Graphics Reference
In-Depth Information
Figure 6.16.
Computing the distance between
two lines.
orthogonal to both
v
and
w
(Theorem 4.5.12 in [AgoM04]). Expanding the two con-
ditions
AB
•
v
= 0 and
AB
•
w
= 0 reduces to the equations
t
w•v
-
s
v•v
= -
PQ•v
t
w•w
-
s
v•w
= -
PQ•w
with the indicated solutions. By the Cauchy-Schwarz inequality, the denominator D
is zero precisely when the vectors
v
and
w
are parallel.
6.7
Area and Volume Formulas
This section contains some more useful formulas. The “proofs” of these formulas will
rely on simple-minded geometric observations and will not be very rigorous. For rig-
orous proofs one would need to use a theory of areas and volumes. The most elegant
approach would be via differential forms. See [Spiv65] or Section 4.9 in [AgoM05].
Finally, the formulas below are “mathematical” formulas. For efficient ways to
compute them see [VanG95].
6.7.1
Formula.
The area A of a parallelogram defined by two vectors
u
and
v
in
R
2
is given by
u
v
Ê
Ë
ˆ
¯
A =¥=
uv
det
.
(6.27)
Proof.
The first equality follows from properties of the cross product and the fact
that A is the product of the height of the parallelogram and the length of its base. See
Figure 6.17(a). The second follows from direct computation of the cross product and
the indicated determinant.
6.7.2
Formula.
The volume V of a parallelopiped defined by vectors
u
,
v
, and
w
in
R
3
is given by
