Game Development Reference

In-Depth Information

Figure 2-11. Tangential and centripetal acceleration

The formula for the magnitude of centripetal acceleration,
a
n
, is:

a
n
= v
2
/r

where
v
is the tangential velocity. Substituting the equation for tangential velocity into

this equation for centripetal acceleration gives the following alternative form:

a
n
= r ω
2

In two dimensions you can easily get away with using these scalar equations; however,

in three dimensions you'll have to use the vector forms of these equations. Angular

velocity as a vector is parallel with the axis of rotation. In
Figure 2-10
the angular velocity

would be pointing out of the page directly at you. Its sense, or direction of rotation, is

determined by the
right hand rule
. If you curl the fingers of your right hand in an arc

around the axis of rotation with your fingers pointing toward the direction in which

the body is rotating, then your thumb will stick up in the direction of the angular velocity

vector.

If you take the vector cross product (refer to the sidebar
“Vector Cross Product” on page

67
for background and to
Appendix A
for a review of vector math) of the angular velocity

vector and the vector from the axis of rotation to the particle under consideration, you'll

end up with the linear, tangential velocity vector. This is written as follows:

v
=
ω
×
r