x - 27*#1 -
>                   6*#1^2 + 13*#1^3 & ,
>               2]^2 +
>         93*Root[5*x - 27*#1 -
>                   6*#1^2 + 13*#1^3 & ,
>               2]^3), (1/20)*
>      (13 + 117*Root[
>             5*x - 27*#1 - 6*#1^2 +
>                 13*#1^3 & , 3] -
>         123*Root[5*x - 27*#1 -
>                   6*#1^2 + 13*#1^3 & ,
>               3]^2 +
>         93*Root[5*x - 27*#1 -
>                   6*#1^2 + 13*#1^3 & ,
>               3]^3)}
>
>
> y1[x_] = Select[%,
>    Chop[N[# /. x -> p[0][[1]]]] == p[0][[2]] &&
>      Chop[N[# /. x -> p[3][[1]]]] == p[3][[2]] &][[1]]
>
>
> (1/20)*(13 +
>       117*Root[5*x - 27*#1 -
>               6*#1^2 + 13*#1^3 & ,
>           2] - 123*
>         Root[5*x - 27*#1 - 6*#1^2 +
>                 13*#1^3 & , 2]^2 +
>       93*Root[5*x - 27*#1 -
>                 6*#1^2 + 13*#1^3 & , 2]^
>           3)
>
>
> Bob Hanlon
>
>
> On Fri, Jan 10, 2014 at 6:19 PM, E. Martin-Serrano <
> eMartinSerrano at telefonica.net> wrote:
>
> > Hi Bob,
> >
> > Thank you very much for your answer, but I am afraid, I am unable to see
> > how
> > from:
> >
> > Select[
> >    First /@ (bcurve[[2]] /.
> >       Solve[{x == bcurve[[1]], 0 <= x <= 4}, t]),
> >    Chop[N[# /. x -> p[0][[1]]]] == p[0][[2]] &&
> >      Chop[N[# /. x -> p[3][[1]]]] == p[3][[2]] &][[1]]
> >
> > We get the roots expression:
> >
> > (1/20)*(13 + 117*Root[5*x - 27*#1 - 6*#1^2 + 13*#1^3 & ,
> >           2] - 123*Root[5*x - 27*#1 - 6*#1^2 + 13*#1^3 & , 2]^2 +
> >       93*Root[5*x - 27*#1 - 6*#1^2 + 13*#1^3 & , 2]^3)
> >
> > Since
> >
> > Solve[{x == bcurve[[1]], 0 <= x <= 4}, t]
> >
> > Yields just  the empty  list '{}' (no possible solutions), and therefore
> > nothing can be selected.
> >
> >  In fact from
> >
> > y1[x_] = Select[
> >    First /@ (bcurve[[2]] /.
> >       Solve[{x == bcurve[[1]], 0 <= x <= 4}, t]),
> >    Chop[N[# /. x -> p[0][[1]]]] == p[0][[2]] &&
> >      Chop[N[# /. x -> p[3][[1]]]] == p[3][[2]] &][[1]]
> >
> > the result that I obtain is
> >
> > 1[[1]]
> >
> > The same with the alternative solution expressed with radicals.
> >
> > From this, it is clear that I am missing something fundamental.
> >
> > E. Martin-Serrano
> >
> >
> >
> > -----Mensaje original-----
> > De: Bob Hanlon [mailto:hanlonr357 at gmail.com]
> > Enviado el: viernes, 10 de enero de 2014 8:49
> > Para: mathgroup at smc.vnet.net
> > Asunto: Re: Elasticity functions for Bezier and rectangular
> > forms
> >
> >
> > pts = {p[0], p[1], p[2], p[3]} =
> >    Rationalize[{{0, .65}, {1.8, 2.6}, {4, 2.5}, {4, 5}}]=
;
> >
> >
> > n = Length[pts] - 1;
> >
> >
> > bcurve = Sum[Binomial[n, k]*(1 - t)^(n - k)*t^k p[k],
> >     {k, 0, n}] // Simplify;
> >
> >
> > Solving for solution expressed as Root objects
> >
> >
> > y1[x_] = Select[
> >    First /@ (bcurve[[2]] /.
