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Elasticity functions for Bezier and rectangular forms

  • To: mathgroup at smc.vnet.net
  • Subject: [mg132168] Elasticity functions for Bezier and rectangular forms
  • From: "E. Martin-Serrano" <eMartinSerrano at telefonica.net>
  • Date: Thu, 9 Jan 2014 01:50:10 -0500 (EST)
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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 and
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 the actual
Bezier curve bcurve yet.*)



(*In other words the apparent direct procedure to convert a parametric 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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