Vectors - Mathematics for Computer Graphics - page 29

6
Vectors
Vectors are a relatively new arrival to the world of mathematics, dating only
from the 19th century. They provide us with some elegant and powerful tech-
niques for computing angles between lines and the orientation of surfaces.
They also provide a coherent framework for computing the behaviour of dy-
namic objects in computer animation and illumination models in rendering.
We often employ a single number to represent quantities that we use in our
daily lives such as, height, age, shoe size, waist and chest measurements. The
magnitude of this number depends on our age and whether we use metric or
imperial units. Such quantities are called scalars . In computer graphics scalar
quantities include colour, height, width, depth, brightness, number of frames,
etc.
On the other hand, there are some things that require more than one
number to represent them: wind, force, weight, velocity and sound are just
a few examples. These cannot be represented accurately by a single number.
For example, any sailor knows that wind has a magnitude and a direction.
The force we use to lift an object also has a value and a direction. Similarly,
the velocity of a moving object is measured in terms of its speed (e.g. miles
per hour) and a direction such as north-west. Sound, too, has intensity and a
direction. These quantities are called vectors . In computer graphics, vectors
are generally made of two or three numbers, and this is the only type we will
consider in this chapter.
Mathematicians such as Caspar Wessel (1745-1818), Jean Argand (1768-
1822) and John Warren (1796-1852) were simultaneously exploring complex
numbers and their graphical representation. In 1837, Sir William Rowan
Hamilton (1788-1856) made his breakthrough with quaternions. In 1853,
Hamilton published his topic Lectures on Quaternions in which he described
terms such as vector, transvector and provector . Hamilton's work was not