> >       Solve[{x == bcurve[[1]], 0 <= x <= 4}, t]),
> >    Chop[N[# /. x -> p[0][[1]]]] == p[0][[2]] &&
> >      Chop[N[# /. x -> p[3][[1]]]] == p[3][[2]] &][[1]]
> >
> >
> > (1/20)*(13 + 117*Root[5*x - 27*#1 - 6*#1^2 + 13*#1^3 & ,
> >           2] - 123*Root[5*x - 27*#1 - 6*#1^2 + 13*#1^3 & , 2]^2 +
> >       93*Root[5*x - 27*#1 - 6*#1^2 + 13*#1^3 & , 2]^3)
> >
> >
> >
> > Solving for solution expressed with radicals
> >
> >
> > y2[x_] = Select[(First /@ (bcurve[[2]] /.
> >          Solve[{x == bcurve[[1]], 0 <= x <= 4}, t])) //
> >      ToRadicals //
> >     FullSimplify,
> >    Chop[N[# /. x -> p[0][[1]]]] == p[0][[2]] &&
> >      Chop[N[# /. x -> p[3][[1]]]] == p[3][[2]] &][[1]]
> >
> >
> > (1/20)*(-((465*x)/13) + (1/4394)*(-245386 +
> >            (30482562*(-1)^(1/3)*2^(2/3))/
> >              (718 - 845*x + 13*Sqrt[5]*Sqrt[(-4 + x)*
> >                         (1944 + 845*x)])^(2/3) +
> >            (11676984*(-1)^(2/3)*2^(1/3))/
> >              (718 - 845*x + 13*Sqrt[5]*Sqrt[(-4 + x)*
> >                         (1944 + 845*x)])^(1/3) - 48252*(-1)^(1/3)*2^(2/=
3)*
> >              (718 - 845*x + 13*Sqrt[5]*Sqrt[(-4 + x)*
> >                         (1944 + 845*x)])^(1/3) - 1041*(-1)^(2/3)*2^(1/3=
)*
> >              (718 - 845*x + 13*Sqrt[5]*Sqrt[(-4 + x)*
> >                         (1944 + 845*x)])^(2/3)))
> >
> >
> >
> > Note that y2 is a different branch than that obtained by converting y1 =
to
> > radicals (see also plot below)
> >
> >
> > y1r[x_] = y1[x] // ToRadicals // FullSimplify
> >
> >
> > (1/20)*(-((465*x)/13) + (1/4394)*(-245386 -
> >            (30482562*(-2)^(2/3))/(718 - 845*x +
> >                   13*Sqrt[5]*Sqrt[(-4 + x)*(1944 + 845*x)])^(2/3) -
> >            (11676984*(-2)^(1/3))/(718 - 845*x +
> >                   13*Sqrt[5]*Sqrt[(-4 + x)*(1944 + 845*x)])^(1/3) +
> >            48252*(-2)^(2/3)*(718 - 845*x + 13*Sqrt[5]*
> >                     Sqrt[(-4 + x)*(1944 + 845*x)])^(1/3) +
> >            1041*(-2)^(1/3)*(718 - 845*x + 13*Sqrt[5]*
> >                     Sqrt[(-4 + x)*(1944 + 845*x)])^(2/3)))
> >
> >
> >
> > Expressed as an interpolation function
> >
> >
> > y3 = Interpolation[Table[bcurve, {t, 0, 1, .01}]];
> >
> >
> >
> > Note that y1r is not the proper branch:
> >
> >
> > Grid[
> >  {{Graphics[{BezierCurve[pts], Red,
> >      AbsolutePointSize[4], Point[pts]},
> >     AspectRatio -> 1, Frame -> True, Axes -> False,
> >     PlotLabel -> "Bezier Curve"],
> >    pp = ParametricPlot[BezierFunction[pts][x],
> >      {x, 0, 4}, AspectRatio -> 1, Frame -> True, Axes -> Fals=
e,
> >      PlotLabel -> "BezierFunction",
> >      Epilog -> {Red, AbsolutePointSize[4], Point[pts]}],
> >    ParametricPlot[bcurve, {t, 0, 1},
> >     AspectRatio -> 1, Frame -> True, Axes -> False,
> >     PlotLabel -> "bcurve",
> >     Epilog -> {Red, AbsolutePointSize[4], Point[pts]}]},
> >   {Plot[y1[x], {x, 0, 4}, AspectRatio -> 1,
> >     Frame -> True, Axes -> False,
> >     PlotLabel -> "Root Object (y1)",
> >     Epilog -> {Red, AbsolutePointSize[4], Point[pts]}],
