Graphics Reference
In-Depth Information
=
c 3 (a 1 b 2
a 2 b 1 ) e 1
e 2
e 3 +
c 1 (a 2 b 3
a 3 b 2 ) e 2
e 3
a
b
c
e 1
e 2
= (c 3 (a 1 b 2 a 2 b 1 ) + c 1 (a 2 b 3 a 3 b 2 ) + c 2 (a 3 b 1 a 1 b 3 )) e 1
+
c 2 (a 3 b 1
a 1 b 3 ) e 3
e 1
e 2
e 3
or using a determinant:
a 1
b 1
c 1
a
b
c
=
a 2
b 2
c 2
e 1
e 2
e 3
a 3
b 3
c 3
which is the well-known expression for the volume of a parallelpiped formed by
three vectors.
The term e 1
e 3 is a trivector and implies that the volume is oriented. If
the sign of the determinant is positive, the original three vectors possess the same
orientation of the three basis vectors. If the sign of the determinant is negative, the
three vectors oppose the orientation of the basis trivector.
e 2
6.9 Axioms
One of the features of geometric algebra is that it behaves very similar to the every-
day algebra of reals:
Axiom 6.1 The associative rule:
a ( bc )
=
( ab ) c .
Axiom 6.2 The left and right distributive rules:
+
=
+
a ( b
c )
ab
ac
( b
+
c ) a
=
ba
+
ca .
The next four axioms describe how vectors interact with a scalar λ :
Axiom 6.3
a ) b
=
λ( ab )
=
λ ab .
Axiom 6.4
=
λ(φ a )
(λφ) a .
Axiom 6.5
λ( a
+
b ) = λ a
+ λ b .
Axiom 6.6
+
φ) a
=
λ a
+
φ a .
Search WWH ::




Custom Search