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Re: Re: problem getting the area under a parmetric curve

  • To: mathgroup at
  • Subject: [mg52898] Re: [mg52860] Re: problem getting the area under a parmetric curve
  • From: Daniel Lichtblau <danl at>
  • Date: Thu, 16 Dec 2004 03:40:34 -0500 (EST)
  • References: <cpauof$ipf$> <>
  • Sender: owner-wri-mathgroup at

Roger Bagula wrote:
> Another of these distribution like pairs is:
> {(1-t^7)/(1+t^7),2^1/6*t*(7+35*t^14+21*t^28+t^42)^1/7/(1+t^7)}
> When fitted in the previous area programs,
> my computer doesn't compute the area of the normalization factor.
> n=6, 8 are again a squared circles.
> It appears that these can be done for general n case
> and the odd ones give the distributioin like parametric.
> [...]

I find it quite difficult to figure out what you want to do with this 
pair of functions in 't', or what 'n' might refer to. I'm guessing you'd 
like an implicit form? That's easy. As per another recent thread, just do

exprs = {(1-t^7)/(1+t^7),2^(1/6)*t*(7+35*t^14+21*t^28+t^42)^(1/7)/(1+t^7)};

InputForm[imp = First[GroebnerBasis[{x,y} - exprs, {x,y}, t]]]
Out[33]//InputForm= 2^(1/6) - 2^(1/6)*x^7 - y^7

Note that I altered your second expression so that the fractional 
exponents would be parsed in the way I believe you desired.

The following plots should convince you that the implicit form is correct.

ParametricPlot[exprs, {t,0,4}]

ContourPlot[imp, {x,-1,1}, {y,0,1.2}, Contours->{0},

It might make it easier for other readers to offer ideas (if that's what 
you hope to obtain) if you clarify some remarks. For example:

(i) "When fitted in the previous area programs..."

When what is fitted? Fitted how? What previous programs?

(ii) "[M]y computer doesn't compute the area of the normalization factor."

What exactly are you asking of your computer? (Show a small code snippet 
and explain clearly what you expect vs. what you obtain). Generally 
speaking your computer will do as it is told, and without knowing that 
detail it's difficult to comment or offer suggestions. I should also 
point out that if you are using Mathematica and your expressions are 
constructed in such a way that they will parse with operator precedence 
other than what you intend, that falls under the heading of user error.

(iii) "n=6, 8 are again a squared circles"

Again, what is 'n. Do you mean

exprs[n_] := {(1-t^n),
   2^(1/(n-1))*t*Sum[Binomial[n,j]*t^(n*(j-1)),{j,1,n,2}]^(1/n)} /

If you do not clearly state details such as this, most readers will be 
unable and/or unwilling to guess.

By the way:

ParametricPlot[exprs[6], {t,-10,10}]

InputForm[imp6 = First[GroebnerBasis[{x,y} - exprs[6], {x,y}, t]]]
Out[68]//InputForm= 2^(1/5) - 2^(1/5)*x^6 - y^6

ContourPlot[imp6, {x,-1,1}, {y,-2,2}, Contours->{0},

The plots indicate the closed boxy curve you seem to expect. This closed 
curve arises for even values of the exponent parameter 'n' because we 
lose the pole at t=-1 (so the curve is continuous) and the limits of the 
first coordinate are both -1 as t->+-Infinity, while the limits of the 
second are both 0 (so the curve is closed).

My overall comment is this: the key to good posts is clarity (I don't 
claim to be a paragon in this virtue, by the way). If you do not provide 
sufficient code, or show code that is not  going to parse the way you 
like, or use terminology that is not standard and do not explain it, 
then it is quite difficult to see whatever it is you wish to make known. 
If in addition you have questions, then any lack of clarity in the post 
decreases the likelihood that people might offer useful assistance.

Daniel Lichtblau
Wolfram Research

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