# Degree of freedom (DOF) for 3D transforms?

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I have read that each axis in a coordinate system corresponds to a degree of freedom. That means that specifying a point in 3D can be done using 3 degrees of freedom. But if I would like to transform a point in 3D how many degrees of freemdom would that include? I have tried to find some resources describing DOF but I have not found any, if anyone knows of some sites that describes DOF please post!

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Found these slides:

http://www.cs.cornell.edu/Courses/cs4620/2008fa/lectures/07transforms3d.pdf

On slide 14 it says:

"3D affine transformation has 12 degrees of freedom
- count them by looking at the matrix entries we’re allowed to change"

But when I look at the previous matrices I only get:

3, from translation
3, from scaling
and 4 from rotation.

Where does the last 2 come from? I don't count the "ones" in scaling and translation since they are fixed.

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There are only three DOFs from rotation, not four. The simplest way to see this is that a rotation can be completely specified by a 3-dimensional vector, where the direction of the vector gives the rotation axis, and the length gives the angle of rotation.

The missing three DOFs are from shear transformations. You can shear along each of the three co-ordinate axes so this gives 3 DOFs.

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Ok that would make sense, but then the slides must be wrong then since it does not mention shear.

Regarding rotation. How are the 3 DOF specified? Basically the matrix only contains an angle \theta that is evaluated in sin and cos.

Another thing. I have read that the affine transform preserve distances and angles. But does that also hold for shear?

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A rotation about a particular axis (say the x-axis) will be specified in terms of sin and cos of some angle theta, so yes, there is one DOF in that case. But a general rotation will be a combination of three rotations (one about each axis) so there are 3 different "theta" variables to be given. (This method of specifying rotations is called "Euler angles".)

Affine transformations don't preserve distances and angles in general; rotations and translations do, but scaling and shear do not. Shear does preserve areas (in 2D) or volumes (in 3D) however.

The other type of affine transform is a reflection, but this is effectively the same thing as scaling with scale factor = -1.

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Quote:
 Original post by Stephen_T[...] Shear does preserve areas (in 2D) or volumes (in 3D) however.

In general, affine transformations preserve ratios of areas (in 2D) or volumes (in 3D). The constant that all areas or volumes are multiplied by is the determinant of the matrix.

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