Game Development Reference

In-Depth Information

Vector Differential Calculus

So far we've looked at differentiation purely in terms of a single scalar quantity.

For full three-dimensional physics we need to deal with vector positions rather than

scalars.

Fortunately the simple calculus we've looked at so far works easily in three di-

mensions (although you must be careful: as a general rule, most of the equations for

one-dimensional calculus you find in mathematics reference books cannot be used in

three dimensions).

If the position of the object is given as a vector in three dimensions, then its rate

of change is also represented by a vector. Because a change in the position on one axis

doesn't change the position on any other axes, we can treat each axis as if it were its

own scalar differential.

The velocity and the acceleration of a vector depends only on the velocity and

acceleration of its components:

⎡

⎣

⎤

⎦

a
x

˙

˙

˙

a

=

a
y

˙

a
z

and similarly

⎡

⎤

a
x

¨

¨

⎣

⎦

a

¨

=

a
y

¨

a
z

As long as the formulae we meet do not involve the products of vectors, this

matches exactly with the way we defined vector addition and vector-scalar multi-

plication earlier in the chapter. The upshot of this is that for most formulae involving

the differentials of vectors, we can treat the vectors as if they were scalars. We'll see an

example of this in the next section.

As soon as we have formulae that involve multiplying vectors together, or that

involve matrices, however, things are no longer as simple. Fortunately they are rare in

this topic.

Velocity, Direction, and Speed

Although in regular English we often use
speed
and
velocity
as synonyms, they have

different meanings for physics development. The velocity of an object, as we've seen,

is a vector giving the rate at which its position is changing.

The speed of an object is the magnitude of this velocity vector, and the direction

that the object is moving in is given by the normalized velocity. So

s
d

x

˙

=