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From: hollasch@ENUXHA.EAS.ASU.EDU (Steve Hollasch)
Newsgroups: comp.graphics
Subject: How do YOU compute refraction vectors?
Summary: Here's my method; do you know of a better one?
Message-ID: <9105210459.AA01904@enuxha.eas.asu.edu>
Date: 21 May 91 04:59:18 GMT
Sender: daemon@ucbvax.BERKELEY.EDU
Reply-To: hollasch@enuxha.eas.asu.edu (Steve Hollasch)
Organization: Arizona State University
Lines: 70


    About a month ago I posted a request for a reference on the particular
refraction equation I used, but no one responded.  Could have been
disinterest, but perhaps there are people who haven't heard of it,
either.                                     _          _
                                        P1 |            |
    The equation I use is  R = (D.N)N + -- | D - (D.N)N |, where P1 is the
                                        P2 |_          _|

index of refraction of the material containing the ray origin, P2 is
the index of refraction of the intersected object, N is the surface
normal, and D is the ray direction vector TOWARDS the surface.  Note that
the resulting vector is NOT a unit vector.

    Does anybody know of a more efficient equation?  How do YOU compute
the refraction vector?

    For those of you who are interested, the derivation of the above
equation follows.

//////////////////  WARNING -- DERIVATION AHEAD  /////////////////////

N: Normal vector                    N
                                  /|\
b: "horizontal" component          |
   of vector D.                    |  b
                                   |/........o
a: Negative "vertical"          a /:\      _/
   component of vector D.          :     _/
                                   :   _/
                       P1          :A_/ D
                       ____________:/___________
                                  /:
                       P2        / :
                                /A':
                               /   :
                              /    :
                             /     :
R: Refraction vector     R |/.....\:/ -a
                             \ kb  |
                                   |
                                  \|/ -N


               || b ||     || D - (D.N)N ||
    sin(A)  =  -------  =  ----------------
               || a ||      || - (D.N)N ||

               || kb ||     k || D - (D.N)N ||
    sin(A') =  --------  =  ------------------
               || -a ||        || (D.N)N ||

    By Snell's law,  P1 sin(A) = P2 sin(A'),  so  (P1/P2) sin(A) = sin(A').
Using this equality and combining the above two equations, we get

             k || D - (D.N)N ||      P1 || D - (D.N)N ||
             ------------------  =   -- ----------------  
                || (D.N)N ||         P2  || - (D.N)N ||

Since  || (D.N)N || = || - (D.N)N ||,  this equation simplifies to
k = (P1/P2).  So, solving for the refraction vector R, we arrive at the
following equation:
                                   _          _
                               P1 |            |
    R  =  -a + kb  =  (D.N)N + -- | D - (D.N)N |
                               P2 |_          _|

______________________________________________________________________________
Steve Hollasch                /      Arizona State University (Tempe, Arizona)
hollasch@enuxha.eas.asu.edu  /  uunet!mimsy!oddjob!noao!asuvax!enuxha!hollasch