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