# Converting to and from local and world 3D coordinate spaces?

I've been following a guide I found here (http://knol.google.com/k/matrices-for-3d-applications-view-transformation) on constructing a matrix that will allow me to 3D coordinates to an object's local coordinate space, and back again. I've tried to implement these two matrices using my object's look, side, up and location vectors and it seems to be working for the first three coordinates. I'm a little confused as to what I should expect for the w coordinate.

Here are couple of examples from the print outs I've made of the matricies that are constructed. I'm passing a test vector of [9, 8, 14, 1] each time to see if I can convert both ways:

Basic example:

localize matrix:
Matrix: 0.000000    -0.000000   1.000000    0.000000
0.000000    1.000000    0.000000    0.000000
1.000000    0.000000    0.000000    0.000000
5.237297    -45.530716  11.021271   1.000000
globalize matrix:
Matrix: 0.000000    0.000000    1.000000    0.000000
-0.000000   1.000000    0.000000    0.000000
1.000000    0.000000    0.000000    0.000000
-11.021271  -45.530716  -5.237297   1.000000

test:
Vector4f(9.000000, 8.000000, 14.000000, 1.000000)
localTest:
Vector4f(14.000000, 8.000000, 9.000000, -161.812256)
worldTest:
Vector4f(9.000000, 8.000000, 14.000000, -727.491455)


More complicated example:

localize matrix:
Matrix: 0.052504    -0.000689   -0.998258   0.000000
0.052431    0.998260    0.002068    0.000000
0.997241    -0.052486   0.052486    0.000000
58.806095   2.979346    -39.396252  1.000000
globalize matrix:
Matrix: 0.052504    0.052431    0.997241    0.000000
-0.000689   0.998260    -0.052486   0.000000
-0.998258   0.002068    0.052486    0.000000
-42.413120  5.975957    -56.419727  1.000000

test:
Vector4f(9.000000, 8.000000, 14.000000, 1.000000)
localTest:
Vector4f(-13.508600, 8.486917, 9.290090, 2.542114)
worldTest:
Vector4f(9.000190, 7.993863, 13.990230, 102.057129)


As you can see in the more complicated example, the coordinates after converting both ways loose some precision, but this isn't a problem. I'm just wondering how I should deal with the last (w) coordinate? Should I just set it to 1 after performing the matrix multiplication, or does it look like I've done something wrong?

• kudos to ya ... that just went way over my head :) – War Mar 18 '13 at 16:21

It looks like you are confusing rows and columns in your matrices, either in the way your load or store them, or when you perform the matrix×vector multiplication. The w coordinate should always yield 1 with the matrices you are using.

• Hi - thanks for the reply. My matrix is in row vector notation and the vectors that I'm multiplying are column vectors. I've multiplied the same matrices and vectors in matlab and get the same answers. – James Bedford Mar 6 '11 at 14:15
• Ahhh I've just tried the tanspose of this matrix in Matlab multiplied with the vector and the w coordinate yields 1, so when I'm constructing my transformation matrices I must be using the wrong matrix notation. – James Bedford Mar 6 '11 at 14:21