Linear Transformation

This page was last modified 2010-11-03 10:05:34 by Puchu.Net user Choco. (Show history)

1 Matrix Representation
2 Matrix Multiplication
3 Transforms
4 References

Matrix Representation

While working with libraries like OpenGL, matrices are represented as a 16-element array. There are two representations, row- or column-major and this indicates how elements are stored and manipulated. So if a translation matrix looks like this:

$\mathbf{T}=\left(\begin{array}{cccc} 1 & 0 & 0 & t_{x} \\ 0 & 1 & 0 & t_{y} \\ 0 & 0 & 1 & t_{z} \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$

For OpenGL, which uses column-major style, elements $t_{x}$ , $t_{y}$ , and $t_{z}$ are accessed with indices 13, 14, and 15. In a system that uses row-major style, these same elements will be accessed with indices 3, 7, and 11. The code fragments below uses column-major style.

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Matrix Multiplication

Finding matrix product of two $4 \times 4$ matrices $A$ and $B$ involves calculating for each element:

$(AB)_{i,j} = \sum_{k=1}^4 A_{ik}B_{kj}$

In C code this translates to a function like:

void Multiply( const float* A, float* B )
{
  float t[16];

  t[0]  = A[0]  * B[0] + A[1]  * B[4] + A[2]  * B[8]  + A[3]  * B[12];
  t[1]  = A[0]  * B[1] + A[1]  * B[5] + A[2]  * B[9]  + A[3]  * B[13];
  t[2]  = A[0]  * B[2] + A[1]  * B[6] + A[2]  * B[10] + A[3]  * B[14];
  t[3]  = A[0]  * B[3] + A[1]  * B[7] + A[2]  * B[11] + A[3]  * B[15];

  t[4]  = A[4]  * B[0] + A[5]  * B[4] + A[6]  * B[8]  + A[7]  * B[12];
  t[5]  = A[4]  * B[1] + A[5]  * B[5] + A[6]  * B[9]  + A[7]  * B[13];
  t[6]  = A[4]  * B[2] + A[5]  * B[6] + A[6]  * B[10] + A[7]  * B[14];
  t[7]  = A[4]  * B[3] + A[5]  * B[7] + A[6]  * B[11] + A[7]  * B[15];

  t[8]  = A[8]  * B[0] + A[9]  * B[4] + A[10] * B[8]  + A[11] * B[12];
  t[9]  = A[8]  * B[1] + A[9]  * B[5] + A[10] * B[9]  + A[11] * B[13];
  t[10] = A[8]  * B[2] + A[9]  * B[6] + A[10] * B[10] + A[11] * B[14];
  t[11] = A[8]  * B[3] + A[9]  * B[7] + A[10] * B[11] + A[11] * B[15];

  t[12] = A[12] * B[0] + A[13] * B[4] + A[14] * B[8]  + A[15] * B[12];
  t[13] = A[12] * B[1] + A[13] * B[5] + A[14] * B[9]  + A[15] * B[13];
  t[14] = A[12] * B[2] + A[13] * B[6] + A[14] * B[10] + A[15] * B[14];
  t[15] = A[12] * B[3] + A[13] * B[7] + A[14] * B[11] + A[15] * B[15];

  // result overwrites B
  memcpy(B, t, sizeof(float) * 16);
}

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Transforms

Apply transforms is the same as multiplying with the corresponding transformation matrices.

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Translate

To move by a translation vector $(t_{x}, t_{y}, t_{z}, 0)$ :

$\mathbf{T}=\left(\begin{array}{cccc} 1 & 0 & 0 & t_{x} \\ 0 & 1 & 0 & t_{y} \\ 0 & 0 & 1 & t_{z} \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$

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Scale

To scale by a factor in each axis:

$\mathbf{S}=\left(\begin{array}{cccc} s_{x} & 0 & 0 & 0 \\ 0 & s_{y} & 0 & 0 \\ 0 & 0 & s_{z} & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$

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Rotate

Basic rotations about the x, y, and z axes:

$\mathbf{R_{x}}=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & cos \phi & - sin \phi & 0 \\ 0 & sin \phi & cos \phi & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ rotates the y-axis towards the z-axis

$\mathbf{R_{y}}=\left(\begin{array}{cccc} cos \phi & 0 & sin \phi & 0 \\ 0 & 1 & 0 & 0 \\ - sin \phi & 0 & cos \phi & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ rotates the z-axis towards the x-axis

$\mathbf{R_{z}}=\left(\begin{array}{cccc} cos \phi & - sin \phi & 0 & 0 \\ sin \phi & cos \phi & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$ rotates the x-axis towards the y-axis

To rotate $\phi$ radians about an arbitrary, normalized axis $\mathbf{r}$ :

$\mathbf{R}=\left(\begin{array}{cccc} (\mathbf{c}+(1-\mathbf{c})r_{x}^{2}) & ((1-\mathbf{c})r_{x}r_{y}-r_{z}\mathbf{s}) & ((1-\mathbf{c})r_{x}r_{z}+r_{y}\mathbf{s}) & 0 \\ ((1-\mathbf{c})r_{x}r_{y}+r_{z}\mathbf{s}) & (\mathbf{c}+(1-\mathbf{c})r_{y}^{2}) & ((1-\mathbf{c})r_{y}r_{z}+r_{x}\mathbf{s}) & 0 \\ ((1-\mathbf{c})r_{x}r_{z}+r_{y}\mathbf{s}) & ((1-\mathbf{c})r_{y}r_{z}+r_{x}\mathbf{s}) & (\mathbf{c}+(1-\mathbf{c})r_{z}^{2}) & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right) $

where

$\mathbf{c} = cos \phi$ and $\mathbf{s} = sin \phi$

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Shear

To apply shear along x-, y-, or z-axis, modify the coefficients below:

$\mathbf{Sh}=\left(\begin{array}{cccc} 1 & Sh_{xy} & Sh_{xz} & 0 \\ Sh_{yx} & 1 & Sh_{yz} & 0 \\ Sh_{zx} & Sh_{zy} & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$

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References

Akenine-Moller, Tomas and Eric Haines, Real-Time Rendering 2nd Edition, A K Peters: 42-43, 725.

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Contents

Matrix Representation

Matrix Multiplication

Transforms

Translate

Scale

Rotate

Shear

References