[cairo] Uniform Line Widths

Behdad Esfahbod behdad at behdad.org
Tue Apr 15 13:44:51 PDT 2008 

On Tue, 2008-04-15 at 12:34 -0800, Bobby Salazar wrote:
> Thanks Behdad, it works perfectly!

'welcome.  This function also exists in cairo-matrix.c.   May be a good
candidate to make public.  Another good candidate is the
get_scale_factor that is mostly used for text.  Pango duplicates that
function and exposes it publicly.

behdad


> > -----Original Message-----
> > From: behdad at behdad.org
> > Sent: Tue, 15 Apr 2008 15:58:44 -0400
> > To: bobby8934 at inbox.com
> > Subject: Re: [cairo] Uniform Line Widths
> > 
> > On Mon, 2008-04-07 at 15:42 -0400, Bobby Salazar wrote:
> >> 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.
> > 
> > Found a bug in delta computation.  Also simplified the computations a
> > bit.  You can find it in cairo/doc/tutorial/src/singular.c now.  Try
> > with git master.
> > 
> > behdad
> > 
> >> 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
> >>>> 
> >>>> ____________________________________________________________
> >>>> FREE 3D MARINE AQUARIUM SCREENSAVER - Watch dolphins, sharks & orcas
> >>>> on
> >>>> your desktop!
> >>>> Check it out at http://www.inbox.com/marineaquarium
> >>>> _______________________________________________
> >>>> cairo mailing list
> >>>> cairo at cairographics.org
> >>>> http://lists.cairographics.org/mailman/listinfo/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
> >> _______________________________________________
> >> cairo mailing list
> >> cairo at cairographics.org
> >> http://lists.cairographics.org/mailman/listinfo/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
-- 
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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