In article <403e78b2@news.povray.org>, Wolfgang Wieser <wwi### [at] gmxde> 
wrote:

> The idea is that in case we cross the isosurface threshold and 
> one point is outside (EP1) while the other one is inside (EP2), 
> then the average of the two depth values should give a better 
> estimate for the actual intersection than merely using EP2. 

Linear interpolation is a pretty common last step in the bisection 
algorithm...I did it for the blob2 object, I wasn't aware the isosurface 
object didn't do it.

This is also close to the regula falsi or false secant method. The idea 
with this method is that when you have a pair of points that bracket a 
root, approximate the function as linear within that interval and use 
the computed root as the new splitting point. It is similar to Newton's 
method, but doesn't require a derivative of the function. It converges 
faster than bisection in most cases, but is only guaranteed to converge 
if you start out with a bracketed root. I've considered using a variant 
of this as a second stage to the isosurface root solver...use bisection 
to locate the roots to within a certain crude precision, and then use 
regula falsi to refine the roots.

-- 
Christopher James Huff <cja### [at] earthlinknet>
http://home.earthlink.net/~cjameshuff/
POV-Ray TAG: <chr### [at] tagpovrayorg>
http://tag.povray.org/

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