<div dir="ltr"><div>The downside is that this only applies to spherical celestial coordinate systems.</div><div>And I maintain that it is a very bad idea to allow self-intersecting polygons.</div><div><br></div><div> - Arnold<br></div></div><div class="gmail_extra"><br clear="all"><div><div class="gmail_signature" data-smartmail="gmail_signature"><div dir="ltr">-------------------------------------------------------------------------------------------------------------<br>Arnold H. Rots Chandra X-ray Science Center<br>Smithsonian Astrophysical Observatory tel: +1 617 496 7701<br>60 Garden Street, MS 67 fax: +1 617 495 7356<br>Cambridge, MA 02138 <a href="mailto:arots@cfa.harvard.edu" target="_blank">arots@cfa.harvard.edu</a><br>USA <a href="http://hea-www.harvard.edu/~arots/" target="_blank">http://hea-www.harvard.edu/~arots/</a><br>--------------------------------------------------------------------------------------------------------------<br><br></div></div></div>
<br><div class="gmail_quote">On Thu, Jun 14, 2018 at 9:13 AM, Francois-Xavier PINEAU <span dir="ltr"><<a href="mailto:francois-xavier.pineau@astro.unistra.fr" target="_blank">francois-xavier.pineau@astro.unistra.fr</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><span class=""><br>
Le 13/06/2018 à 23:58, Markus Nullmeier a écrit :<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
Hello list,<br>
<br>
On 13/06/18 10:05, Francois-Xavier PINEAU wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
Example: if we take the case of a simple 4 vertices's polygon having a<br>
butterfly shape (i.e. having two crossing great-circle arcs), then the<br>
inside of one "wing" is in the counter-clockwise sense while the inside<br>
of the other "wing" is in the clockwise sense.<br>
</blockquote>
Well, a definition that allows polygon edges to intersect would be<br>
unsound, probably in several ways. One of them is the mentioned<br>
inability to describe the complement of a polygon as a single polygon.<br>
</blockquote></span>
Ok, a polygon (defined by a list of successive vertices) defines two complementary areas on the unit sphere.<br>
To know which one is considered as being inside the polygon, why not taking a reference point like the North pole (N):<br>
<br>
If N is on the left side of the first great-arc of the polygon (mathematically, if ((vertex1 cross_product vertex2) scalar_product north pole) > 0), then N is in the area considered as being inside the polygon, else it is in the outside area.<br>
<br>
<br>
I see one possible complication:<br>
if N is a vertex or on an edge of the polygon, then:<br>
- the first edge must not contain N<br>
- if N is considered as inside: the inside is defined by the area(s) on the left part of the edge(s) containing N<br>
- if N is considered as outside: the inside is the complement of the area considering N inside<br>
<br>
This simple definition is able to deal with both simple and self-intersecting polygons.<br>
<br>
In attachment, 3 large polygons (2 self intersecting, and one having a surface area > 2*pi): I adapted the following 10 lines of code (in 2d) to the spherical case (<a href="https://wrf.ecse.rpi.edu//Research/Short_Notes/pnpoly.html" rel="noreferrer" target="_blank">https://wrf.ecse.rpi.edu//Res<wbr>earch/Short_Notes/pnpoly.html</a>)<wbr>.<br>
<br>
Kind regards,<br>
<br>
<br>
François-Xavier<br>
<br>
<br>
<br>
<br>
<br>
</blockquote></div><br></div>