Game Development Reference

In-Depth Information

Figure 3-3. Force and torque

We said earlier that you find the magnitude of torque by multiplying the magnitude of

the applied force times the perpendicular distance between the axis of rotation and the

line of action of the force. This calculation is easy to perform in two dimensions where

the perpendicular distance (
d
in
Figure 3-3
) is readily calculable.

However, in three dimensions you'll want to be able to calculate torque by knowing only

the force vector and the coordinates of its point of application on the body relative to

the axis of rotation. You can accomplish this by using the following formula:

M
=
r
×
F

The torque,
M
, is the vector cross product of the position vector,
r
, and the force vector,

F
.

In rectangular coordinates you can write the distance, force, and torque vectors as fol‐

lows:

r
= x
i
+ y
j
+ z
k

F
= F
x
i
+ F
y
j
+ F
z
k

M
= M
x
i
+ M
y
j
+ M
z
k

The scalar components of
r
(
x
,
y
, and
z
) are the coordinate distances from the axis of

rotation to the point of application of the force,
F
. The scalar components of the torque

vector,
M
, are defined by the following:

M
x
= y F
z
- z F
y

M
y
= z F
x
- x F
z