How to calculate the start and end point of an elliptical arc from given angles.

[Regina Henschel <regina.henschel@libreoffice.org>]

Hi all,

the command U (angle-ellipse) and T (angle-ellipseto) draw an elliptical 
arc from the ellipse defined by center (x|y) and size (w,h), and 
start-angle t0 and end-angle t1. Example:
<draw:custom-shape
  draw:style-name="gr1" draw:text-style-name="P1" draw:layer="layout"
  svg:width="6cm" svg:height="2cm" svg:x="6cm" svg:y="6.5cm">
     <text:p/>
     <draw:enhanced-geometry
       svg:viewBox="0 0 6000 2000"
       draw:type="non-primitive"
       draw:enhanced-path="U 3000 1000 6000 2000 30 315 N"/>
</draw:custom-shape>

But the specification does not describe, how to calculate the start and 
end point. I have tested it with some applications and found, that 
actually two methods are used.

(1) Used by LibreOffice 4.0 and PowerPoint 2013 Preview
Draw an angle with the given value (here 30deg and 315deg). Calculate 
the intersection of the leg of the angle with the full ellipse (here 
center (3000|1000) and size (6000|2000)).

(2) Used by Apache OpenOffice and Calligra Stage
Draw an angle with the given value. Draw a circle from maximum radius. 
Intersect leg of the angle with the circle and remember the point. Scale 
the circle (and thereby transform the point) in height (or width) to fit 
the given ellipse. Take the transformed point as start or end point 
respectively.

I have no proposal, because both methods are reasonable.

The attached file "source.odp" contains the document itself. The 
attached file "Diagnose.odg" contains screenshots and additional lines, 
to determine how the start and end point is calculated.

The screenshots show some additional problems:
(A) The third and forth parameter are sometimes interpreted as radius, 
sometime as diameter. That is already tracked in 
https://tools.oasis-org/issues/browse/OFFICE-3711
(B) PowerPoint starts the angle not at the x-axis, but at the y-axis. 
That is a general problem of insufficient angle definitions in the 
specification. I think, the angle should start at the positive x-axis, 
when not explicitly something else is specified.
(C) The applications agree in which direction they draw the arc. But it 
looks not like "clockwise", although "clockwise" is specified.
(D) The applications agree use the angle direction as positive from 
positive x-axis to negative y-axis. That is against mathematical 
definition. Again a problem of insufficient angle definitions in the 
specification.
I think, that B,C,D need a separate discussion.


Kind regards
Regina

Attachment: source.odp
Description: application/vnd.oasis.opendocument.presentation

Attachment: Diagnose.odg
Description: application/vnd.oasis.opendocument.graphics