So there are alot of tutorials about rotation matrices, but what about transform matrices in general. How do you calculate a translation or a scaling matrix. Can you simply multiply translation, rotation and scaling matrices together to produce a single transform matrix? Why do OpenGL and D3D use 4x4 matrices? Wouldn''t it be simpler to do this in 3x3?

So is there much point in using 4x4 matrices if I wanted to implement my own internal matrix system, apart from compatibility.

What would keep a 3x3 matrix to represent translation?

the other advantage of 4x4 matricies besides translation is that the projection transformation can be stored in matrix form. to multiply by a 4x4 matrix, a vector must expand to 4 dimensions, its 4th element being a one (identity). After the transformation, the vector is transformed back into a 3 element vector by dividing by the 4th element, which may no longer be a one if matrix modified it.

Yeah, for some reason a lot of books/tutorials/papers seem to describe fundamental 3D topics in very vague or confusing ways. They either tend to simplify things to such a degree that you look at something, like a matrix, as a black box where you put numbers in and, like magic, results pop out. Or, they go the other way and describe it in confusing math jargon, with symbols and greek letters and 50 cent words that almost seem designed to confuse instead of educate.

Anyway, enough of that ranting... I''ll try to do my best to describe what a 3x3 and 4x4 matrix are and how to use them.

NOTE: I''ll talk in terms of a right-handed coordinate system. This means that if you stand at the origin, positive X is on your right, positive Y is up, and positive Z goes behind you. A left-handed system(ala DirectX) would be the same except positive Z would go forward from you.

First of all, regardless of how you''re working, the Identity of a 4x4 Matrix always looks like this:

1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1

Next, it''s good to know that matrices can be either row or column based. Column based matrices tend to be the most common, so I''ll use that for my examples. In a colum-based matrix you put the position in the last column, like so:

1 0 0 Pos.x
0 1 0 Pos.y
0 0 1 Pos.z
0 0 0 1

For row-based matrices, the position''s xyz coordinates would be placed in the first three numbers on the bottom row.

Ok, next we''ll back track slightly to a 3x3 matrices. A 3x3 matrix is typically used to describe rotation and/or scale. The Identity of 3x3 matrix looks like this

1 0 0
0 1 0
0 0 1

Ok great, but let''s stop for a second. What do those 9 numbers actually mean? And how do they store rotation and scale? While I seldom see this mentioned, the answers to these questions are actually very intuitive. Those 9 numbers represent points in 3D map onto each 3D axis(X Y Z). In those fancy math-books, they call these "Basis Vectors" Let''s re-write the matrix in this form:

XAxis.x YAxis.x ZAxis.x
XAxis.y YAxis.y ZAxis.y
XAxis.z YAxis.z ZAxis.z

So, of course, in standard world space(Nothing transformed), the x axis points down the x axis, the y down the y, and the z down the z. Which means that the vector that describes the XAxis is (1,0,0), the Y is (0,1,0), and the Z is (0,0,1). So, now we can see quite easily why the identity matrix looks the way that it does.

Ok, so what does this mean exactly? What it means is that if you take something and want to scale it by 3 along the X axis, that the matrix to describe it would look like this:

3 0 0
0 1 0
0 0 1

If you wanted to mirror something along the Y axis, you''d get a matrix that looks like this:

1 0 0
0 -1 0
0 0 1

Again, since all these numbers do is tell you the direction and length of each axis, it means you can also describe rotation with a 3x3 matrix. So, if you are standing at the origin and you turn 90 degrees to your left then the X axis, from your vantage point, would point down the world''s -Z axis. The Y axis would be the same as the world''s, and the Z axis would now point down the world''s positive X axis. We can represent that 90 degree rotation with this matrix:

0 0 1
0 1 0
-1 0 0

You can combine rotation and scale as well. So, if you did that same rotation and did the same scale by 3 in X that we did above, you''d get this matrix:

0 0 1
0 1 0
-3 0 0

Again, these 3 sets of 3 numbers simply represent how the XYZ coordinates map into another set of XYZ coordinates. In the examples above, we''ve mapped your viewpoint to that of the world.

Now, the intersting thing to be aware of here is that so far we''ve kept each axis'' direction the same relative to the other two axis(except in the mirror on Y example). The fact of the matter is that for most normal transforms this always holds true. However, you can also distort the axis to achieve interesting effects. For example, if you used a matrix where you left the X and Z axis the same as the identity, but bent the Y axis you could skew your object. A matrix to do that would look something like this:

1 1 0
0 1 0
0 0 1

One thing to note in the matrix above is that if you calculate the length of the Y vector(1,1,0) it is not 1. So, in this example, we skewed and scaled instead of just skewing. I just used 1''s and 0''s to make it easier to read

Ok, so that''s how 3x3 matrices work effectively. Just to recap again, a 3x3 matrix is a set of 3 vectors that describe how 3D values in one "space" transform into the 3 vectors in another "space".

Ok, next let''s quickly gloss over something they call "homogenous coordinates". In 3D space, we represent a point as 3 numbers(XYZ). We also represent a vector as 3 numbers(XYZ). However, in reality, vectors and points are not exactly the same thing. A vector describes a direction, but has no position. In contrast, a point has a position but no direction. Normally, although the two look exactly the same when written, we tend to do the appropriate thing with them based on the context we use them in.

However, if we move into 4D homogenous space, then there is a visible difference. In this space, we add a new component "w". A vectors w component is 0, and a point''s w component is 1. So, in this space, if we would describe a point and a vector like so:

Point( 2, 3, 7, 1 )
Vector( 2, 3, 7, 0 )

Also, you''ll notice that in a non-perspective-distorted 4x4 Matrix the bottom row is always ( 0 0 0 1 ). That''s because the bottom row is the "w" component. And since the first three columns are the XYZ vectors, their w component is 0. And since the last row represent''s position, which is a point, it''s w component is 1.

So, now we finally get to the meat of it all. Basically, a 4x4 matrix lets you represent a mapping of each of the 3 vectors(XYZ) as well as an offset position. Like so:

XAxis.x YAxis.x ZAxis.x OffsetPoint.x
XAxis.y YAxis.y ZAxis.y OffsetPoint.y
XAxis.z YAxis.z ZAxis.z OffsetPoint.z
XAxis.w YAxis.w ZAxis.w OffsetPoint.w

When you look at it like that, hopefully you''ll find that it''s actually pretty easy to understand

Well, I didn''t mean for this to get this long, but hopefully it helped!

-John