Pro OpenGL ES for iOS

36 CHAPTER 2: All That Math Jazz^

Mathematicians are always fond of expressing things in the most compact form

possible. So 2D rotations can be ‘‘simplified’’ using matrix notation:

⎥

⎦

⎤

⎢

⎣

⎡ −

=

sin( ) cos( )

cos( ) sin( )

a a

a a

Ra

Note One of the most overused words in Star Trek is matrix. Pattern-matrix here, buffer-

matrix there—“Number One, I have a headache-matrix and need to take a nap-matrix.” (And

don’t get me started on the use of protocol in 24 .) Every self-respecting Star Trek drinking

game (as if any drinking game would be self-respecting) should use matrix in its selection of

words.

Ra is shorthand for our 2D rotation matrix. Although matrices might look busy, they are

actually pretty straightforward and easy to code because they follow precise patterns. In

this case, x and y can be represented as a teeny matrix:

⎥

⎦

⎤

⎢

⎣

⎡

⎥

⎦

⎤

⎢

⎣

⎡ −

⎥=

⎦

⎤

⎢

⎣

⎡

y

x

a a

a a

y

x

sin( ) cos( )

cos( ) sin( )

'

'

Translations can also be encoded in a matrix form. Because translations are merely

moving the point around, the translated values of x and y come from adding the amount

of movement to the point. What if you wanted to do a rotation and a translation on the

same object? The translation matrix requires just a tiny bit of nonobvious thought. Which

is the right one, the first or second shown here?

⎥

⎦

⎤

⎢

⎣

⎡

=

Tx Ty

T

1 1

or

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

=

1

0 1 0

1 0 0

Tx Ty

T

The answer is obviously the second one, or maybe it’s not so obvious. The first one

ends up as the following, which doesn’t make much sense:

x′=x+yTx and y'=x+yTy

So, to create a matrix for translation, we need a third component for our 2D point,

commonly written as (x,y,1), as is the case in the second expression. Ignoring where the

1 comes from for a moment, notice that this can be easily reduced to the following:

x′=x+Tx and y′=y+Ty

The value of 1 is not to be confused with a third dimension of z; rather, it is a means

used to express an equation of a line (in 2D space for this example) that is slightly

different from the slope/intercept we learned in grade school. A set of coordinates in this