> >    Plot[y2[x], {x, 0, 4}, AspectRatio -> 1,
> >     Frame -> True, Axes -> False,
> >     PlotLabel -> "Radicals (y2)",
> >     Epilog -> {Red, AbsolutePointSize[4], Point[pts]}],
> >    Plot[y3[x], {x, 0, 4}, AspectRatio -> 1,
> >     Frame -> True, Axes -> False,
> >     PlotLabel -> "Interpolation of bcurve (y3)",
> >     Epilog -> {Red, AbsolutePointSize[4], Point[pts]}]},
> >   {Plot[y1r[x], {x, 0, 4}, AspectRatio -> 1,
> >     PlotStyle -> Red, Frame -> True, Axes -> False,
> >     PlotLabel -> "Root converted to Radicals (y1r)"],
> >    "", ""}}]
> >
> >
> >
> > Bob Hanlon
> >
> >
> >
> >
> > On Thu, Jan 9, 2014 at 1:50 AM, E. Martin-Serrano <
> > eMartinSerrano at telefonica.net> wrote:
> >
> > >
> > > Hi,
> > >
> > >
> > >
> > > (*I have the Bezier curve*)
> > >
> > >
> > >
> > > pts = {p[0], p[1], p[2], p[3]} = {{0, .65}, {1.8, 2.6=
}, {4, 2.5}, {4,
> > > 5}};
> > >
> > >
> > >
> > > n = Length[pts] - 1;
> > >
> > >
> > >
> > > bcurve = Sum[Binomial[n, i]* (1 - t)^(n - i) *t ^i  p[i] , {i=
, 0, n}];
> > >
> > >
> > >
> > > (* the plot range for the resulting Bezier curve is obviously
> > > {p[0][[1]], p[3][[1]} ->  {0,4} *)
> > >
> > >
> > >
> > > (*Then, I want to transform the above parametric Bezier curve into =
a
> > > rectangular {x,y} curve one by solving curve[[1]] x= x(t) for t a=
nd
> > > plugging  the result t-> x(t) into curve[[2]]  or  y = y(t).*)
> > >
> > >
> > >
> > > sols= Solve[x == bcurve[[1]], t, Reals],
> > >
> > >
> > >
> > > bcurve[[2]]/.%
> > >
> > >
> > >
> > > yvalues = (% /. x -> #) & /@ Range[0, 4]
> > >
> > >
> > >
> > > (*For testing purposes, by visual inspection I discard directly the
> > > resulting yvalues which overflow the plot range interval {0,4}  and
> > > select only those values of y which are members of the interval.*)
> > >
> > >
> > >
> > > #[[2]]&/@yvalues
> > >
> > >
> > >
> > > (*But that done, none of these selected  {x,y} points belong to t=
he
> > > actual Bezier curve bcurve yet.*)
> > >
> > >
> > >
> > > (*In other words the apparent direct procedure to convert a parametri=
c
> > > curve into rectangular one does not seem to work here.*)
> > >
> > >
> > >
> > > (*Remark: I need the basic Bezier curve bcurve to be reshapable
> > > (dynamic) and be able to calculate the elasticity  point function of
> > > each new resulting function as the Bezier curve bcurve gets reshaped=
,
> > > and for some reason I am not being able of  getting directly the
> > > elasticity point function of bcurve = Sum[Binomial[n, i]* (1 - t)=
^(n -
> > > i) *t ^i  p[i] , {i, 0, n}]; no problem for any other similar=
 (or
> > > approximate) function given in rectangular form*) ;
> > >
> > >
> > >
> > > Any help will be welcome.
> > >
> > >
> > >
> > > E. Martin-Serrano
> > >
> > >
> > >
> > > P.S. For the sake of clarity I have enclosed the text in  as in
> > > (*text*) and symbols or experssions within text as in symbol (when
> > > necessary).
> > >
> > >
> > >
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