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

[John Doty <jpd@xxxxxxxxx>]

On Jul 14, 2011, at 1:09 PM, DJ Delorie wrote:

> 
>> The linear "stretching" transformation (x', y') = (a*x, b*y) applied
>> to a circle yields an ellipse whose axes are parallel to the
>> coordinate axes.
> 
> Yes, *unless* you're including the start/end angles in that
> transformation, instead of applying them afterwards.
> 
> http://www.delorie.com/pcb/tmp/stretched-arc.pcb
> 
> This shows two 45-degree Arcs.  One has width==height, the other does
> not.  Note that the end angle on the stretched arc is 45, but the
> arc does not end at the 45 degree marker.

Linear stretching doesn't generally preserve angles. Still, the result of stretching a circle is an ellipse. Indeed, the more general statement that stretching an ellipse *in any direction* yields an ellipse is true.

Since stretching does not preserve angles, the operation of selecting an arc from an ellipse by angle does not commute with stretching. Thus, your example.

This is closely related to math that confused the great genius Johannes Kepler for quite a while. We're still somewhat burdened by artifacts of his path to understanding: look up "eccentric anomaly" sometime.

John Doty              Noqsi Aerospace, Ltd.
http://www.noqsi.com/
jpd@xxxxxxxxx




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