Game Development Reference
In-Depth Information
depicts, you can establish a domain that consists of the set {1, 2, 3, 4}. You can
establish a range that consists of the set {y, z}.
Cartesian Coordinates
When Rene Descartes (1596-1650) invented the system of mapping coordinates
that bears his name, he was lying ill in bed, watching a fly negotiate the
space above him. To be able to determine the location of the fly as it buzzed above
him, Descartes imagined a flattened world in which the fly moved around on a
plane that roughly corresponded in dimensions to the ceiling.
To make it so he could trace the path of the fly on the plane, the mathematician
drew axes that divided the ceiling area into four quadrants. He labeled the
quadrants using Roman numerals, as shown in Figure 4.9. At the intersection he
placed the origin of his mapping. The axes he divided into equal segments
identified with positive and negative values. The vertical axis he labeled as the
y
axis. The horizontal axis he labeled as the
x
axis.
The horizontal axis of the Cartesian plane represents domain values. The
x
axis
of the Cartesian plane represents range values. Each point on the Cartesian
plane is a coordinate, and each coordinate consists of an ordered pair of values.
The
x
value always occurs first in an ordered pair. The
y
value always occurs
second.
Mathematicians sometimes refer to the first number in an ordered pair as the
abscissa
. They sometimes refer to the second number as the
ordinate
. Each
number is a coordinate. The ordered pair designates a point within the quadrants
of the coordinate system. At the center of the system, where the two axes
intersect, you find the
origin
of the system.
If you picture the system upright, the quadrants of the Cartesian plane are set
up so that quadrant I is in the upper right. You then move counter-clockwise
through the quadrants. Quadrant II is on the top left. Quadrant III is on
the bottom on the left. Quadrant IV lies on the bottom on the right. If you
trace the flight of the fly, you can discern that the positive and negative values
of the ordered pairs take on a specific pattern. Table 4.1 provides a summary of
this pattern.
