gl modelview matrices


Just a quick question. Why does the modelview matrix have a 4th column for transformations. It was my understanding that you can define transformations of point in 3d space by using a 3x3 matrix.

Also what is the 4th row for?

Thanks For Clearing this up. If possible could someone please point me to some detailed clear docs so I can show my maths teacher as he is also interested.

Many Thanks

You can only define combinations of scaling and rotation using a 3x3 matrix.

For translations, you have to use something called “homogenous coordinates”, that is, each point in 3d space is represented by a straight line in 4d space by intersecting the line with the w=1 hyperplane (that is, divide each component by w).

All important 3d transformations including translation and projection can be done with a 4d matrix in homogenous coordinates.

Do a google search for “homogenous coordinates”, and you’ll find a lot of detailed papers on this topic.

Well, perhaps it could be explained simplier:
x’ = ax + by + cz + dw
This is transformation of x coordinate of vertex defined as (x, y, z, w) by correspnding matrix row (a, b, c, d). I don’t use the a1, a2, a3 notation, because we will focus on just one row of this matrix to transform just x coordinate of vertex.

If you want to scale then you have:
(a, b, c, d) = (scale, 0, 0, 0)
x’ = scalex + 0y + 0z + 0w
Which gives: x’ = scale * x

If you want to rotate then you have:
(a, b, c, d) = (cos(angle), -sin(angle), 0, 0)
x’ = cos(angle)x - sin(angle)y + 0z + 0w
Which gives: x’ = cos(angle)*x - sin(angle)*y

If you want to translate you have:
(a, b, c, d) = (1, 0, 0, x_translation)
x’ = 1x + 0y + 0z + x_translationw
Which gives: x’ = x + x_translation*w
Since w coordinate of every vertex is usually set to 1 you get:
x’ = x + x_translation

If you define a vertex with .w=0 then it will be immune to glTranslate, but will react to glRotate/glScale.