Archemedies Spiral

rhimmelblau

Hi I'm working on a project where I need to develop a flight path to cover a circular area. I was thinking of having the plane follow an archenemies spiral. I found that the general equation in polar is r=a(theta)^1/n
My question is if I have a specific distance I want each spiral to be from the last how do I input that into the equation.
Also is there a Cartesian equation for the spiral?
Edit: Also how does one calculate the arc length of the spiral?

tadchem

The equation for the Archimedes spiral is:
R = a*theta
Each turn is separated from the last (and the next) by a distance (measured radially) of
d = a*(2*pi)
http://mathworld.wolfram.com/ArchimedesSpiral.html
There is no Cartesian Equation because it is not a single-valued function in cartesian space; there is no single value of x (or y) that can be associated with a given value of y (or x).


P.S. Don't tell your archenemies. Make them figure it out for themselves.

rhimmelblau

So if I have a radius that I need to search and a distance each successive turn should be from the last, then I can use the arc length equation s=0.5*a[theta*sqrt(1+theta^2)+ln(theta+sqrt(1+theta^2))].
So I would plug in "a" equal to my distance/2*pi,
Then theta would be how many turns I go around the circle, which I can find by adding up the incremental distances between the spirals until I reach the radius of the search area.

Correct me if you see any flaws in my logic.