Graphics Reference
In-Depth Information
widely accepted until 1881, when the American mathematician Josiah Gibbs
(1839-1903) published his treatise
Vector Analysis
, describing modern vector
analysis.
6.1 2D Vectors
In computer graphics we employ 2D and 3D vectors. In this chapter we
first consider vector notation in a 2D context and then extrapolate the ideas
into 3D.
6.1.1 Vector Notation
A scalar such as
x
is just a name for a single numeric quantity. However,
because a vector contains two or more numbers, its symbolic name is printed
using a
bold
font to distinguish it from a scalar variable. Examples are
n
,
i
and
Q
.
When a scalar variable is assigned a value we employ the standard algebraic
notation
x
=3
However, when a vector is assigned its numeric values, the following notation
is used:
n
=
3
4
which is called a
column
vector. The numbers 3 and 4 are called the
compo-
nents
of
n
, and their position within the brackets is significant. A
row
vector
transposes the components horizontally,
n
=[34]
T
where the superscript
T
reminds us of the transposition.
6.1.2 Graphical Representation of Vectors
Because vectors have to encode direction as well as magnitude, an arrow
could be used to indicate direction and a number to specify magnitude. Such
a scheme is often used in weather maps. Although this is a useful graphical
interpretation for such data, it is not practical for algebraic manipulation.
Cartesian coordinates provide an excellent mechanism for visualizing vec-
tors and allowing them to be incorporated within the classical framework of
mathematics. Figure 6.1 shows a vector represented by a short line segment.
The length of the line represents the vector's magnitude, and the orienta-
tion defines its direction. But as you can see from the figure, the line does
not have a direction. Even if we attach an arrowhead to the line, which is
standard practice for annotating vectors in topics and scientific papers, the
arrowhead has no mathematical reality.