Game Development Reference

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3.

Prove Equation (4.35).

4.
Let
N
be the normal vector to a surface at a point
P
, and let
S
and
T
be tan-

gent vectors at the point
P
such that

ST N
. Given an invertible 3

×=

×

3

ma-

)
(

)

(

)

(

)

(

−

1T

(

)

trix
M
, show that

MS MT M M S T
, supporting the fact

that normals are correctly transformed by the inverse transpose of the matrix

M
. [
Hint.
Use Equation (2.25) to write the cross product

×

=

det

×

(

)

(

)

MS

×

MT
as

(

)

(

)

0

−

MS

MS

z

y

(

)

(

)

(

)

(

)

.

MS

×

MT

=

MS

0

−

MS

MT

z

x

(

)

(

)

−

MS

MS

0

y

x

Then find a matrix
G
such that

(

)

(

)

0

−

SS

0

−

MS

MS

z

y

z

y

(

)

(

)

G

S

0

−=

S

MS

0

−

MS

M

,

z

x

z

x

(

)

(

)

−

SS

0

−

MS

MS

0

y

x

y

x

)
(

)

(

−

1T

and finally use Equation (3.65) to show that

G

=

det

MM

.]

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