Matrix
Matrix
Definition
A Matrix is typically used as a linear transformation to map vectors to vectors. That is: Y = MX where X is a Vector and M is a Matrix applying any or all transformations (scale, rotate, translate).
There are a few special matrices:
zero matrix is the Matrix with all zero entries.
| 0 | 0 | 0 |
|---|---|---|
0 |
0 |
0 |
0 |
0 |
0 |
The Identity Matrix is the matrix with 1 on the diagonal entries and 0 for all other entries.
| 1 | 0 | 0 |
|---|---|---|
0 |
1 |
0 |
0 |
0 |
1 |
A Matrix is invertible if there is a matrix M-1 where MM-1 = M-1 = I.
The transpose of a matrix M = [mij] is MT = [mji]. This causes the rows of M to become the columns of MT.
| 1 | 1 | 1 | 1 | 2 | 3 | |
|---|---|---|---|---|---|---|
2 |
2 |
2 |
⇒ |
1 |
2 |
3 |
3 |
3 |
3 |
1 |
2 |
3 |
A Matrix is symmetric if M = MT.
| X | A | B |
|---|---|---|
A |
X |
C |
B |
C |
X |
Where X, A, B, and C equal numbers
jME includes two types of Matrix classes: Matrix3f and Matrix4f. Matrix3f is a 3x3 matrix and is the most commonly used (able to handle scaling and rotating), while Matrix4f is a 4x4 matrix that can also handle translation.
Transformations
Multiplying a Vector with a Matrix allows the Vector to be transformed. Either rotating, scaling or translating that Vector.
Scaling
If a diagonal Matrix, defined by D = [dij] and dij = 0 for i != j, has all positive entries it is a scaling matrix. If di is greater than 1 then the resulting Vector will grow, while if di is less than 1 it will shrink.
Rotation
A rotation matrix requires that the transpose and inverse are the same matrix (R-1 = RT). The rotation matrix R can then be calculated as: R = I + (sin(angle)) S + (1 - cos(angle)S2 where S is:
| 0 | u2 | -u1 |
|---|---|---|
-u2 |
0 |
u0 |
u1 |
-u0 |
0 |
Translation
Translation requires a 4x4 matrix, where the Vector (x,y,z) is mapped to (x,y,z,1) for multiplication. The Translation Matrix is then defined as:
| M | T |
|---|---|
ST |
1 |
where M is the 3x3 matrix (containing any rotation/scale information), T is the translation Vector and ST is the transpose Vector of T. 1 is just a constant.