Re: gEDA-user: pcb GL can't render stretched arcs

[karl@xxxxxxxxxxx (Karl Hammar)]

Igor Lopez:
> 2011/7/16 Andrew Poelstra <asp11@xxxxxx>:
> > There is a fairly informative discussion of this problem on SO:
> > http://stackoverflow.com/questions/2945337/how-to-detect-if-an-ellipse-intersectscollides-with-a-circle
> 
> I had a look and found one algebraic solution close to the one I have proposed.
> 
> > The correct methods given there generally require root-solvers,
> 
> Why is the algebraic solution not correct?

Because of false assumptions about the "inner" and "outer" arcs, see
below.

> > Since we rarely care about the exact distance, only "is it between
> > 0 and Radius" it is probable that we could use a bisection method
> > with very few iterations to get an answer.
> 
> I am convinced we do not need any iterations or equation solvers.
> We just need to check with the transformed point coordinates
> in the equations for the two arcs where one is the inner arc and
> the other is the outer arc where the point radius has been subtrcacted
> respectively added to the arc lenghts(width, height).
> If the point intersects the equation for the inner arc is <= 0 and
> the equation for the outer arc is >= 0 and since it is not a complete
> ellipse one need to check that the angle between point and
> arc centerpoint is within the arcs limiting angles which is not
> to difficult if the limiting angles are known.

You are assuming that the two arcs are ellipses, which they are not.

E.g. if you draw an ellipse with a thick pen, the inner space will
look more like an almond than an ellipse.

Take an ellipse with a = 5 and b = 4, and add a thickness = 1 to the
perimeter.

The point Pb = (xb,yb) = (3, 16/5) = (3, 3.2) is on the ellipse.

Go out 1 (thickness) from that point in the normal direction (see e.g. 
[1]), i.e. along the line 

 y - 16/5 = mn * (x - 3),
 mn = a*a*yb / (b*b*xb) = (25*16/5) / (16*3) = 5/3

you will come to the point

 Pt = (xt, yt) = (3+3/k, 16/5 + 5/k) ~= (3.51, 4.06)
 where k = sqrt(34), 5.83 < k < 5.84

Now, check if Pt lies on the ellipse:

 x^2/at^2 + y^2/bt^2 = 1, where at = a+1 = 6, bt = b+1 = 5

 (3+3/k)^2 / 36 + (16/5+5/k)^2 / 25 ~= 1.0016 > 1

i.e. the new arc is not an ellipse.

///

If this error is small enougth to ignore, fine, but what error do we
tolerate?

Regards,
/Karl Hammar

[1] http://de.wikipedia.org/wiki/Ellipse#Normalengleichung_.28kartesische_Koordinaten.29

-----------------------------------------------------------------------
Aspö Data
Lilla Aspö 148
S-742 94 Östhammar
Sweden
+46 173 140 57




_______________________________________________
geda-user mailing list
geda-user@xxxxxxxxxxxxxx
http://www.seul.org/cgi-bin/mailman/listinfo/geda-user