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Re: Spirals and arc length

  • To: mathgroup at smc.vnet.net
  • Subject: [mg41292] Re: Spirals and arc length
  • From: "Bill Bertram" <wkb at ansto.gov.au>
  • Date: Tue, 13 May 2003 04:20:33 -0400 (EDT)
  • Organization: Australian Nuclear Science and Technology Organisation
  • References: <b9n9ki$8sl$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

"DIAMOND Mark" <noname at noname.com> wrote in message
news:b9n9ki$8sl$1 at smc.vnet.net...
> Please excuse the double posting, but I am interested in both the
> mathematics and a Mathematica approach to the following problem.
>
> Simply put, I wish to find the polar coordinates of a point that has been
> moved along a spiral arc.
> If I have a point (theta0,r0) on a spiral r=a Exp(b*theta), and I travel
> along the spiral arc some distance (delta), then what are the polar
> coordinates of the new point?
>

Mark,
If r is a function of theta, r(theta). the Cartesian coordinates (x, y) of a
point (r,theta) are

x = r(theta)*cos(theta), y = r(theta)*sin(theta). An element of length along
the curve is given by

(ds)^2 = (dx)^2 + (dy)^2  which can be rewritten in terms of theta by using

dx = (dx/dtheta)*dtheta and dy = (dy/dtheta) dtheta.

So, what you end up with is a first order differential equation of the form,

ds/dtheta = SomeFunction(theta)
 which can then solve analytically if you're lucky, else numerically.

Cheers,
  Bill









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