Re: Spirals and arc length
- To: mathgroup at smc.vnet.net
- Subject: [mg41284] Re: [mg41271] Spirals and arc length
- From: David Terr <dterr at wolfram.com>
- Date: Tue, 13 May 2003 04:18:08 -0400 (EDT)
- References: <200305120458.AAA09126@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
DIAMOND Mark wrote: >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? > >I would really like a few different things if anyone can help; not >necessarily in priority order ... >(1) a simple expression for the answer; >(2) an explanation that I can follow and apply to a spiral of a different >form, say Archimaedian or hyperbolic; >(3) a Mathematica approach to *deriving* the appropriate expression. > >This may be too much to ask, but I have tried tackling the problem myself >and even having read the mathworld entries on the various spirals (and >arc-length), I'm not sure where to begin. > >Cheers, > >-- >Mark R Diamond >Vision Research Laboratory >The University of Western Australia >email: FirstNameFollowedbySurnameInitialAtpsy.edu.au > > > > I don't think Mathematica is the best tool for solving this problem. To do so, the best place to start is the arc length formula in polar coordinates: ds^2 = dr^2 + r^2 d\theta^2. Integrating this formula over a curve given in polar coordinates gives you its length. I think you'll find it isn't too hard to do so for the spirals you're interested in. David
- References:
- Spirals and arc length
- From: "DIAMOND Mark" <noname@noname.com>
- Spirals and arc length