C[omp]ute

From: "andrew cooke" <andrew@...>

Date: Sat, 15 Sep 2007 16:16:32 -0400 (CLT)

I'm modelling a set of colliding lines (rods in 2D) and using Napito to
plot their trajectories.

Modelling individual lines is easy; the problem is reliably detecting
collisions.  In particular, I need to find the *first* collision amongst
all lines (since the movements change after a collision I am simply
restarting after that point - a future optimisation might do something
more sophisticated).

So this is some kind of search.  The trouble is that I don't have an
analytic solution for collisions.  Instead I have a fairly efficient but
pessimistic (it may give false positives) method for determining whether
any two lines touch within a given interval and an iterative method for
determining the time of intersection within some interval that is only
reliable in the limit of small intervals.

Finding the first possible pair of lines ("candidates") is easy - just use
the pessimistic method over progressively smaller time intervals.  But the
"limit of small intervals" makes everything uncertain and possibly
expensive - if you narrow down on an interval to search, and then find
that that it was unsuccessful (which is possible, since the initial
process is pessimistic), then what do you do?  It seems that you are
forced into a fairly detailed systematic search over time (and of
necessarily small intervals).

Fortunately, I can implement the candidate test in a way that makes it
progressively less pessimistic as the time interval decreases (it's
asymptotically correct).  So an explicit search is not that expensive. 
What becomes an issue then is the efficient broadening of the search to
include other candidates if the initial "best" fails.

This suggests that the best approach is to use a depth first search in
time (consider the tree that successively divides intervals in half),
discarding lines when they are not part of the overlapping group as you
descend, pruning when there are no candidates, and using the iterative
method as a test only at the very bottom "leaf" intervals.

In retrospect that seems obvious, but it's taken me a heck of a time to
see it clearly.  I keep being tempted to use the iterative method sooner
(when it works it converges rapidly, but when it fails I am left with no
way to fold that knowledge back into the search).

I guess there may be a future optimisation which uses the iterative
approach in some kind of speculative manner.  Perhaps when I am more
confident about its properties (it's just an analytic solution to a linear
approximation).

Andrew