Game Engine Architecture

142 4. 3D Math for Games

a+ b

• b b

a–b a

Figure 4.6. Vector addition and subtraction.

The scale factor can be diff erent along each axis. We call this nonuniform

scale , and it can be represented as the component-wise product of a scaling vector

s and the vector in question, which we’ll denote with the ⊗ operator. Techni-

cally speaking, this special kind of product between two vectors is known as

the Hadamard product. It is rarely used in the game industry—in fact, nonuni-

form scaling is one of its only commonplace uses in games:

(4.1)

As we’ll see in Section 4.3.7.3, a scaling vector s is really just a compact way to

represent a 3 × 3 diagonal scaling matrix S. So another way to write Equation

(4.1) is as follows:

4.2.4.2. Addition and Subtraction

The addition of two vectors a and b is defi ned as the vector whose components

are the sums of the components of a and b. This can be visualized by placing the

head of vector a onto the tail of vector b—the sum is then the vector from the

tail of a to the head of b:

a + b = [ (ax + bx), (ay + by), (az + bz) ].

Vector subtraction a – b is nothing more than addition of a and –b (i.e., the

result of scaling b by –1, which fl ips it around). This corresponds to the vector

v 2 v

v

Figure 4.5. Multiplication of a vector by the scalar 2.

sa⊗=(sa sa saxxyyzz, , ).

00

[ ] 0 0 [ ].

00

x

x y z y xxyyzz

z

s

aaa s sasasa

s

⎡⎤

==⎢⎥

⎢⎥

⎢⎥⎣⎦

aS