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Unformatted text preview: tion • The equation can be simpliﬁed if we assume that Ai is
small compared with r. !
• Thethis is the case, then wewe assume that A is small
If equation can be simpliﬁed if can treat the inner integral as
i
constant over
compared with r. the surface of Ai. !
If this is the case, then we can treat the inner integral as constant
over the surface of Ai . Fij = 1
Ai  Z Ai Z cos Aj cos
⇡r2
i j dAj dAi Graphics Lecture 13: Slide 22!
. Simplifying form factors "
Simplifying form factors
• With this assumption the outer integral evaluates to Ai
(i.e. the area of Ai). !
With this assumption the outer integral evaluates to Ai  (ie the
• Hence we
area of Ai ). can write the integral as: !
Hence we can write the integral as:
Z
cos i cos
Fij =
⇡r2
Aj Graphics Lecture 13: Slide 23! j dAj Further simplifying
Further simplifying Further simplifying " We assumed that the radius is large compared with patch Ai
We assumed that the radius is large compared with patch Ai . .
Shouldalso be reasonable to assume it is large compared tothe
also be
to assume it is
the
Shouldassumedreasonableradius is largelarge compared topatch
We
that the
compared with
size of A.
size .of Aj j . also be reasonable to assume it is large
A Should
i compared to the size of Aj. !
Hence the integrand of
Hence the integrand of
Hence the integrand ofZ
!
Z
cos cos
cos i icos j j dA
!
F=
Fijij=
dAj j
2
⇡r2
A
⇡r
Aj j
!
can similarly considered constant over
can similarly be be considered constantAj
can similarly be considered constant over Aover Aj
j
!
So weget the approximation
So wewe get the approximation !
So get the approximation
cos cos  A 
cos i icos j jAj j 
F=
Fijij=
⇡ 22
⇡ rr
Graphics Lecture 13: Slide 24!
. 20 29
20 / / 29 . Thehe Hemicube method "
T Hemicube method
Using bounding hemisphere it it can shown that all patches that
Using a a bounding hemisphere can bebe shown that all patches that
p...
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This document was uploaded on 03/26/2014.
 Spring '14

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