[cairo] Uniform Line Widths
Behdad Esfahbod
behdad at behdad.org
Tue Apr 15 12:58:44 PDT 2008
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
> >>
> >> ____________________________________________________________
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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
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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
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