Scalars and Vectors - Vector Analysis for Computer Graphics - page 4

We can move from A to C directly or indirectly. The direct path is − AC, and the indirect path
is via B, i.e., − AB

+ − BC. Since both paths take us from A to C, we declare them equivalent, even
though the Euclidean distances are not the same. Thus, we can state

− AC

= − AB + − BC (1.1)

or, going the opposite way,

− − BC
You may have noticed from Eq. (1.1) that the letters A and C in − AC are the first and last letters
in − AB

− − AC =− − AB

+ − BC. The Bs have effectively been cancelled. This pattern occurs frequently when we
manipulate these labels.

D

BD

DC

B

AD

BC

AB

C

AC

A

Figure 1.5.

Finally, let's add a fourth point, D, as shown in Fig. 1.5. The line segments connecting A B,
and C have not changed, but we have introduced line segments − AD − BD, and − DC, which open
up paths between any two points directly or indirectly. Although there may only be one direct
route, there can be many indirect routes. For example, we can move from A to B in two ways:

A to C and then C to B

or

A to D and then D to B

But wait! We can also go from A to B via C and then D. But hold on! We can also go from
A to B via D and then C, i.e., A to D D to C and then C to B.
They are all valid paths, and as more and more points are added, the number of indirect
paths increases. But should we worry about all these indirect routes? Well, the answer is no.
What is important is that we have a way of annotating a route, and this is influenced by the
labels originally assigned to the line segments. For example, if we want to move from Q to P
but the connecting line segment is labeled − PQ, then we use − − PQ to represent − QP. When we