Re: help about klotoid-integral-solution
- To: mathgroup@smc.vnet.net
- Subject: [mg12527] Re: help about klotoid-integral-solution
- From: Paul Abbott <paul@physics.uwa.edu.au>
- Date: Sat, 23 May 1998 18:10:55 -0400
- Organization: University of Western Australia
- References: <6jomuk$kap@smc.vnet.net>
eckhard_FABIAN wrote:
> it would be great if anyone can help with the solution of the integrals
> for x and y of the klothoid: x=a*sqrt(pi)*INT(cos(pi*u*u/2)du,u=0...t)
> y=a*sqrt(pi)*INT(sin(pi*u*u/2)du,u=0...t) with t=s/(a*sqrt(pi))
> s=OM Distance on the curve from origin O to point M on the curve
> and a>0
> The point O is the centre of symmetry of the curve, it has the
> asymptotic points
> A(a*sqrt(pi)/2,a*sqrt(pi)/2) and B(-a*sqrt(pi)/2,-a*sqrt(pi)/2) These
> formula are ref. in the book
> BRONSTEIN/SEMENDJAJEW TASCHENBUCH DER MATHEMATIK I need formula to
> calculate the x and y -coordinates depending on the increasing distance
> on the curve. Thanks a lot for an answer via email.
Mathematica can compute both the integrals you need:
In[1]:= x[s_, a_] = a*Sqrt[Pi]*Integrate[Cos[Pi*u^2/2], {u, 0,
s/(a*Sqrt[Pi])}]
Out[1]= a*Sqrt[Pi]*FresnelC[s/(a*Sqrt[Pi])]
In[2]:= y[s_, a_] = a*Sqrt[Pi]*Integrate[Sin[Pi*u^2/2], {u, 0,
s/(a*Sqrt[Pi])}]
Out[2]= a*Sqrt[Pi]*FresnelS[s/(a*Sqrt[Pi])]
The answer is expressed in terms of the (non-elementary) Fresnel C and S
functions. There is no simpler way to express these integrals.
Mathematica can compute these functions to arbitary precision and also
produce parametric plots of the clothoid.
Cheers,
Paul
____________________________________________________________________
Paul Abbott Phone: +61-8-9380-2734
Department of Physics Fax: +61-8-9380-1014
The University of Western Australia Nedlands WA 6907
mailto:paul@physics.uwa.edu.au AUSTRALIA
http://www.pd.uwa.edu.au/~paul
God IS a weakly left-handed dice player
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