Game Development Reference
In-Depth Information
The position of the right triangle you create when you generate a figure using
linear equations gives you a point of reference that centers on the origin of the
Cartesian plane. For a triangle you depict the angle according to the standard
model discussed in the previous section; this is angle A. A convenient way to label
an angle is to use the Greek letter theta,
. Accordingly, the angle of
is always
relative to the run and rise of the triangle.
As Figure 12.3 illustrates, if you work within the first coordinate of the Cartesian
plane, as the angle of
increases, the slope of the line becomes more pronounced.
The length of the opposite side increases. The length of the adjacent side
decreases. As you saw when you investigated slopes, the ratio of the adjacent to
the opposite side allows you to determine a value for
.
Circles and the Sine of 2
You can transfer a right triangle you construct in a Cartesian coordinate system
to a circle. When you translate the triangle to a circle, the center of focus remains
on the angle that characterizes the origin (
). As Figure 12.4 illustrates, the
hypotenuse of the triangle provides a way to establish the radius of the
circle. Along with the hypotenuse of the circle, you can also establish a chord
on the circle. The chord in this instance is one that is twice the length of
the opposite side. To create such a chord, you can flip the right triangle down-
ward, from quadrant I into quadrant IV. You identify the chord by using the
term 2
.
If you take half the distance of the cord, you have a line that you define using
alone. This line is known as the sine of
. The sine of theta, then, corresponds to
the ratio that you establish between the distance of the hypotenuse and the
distance of the opposite side.
y
r
sin
y ΒΌ
If you want to determine the sine of
as shown in Figure 12.4, you can put to
work the information the graph provides. Accordingly, if you work within
quadrant I of the Cartesian plane, you can see that the coordinate pair that
describes the point on the circumference of the circle for the line is (4,3).
To calculate the length of the line that extends from the center of the circle (or
the origin of the Cartesian plane), you use the Pythagorean theorem. Following
