Day: May 15, 2009

Mowing video moved

Jumpcut is closing, so I needed to move this video to youtube. This relates to my earlier posts here
http://www.anderswallin.net/2007/12/mowing-tactics/
and here
http://www.anderswallin.net/2007/06/an-emergent-spiral/

When I find time to work on this next, there are many ideas for improvements: How to specify only climb/conventional milling (allowing only the right or left side of the cutter to be used). Using a variable step length for the simulation. Simulating dynamics of the macing (controlling the tool with a trajectory generator with acceleration/speed limits etc). How to implement rapid feed between cutting moves? how to choose among many allowed starting points for the cut? Should this use an adaptive resolution model, like a quad-tree? How should G-code be output, a filter which outputs G-code within a specified tolerance of the simulated path would probably be best?

Uniform random points in a circle using polar coordinates

I need this seldom enough to forget how it's done - but then it's annoying to have to think/google for the solution again when I do need it... So I'll document here.

The task is to generate uniformly distributed numbers within a circle of radius R in the (x,y) plane. At first polar coordinates seems like a great idea, and the naive solution is to pick a radius r uniformly distributed in [0, R], and then an angle theta uniformly distributed in [0, 2pi]. BUT, you end up with an exess of points near the origin (0, 0)!  This is wrong because if we look at a certain angle interval, say [theta, theta+dtheta], there needs to be more points generated further out (at large r), than close to zero. The radius must not be picked from a uniform distribution, but one that goes as

pdf_r = (2/R^2)*r

That's easy enough to do by calculating the inverse of the cumulative distribution, and we get for r:

r = R*sqrt( rand() )

where rand() is a uniform random number in [0, 1]. Here is a picture:

fig2

some matlab code here.

The thinking for generating random points on the surface of a sphere in 3D is very similar. If I get inspired I will do a post on that later, meanwhile you can go read these lecture notes.