Graphics Reference
In-Depth Information
Fig. 6.1
Two line segments
a
and
b
separated by
+
β
6.3 Trigonometric Foundations
Figure
6.1
shows two line segments
a
and
b
with coordinates
(a
1
,a
2
)
and
(b
1
,b
2
)
respectively. The lines are separated by an angle
β
, and it is a trivial exercise to
show that
ab
sin
β
=
a
1
b
2
−
a
2
b
1
which equals the area of the parallelogram formed by
a
and
b
. What is interesting is
that reversing the relative orientation of the lines such that
b
is rotated
−
β
relative
to
a
makes
ab
sin
β
=−
(a
1
b
2
−
a
2
b
1
)
which means that this is antisymmetric due to the sine function.
We know from the definition of the scalar product of vectors that
ab
cos
β
=
a
1
b
1
+
a
2
b
2
which remains unaltered if the relative orientation of the lines is reversed, which
means that this is symmetric due to the cosine function.
6.4 Vectorial Foundations
If we form the algebraic product of two 2D vectors
a
and
b
we have:
a
=
a
1
i
+
a
2
j
b
=
b
1
i
+
b
2
j
a
1
b
1
i
2
a
2
b
2
j
2
ab
=
+
+
a
1
b
2
ij
+
a
2
b
1
ji
.
(6.1)
It is clear from (
6.1
) that
a
1
b
1
i
2
a
2
b
2
j
2
+
has something to do with
ab
cos
β
, and
a
1
b
2
ij
a
2
b
1
ji
has something to do with
ab
sin
β
. The product
ab
creates the terms
i
2
,
j
2
,
ij
and
ji
, which are resolved as follows.
+
