Dan Connelly wrote:
> Warp wrote:
>> This is the reason why only rotation, scaling, skewing (which is actually
>> a combination of rotation and uneven scaling) and translation is 
>> possible.
>>

> I'm a bit confused about the skew argument.  There's

> matrix:
> 
> 3 degrees of freedom: transformation

That should be translation.

> 3 degrees of freedom: scaling
> 3 degrees of freedom: rotation
> 
> So this leaves 3 more degrees of freedom..... which are skew.
> 

> (right-hand coordinates)
> 
> | ax  0  0 |
> | 0  ay  0 | : scaling
> | 0  0  az |
> 
> | cz  sz  0 |

> | 0   0   1 |   (permute for x, y)
> 
> So between scaling and rotation, I'm constrained to
> antisymmetric matrixes.
> 
> A typical shear matrix:
> 
> | uz  vz  0 |

> | 0   0   1 |   (permute for x, y)
> 
> So this gives me 9 parameters:
> ax, ay, az, sx, sy, sz, vx, vy, vz
> 
> the same number of parameters needed to specify the full matrix.
> 

After thinking more about this, I realized my mistake.
Matrix multiplication fails to preserve symmetry, only addition.
So given that, it's possible to generate a symmetric component (the
offdiagonal terms generating shear) in addition to a antisymmetric
component (the off-diagonal terms generating rotation) from rotation
and scaling together, as long as the scaling is non-uniform, as Warp
said.  My linear algebra is too rusty....



| 0 1 | |-1 0 |       | 0 1 |
|-1 0 | | 0 1 |   =   | 1 0 |

And my degrees of freedom argument follows only if one is allowed a single
rotation and a single scaling.  But generating pure shear requires two rotations.

Sorry....

Dan

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