Graphics Reference
In-Depth Information
It is obvious that a
c , which confirms that the associative law does not hold
for the vector triple product. So the answer to the second question we raised at the beginning
of this section is no.
From what we have discovered above, there does not seem to be any geometric significance
to the vector triple product, so the answer to the third question is also no.
Those who wish to examine the derivations for Eqs. (2.20) and (2.21) will find them in
Appendix B.
Before concluding this chapter, let's explore a simple application for the scalar triple product.
One popular method for detecting back-facing polygons is to compute the angle between the
polygon's surface normal and the vector representing the observer's line of sight. This is shown
in Fig. 2.28.
×
b
×
c
=
a
×
b
×
n
θ
a
Figure 2.28.
If the angle between vectors a and n equals, or exceeds 90 , the polygon is invisible. But say
such a normal was not available. Then the scalar triple product could be used instead.
First, we must know the order of the polygon's vertices as viewed from the outside. Let's
assume that this is counter-clockwise. We then establish two vectors, b and c , such that their
cross product points outwards. For example, in Fig. 2.29 the vertices A B, and C would produce
the vectors b
= AB and c
= AC. We create a third vector a pointing towards the observer
= AO. This creates a tetrahedron whose volume can be positive, zero,
or negative, depending on the orientation of the observer and polygon.
located at O, such that a
B
b
C
c
A
a
O
Figure 2.29.
The tetrahedron's volume is given by
x a
y a
z a
1
6 abc
=
1
6 a
·
b
×
c
=
1
6
x b
y b
z b
x c
y c
z c
 
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