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MathGroup Archive 2000

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Re: Area of an Epicycloid ???

  • To: mathgroup at smc.vnet.net
  • Subject: [mg21397] Re: Area of an Epicycloid ???
  • From: "Allan Hayes" <hay at haystack.demon.co.uk>
  • Date: Tue, 4 Jan 2000 02:12:30 -0500 (EST)
  • References: <84plms$lul@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Steve,
Better to use polar coordinates to find the area of the epicycloid

x[t_] = (R + r)*Cos[t] - r*Cos[(R + r)*t/r];
y[t_] = (R + r)*Sin[t] - r*Sin[(R + r)*t/r];

(* it usually helps respondents if you give expressions in Mathematica form:
for example  Cos[t] not cos(t}*)

rad[t_] = Sqrt[ x[t]^2 + y[t]^2] // Simplify

    Sqrt[2*r^2 + 2*r*R + R^2 - 2*r*(r + R)*Cos[(R*t)/r]]


area[R_, r_] = Integrate[rad[t]^2/2, {t, 0, 2Pi}] // Simplify


    (Pi*R*(2*r^2 + 2*r*R + R^2) - r^2*(r + R)*
        Sin[(2*Pi*R)/r])/R


area[1, 1]

    5 Pi

Check: for very small r we would expect to get very near to the area of the
circle:

area[1, 1/1000000]

    (500001000001*Pi)/500000000000


Allan
---------------------
Allan Hayes
Mathematica Training and Consulting
Leicester UK
www.haystack.demon.co.uk
hay at haystack.demon.co.uk
Voice: +44 (0)116 271 4198
Fax: +44 (0)870 164 0565


"Guybrush Threepwood" <guybrush_ at gmx.de> wrote in message
news:84plms$lul at smc.vnet.net...
> hello all !
>
> im just trying to calculate the area between the epicyloid curve and the
big
> circle !
> the problem is that there are some x coordinates which have 2 y values....
> so i cant just simply integrate ! maybe someone of you has a hint for me !
> Someone told me that maybe polar coordinates could be helpfull... what do
ya
> think ?
>
> parameter form formula for epicycloids:
>
> x(t) = (R+r) * cos (t) - r * cos ((R+r)*t / r )
> y(t) = (R+r) * sin (t) -  r * sin ((R+r) *t /r )
>
> thanks in advance & bye
>
> Steve martins !
>
>
>
>
>
>
>
>
>
>
>




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