# [Pixman] [PATCH 01/15] demos/scale: Compute filter size using boundary of xformed ellipse, not rectangle

Oded Gabbay oded.gabbay at gmail.com
Sun Dec 20 06:44:48 PST 2015

```On Thu, Dec 17, 2015 at 11:02 PM, Bill Spitzak <spitzak at gmail.com> wrote:
> Well that was a pain to figure out again (it is tricky to get p and x and y
> cancelled out of the calculation) but does this make any sense:
>
> Calculate bounding box of radius-1 circle in xy transformed to uv:
>
> Transform x,y to u,v by this matrix calculation:
>
>   |u|   |a c| |x|
>   |v| = |b d|*|y|
>
> Horizontal component:
>
>   u = ax+cy (1)
>
> x,y describes a radius-1 circle, p is angle to the point:
>
>   p = 0..2*pi
>   x = cos(p)
>   y = sin(p)
>   x^2+y^2 = 1
>   dx/dp = -sin(p) = -y
>   dy/dp = cos(p) = x
>
> Figure out derivative of (1) relative to p:
>
>   du/dp = a(dx/dp) + c(dy/dp)
>         = -ay + cx
>
> The min and max u are when du/dp is zero:
>
>   -ay + cx = 0
>   cx = ay
>   c = ay/x  (2)
>   y = cx/a  (3)
>
> Substitute (2) into (1) and simplify:
>
>   u = ax + ay^2/x
>     = a(x^2+y^2)/x
>     = a/x (because x^2+y^2 = 1)
>   x = a/u (4)
>
> Substitute (4) into (3) and simplify:
>
>   y = c(a/u)/a
>   y = c/u (5)
>
> Square (4) and (5) and add:
>
>   x^2+y^2 = (a^2+c^2)/u^2
>
> But x^2+y^2 is 1:
>
>   1 = (a^2+c^2)/u^2
>   u^2 = a^2+c^2
>   u = hypot(a,c)
>
> Similarily the max/min of v is at:
>
>   v = hypot(b,d)
>
>
Nice.
to hypot for posterity.

Oded

> On Thu, Dec 17, 2015 at 1:52 AM, Oded Gabbay <oded.gabbay at gmail.com> wrote:
>>
>> On Sat, Dec 12, 2015 at 8:06 PM,  <spitzak at gmail.com> wrote:
>> > From: Bill Spitzak <spitzak at gmail.com>
>> >
>> > This is much more accurate and less blurry. In particular the filtering
>> > does
>> > not change as the image is rotated.
>> > ---
>> >  demos/scale.c | 43 ++-----------------------------------------
>> >  1 file changed, 2 insertions(+), 41 deletions(-)
>> >
>> > diff --git a/demos/scale.c b/demos/scale.c
>> > index d00307e..71c7791 100644
>> > --- a/demos/scale.c
>> > +++ b/demos/scale.c
>> > @@ -55,50 +55,11 @@ get_widget (app_t *app, const char *name)
>> >      return widget;
>> >  }
>> >
>> > -static double
>> > -min4 (double a, double b, double c, double d)
>> > -{
>> > -    double m1, m2;
>> > -
>> > -    m1 = MIN (a, b);
>> > -    m2 = MIN (c, d);
>> > -    return MIN (m1, m2);
>> > -}
>> > -
>> > -static double
>> > -max4 (double a, double b, double c, double d)
>> > -{
>> > -    double m1, m2;
>> > -
>> > -    m1 = MAX (a, b);
>> > -    m2 = MAX (c, d);
>> > -    return MAX (m1, m2);
>> > -}
>> > -
>> >  static void
>> >  compute_extents (pixman_f_transform_t *trans, double *sx, double *sy)
>> >  {
>> > -    double min_x, max_x, min_y, max_y;
>> > -    pixman_f_vector_t v =
>> > -    {
>> > -       { { 1, 1, 1 } },
>> > -       { { -1, 1, 1 } },
>> > -       { { -1, -1, 1 } },
>> > -       { { 1, -1, 1 } },
>> > -    };
>> > -
>> > -    pixman_f_transform_point (trans, &v);
>> > -    pixman_f_transform_point (trans, &v);
>> > -    pixman_f_transform_point (trans, &v);
>> > -    pixman_f_transform_point (trans, &v);
>> > -
>> > -    min_x = min4 (v.v, v.v, v.v, v.v);
>> > -    max_x = max4 (v.v, v.v, v.v, v.v);
>> > -    min_y = min4 (v.v, v.v, v.v, v.v);
>> > -    max_y = max4 (v.v, v.v, v.v, v.v);
>> > -
>> > -    *sx = (max_x - min_x) / 2.0;
>> > -    *sy = (max_y - min_y) / 2.0;
>> > +    *sx = hypot (trans->m, trans->m) / trans->m;
>> > +    *sy = hypot (trans->m, trans->m) / trans->m;
>> >  }
>> >
>> >  typedef struct
>> > --
>> > 1.9.1
>> >
>> > _______________________________________________
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>>