Game Development Reference

In-Depth Information

where
u
is the impulsive torque,
I
is the inertia tensor, and
θ
is the angular velocity, as

before. This is the direct equivalent of equation 7.5, which dealt with linear impulses.

And correspondingly the change in angular velocity
θ
is

θ

I
−
1
u

=

[14.1]

In all these equations
I
should be in world coordinates, as discussed in section 10.2.3.

Impulses behave just like forces. In particular for a given impulse there will be

both a linear component and an angular component. Just as the amount of torque is

given by

τ

=

p
f
×

f

so the impulsive torque generated by an impulse is given by

u

=

p
f
×

g

[14.2]

In our case, for collisions the point of application (
p
f
) is given by the contact

point and the origin of the object:

p
f
=

q

−

p

where
q
is the position of the contact in world coordinates and
p
is the position of

the origin of the object in world coordinates.

14.1.2

R
OTATING
C
OLLISIONS

The effect of the impulse at the collision is to have the points of each object in collision

bounce apart. The movement of the colliding objects at the collision point still follows

the same equations that we met in chapter 7. In other words, if we tracked the two

collision points (one from each object) around the time of the collision, we'd see that

their separating velocity is given by

v
s
=−

cv
s

where
v
s
is the relative velocity of the objects immediately before the collision,
v
s

is the relative velocity after the collision, and
c
is the coefficient of restitution. In

other words, the separation velocity is always in the opposite direction to the closing

velocity, and is a constant proportion of its magnitude. The constant
c
depends on

the materials of both objects involved.

Depending on the characteristics of the objects involved, and the direction of the

contact normal, this separation velocity will be made up of a different degree of lin-

ear and rotational motion. Figure 14.2 shows different objects engaged in the same

collision (again illustrated with an unmoving ground for clarity). In each part of the