Re: [ATM] Raytracing Function

["Donald Good" <donald.good@comcast.net>]

How about this (in the C style)?

float OffsetRt(float a, float x ) {
   if (a<0) Return (sqrt( a*a + x ) - a);
   else Return (x/(sqrt( a*a + x) + a));
}


-----Original Message-----
From: atm-bounces@atmlist.net [mailto:atm-bounces@atmlist.net] On Behalf Of
Jim Burrows
Sent: Friday, September 22, 2006 5:56 PM
To: ATM List
Subject: Re: [ATM] Raytracing Function

At 2006-09-22 11:23 -0400, Richard F.L.R. Snashall wrote:

>One function that keeps recurring in [algebraic] raytracing is what I 
>have come to call an offset root (or offset square root):
>
>     OffsetRt( a, x ) = sqrt( a^2 + x ) - a
>
>     root = root + ( |a| - a )

You will still run into the classical numerical problem:  large - large =
small, with loss of precision.  Sometimes it can be bad enough that even IEE
extended reals (19 digits) will give poor answers.  The solution is
"rationalizing the numerator":

sqrt( a² + x) - a = (sqrt( a² + x) - a)*(sqrt( a² + x) + a)/(sqrt( a² + x) 
+ a) = (a² + x - a²)/(sqrt( a² + x) + a) = x/(sqrt( a² + x) + a),

works fine and is used in lots & lots of places in my ATM programs.

         -- Jim Burrows
         -- mailto://burrjaw@earthlink.net
         -- http://home.earthlink.net/~burrjaw
         -- Seattle N47.4723 W122.3662 (WGS84)  

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