[cairo] Adding cairo_shear and family

[Bill Spitzak]

Paolo Bonzini wrote:
> Bill Spitzak wrote:
>> It might make more sense to make the call do the composite operation 
>> that preserves area:
>>
>> [1+sx*sy sx]
>> [sy 1+sx*sy]
>>
>> Note that if either sx or sy are zero then it is the same as the 
>> proposed call.
> 
> I like this.  But the composite operation that preserves area is
> 
> [sqrt(1+sx*sy)   sx
>  sy              sqrt(1+sx*sy]
> 
> whose determinant is 1.  Your matrix has determinant sx^2 sy^2+sx sy+1.
> 
> For example, with this matrix
> 
>   cairo_shear (cr, sin(a), -sin(a));
> 
> would be the same as cairo_rotate (cr, a) for angle between -90 and 90:
> 
> [sqrt(1-sin^2 a)  sin a                   [cos a    sin a
>  -sin a           sqrt(1-sin^2 a)]   =     -sin a   cos a]
> 
> Paolo

You are right. I just multiplied the x and y shear matrices together, 
the idea being that shear(x,y) would be the same as shear(x,0) and 
shear(0,y) one after the other. Since each step preserves area I thought 
the result would preserve area as well. I then made a mistake in the 
multiplication so that I thought the result was symmetric. The actual 
result is:

[1+sx*sy sx]
[xy      1 ]

or transposed with 1+sx*sy in the lower-right, depending on the order of 
the shear calls. That is not very useful because the xy order matters.

The area-preserving one with sqrt() looks pretty useful to me, however.

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