[cairo] Uniform Line Widths

[Bobby Salazar]

I am experiencing a problem with the code you attached to the below message. Specifically, it doesn't work with certain transformation matrices.

In the function get_singular_values(), if the translation matrix is:

	xx	9.8000000000000
	yx	0.00000000000000
	xy	0.00000000000000
	yy	9.8000000000000
	x0	348.00000000000
	y0	0.00000000000000

Then the function computes invalid numbers for the major and minor singular values.

This is what it calculates:

a = 96.040000000000
b = 96.040000000000
k = 0.00000000000000
a_b = 192.08000000000
delta = -1.#IND000000000

And since delta comes out wrong, everything else that uses it does also.

Do you know what the appropriate fix would be?

> -----Original Message-----
> From: behdad at behdad.org
> Sent: Fri, 22 Feb 2008 07:00:11 -0500
> To: bobby8934 at inbox.com
> Subject: Re: [cairo] Uniform Line Widths
> 
> On Fri, 2008-02-22 at 02:36 -0500, Bobby Salazar wrote:
>> Ok, that is much simpler than I thought, and works good too.
> 
> Good.
> 
>> However, my code is written so that the pre-set line width is correct
>> for the currently applied transformation matrix.
> 
> Well, that can't be the case.  The pen at the applied transformation is
> an ellipse (which causes the non-uniform line width).  Which diameter of
> the ellipse is the desired line width you want?  Most probably the
> smallest.
> 
> Depending on your answer, the solution would be different.  The max and
> min of the diameter/pen-width are twice the semi-major and semi-minor
> axes of the ellipse, which are equal to the singular values of the
> non-translation parts of the matrix.  See:
> 
>   http://en.wikipedia.org/wiki/Semi-major_axis
>   http://www.ee.ic.ac.uk/hp/staff/www/matrix/decomp.html#SVD
> 
> Quoting there: "The non-zero singular values of A are the square roots
> of the non-zero eigenvalues of A⁺ A." where A⁺ is A-transpose.
> 
> Lets see.  Let M be the cairo transformation matrix in question:
> 
>       ⌈xx xy⌉
>   M = |yx yy|
>       ⌊x0 y0⌋
> 
> The non-translation part is:
> 
>   A = ⌈xx xy⌉
>       ⌊yx yy⌋
> 
>   A⁺ A = ⌈xx yx⌉⌈xx xy⌉ = ⌈xx²+yx²     xx*xy+yx*yy⌉
>          ⌊xy yy⌋⌊yx yy⌋   ⌊xx*xy+yx*yy     xy²+yy²⌋
> 
> Name those:
> 
>   B = A⁺ A = ⌈a k⌉
>              ⌊k b⌋
> 
> The eigenvalues of B satisfy:
> 
>   λ² - (a+b).λ + a.b - k² = 0
> 
> The eigenvalues are:
>                __________________
>       (a+b) ± √(a+b)² - 4(a.b-k²)
>   λ = ---------------------------
>                   2
> 
> And the Singular values are the root of λs.
> 
> That's all.  Not as easy and fast as you expected I assume.  See the
> attached code and output.
> 
> 
>> If I then set the transformation matrix to be the identity matrix, the
>> line width is not correct. Is there an equally simple method of
>> mapping a certain line width from any arbitrary transformation matrix
>> into the same pixel sized width it would if the identity matrix were
>> used? I hope that last sentenced makes sense...
> 
> You need to think what the intended pen size in your application is and
> reorder your code to not have to go through this.
> 
> Cheers,
> 
> behdad
> 
> 
> PS. Anyone should feel free to put this on cairographics.org/samples or
> a separate page of its own or whereever that it will be useful!
> 
> 
>> Thanks!
>> 
>>> -----Original Message-----
>>> From: behdad at behdad.org
>>> Sent: Wed, 20 Feb 2008 14:17:11 -0500
>>> To: bobby8934 at inbox.com
>>> Subject: Re: [cairo] Uniform Line Widths
>>> 
>>> On Wed, 2008-02-20 at 10:39 -0800, Bobby Salazar wrote:
>>>> I currently have a function that accepts a cairo context (with a path
>>>> already set) and strokes the path with a pre-set width and color. The
>>>> context can have any arbitrary transformation already applied to it.
>>>> So
>>>> given this situation, how would I ensure that I get a uniform line
>>>> width
>>>> from the stroke even when they may have applied a deforming scale to
>>>> the
>>>> context? Any ideas would be appreciated!
>>> 
>>> Something like:
>>> 
>>>   cairo_save (cr);
>>>   cairo_identity_matrix (cr);
>>>   cairo_set_line_width (cr, width);
>>>   cairo_stroke (cr);
>>>   cairo_restore (cr);
>>> 
>>> 
>>>> Thanks!
>>> 
>>> Have fun with cairo!
>>> --
>>> behdad
>>> http://behdad.org/
>>> 
>>> "Those who would give up Essential Liberty to purchase a little
>>>  Temporary Safety, deserve neither Liberty nor Safety."
>>>         -- Benjamin Franklin, 1759
>> 
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> --
> behdad
> http://behdad.org/
> 
> "Those who would give up Essential Liberty to purchase a little
>  Temporary Safety, deserve neither Liberty nor Safety."
>         -- Benjamin Franklin, 1759