Boxes and Polygons in ADQL/STC. Questions and recommendation.

[Francois Ochsenbein]

Well, a box can't be a limit in RA/Dec is you are far for the Equator;
having a pole within a box would invitably turn the box into a small
circle...

I can't either understand the STC definition of a polygon where 
a vertex may optionnaly be a SmallCircle object: while joining 
point A to point B on a great circle is always possible, it's not
true for a small circle depending on the pole's position.
Moreover joining point A to point B is a joined property of 
the 2 vertices, but Arnold's definition means that the polygon 
has a different shape if you describe the polygon in the opposite
order (great circle lines may turn into small circle lines and
vice-versa). I understand that Arnold wants to describe in a 
single definition both types of lines (great and small circles),
but these are really different geometries.

About the BOX vs polygon, my understanding of STC definition is that
the box sides are great circles (for SPHERICAL flavor) since there
is no possibility of defining smallCircle vertices here.
And compared to the polygon, the 'inside' is perfectly defined
(the part of the sphere which contains the center is the 'inside').

About the BOX orientation: it is undefined only if the box center
is one of the poles.

The four vertices of a box can be computed in a rather straightforward
way if you use the cartesian coordinates 
  (x=cos(Dec)cos(RA), y=cos(Dec)sin(RA), z=sin(Dec), x^2+y^2+z^2=1)
and the rotation matrix between the (RA=0,Dec=0) point and the
box center, which should give the cartesian coordinates of the
vertices
   x = x0*dx -i*(y0/r0)*dy -j*(x0/r0)*z0*dz
   y = y0*dx +i*(x0/r0)*dy -j*(y0/r0)*z0*dz
   z = z0*dx               +j*r0*dz
if (x0, y0, z0) are the cartesian coordinates of the center, 
r0=sqrt(x0^2+y0^2), (i,j) are +/-1 to describe the 4 vertices, 
and the dx dy dz are derived from the box size (a x b)
  dx=cos(b/2)cos(a/2), dy=cos(b/2)sin(a/2), dz=sin(b/2)
[does not work when r0=0, i.e. centered on a pole, in which
case the orientation of the box in undefined]

In the peculiar case dy=0 or dz=0 (one box size is pi radians)
you got only 2 vertices, and it's hard to view this figure
(similar to the area between 2 meridians) as a 'polygon';
if a=b=pi (square of 180deg on each 'side') the 'box' becomes
an hemisphere, as Tom pointed out (the vertices are opposite).
With a=b=2pi (360deg), gou get a single vertice (dx=-1, dy=dz=0,
the opposite of the box center) -- that single vertex represents
the whole sphere (a 'polygon' with a single point?)

Should we recommend that a 'box' is restricted to size<180deg ?

Have a nice week-end :-)
Francois

>
>
>I believe that both BOX definitions would have advantages and clearly
>very different uses:
>
>  - Using "RA-Dec" BOXes one could tile the entire sky without any overlap
>and, as Tom points out, they are extremely simple to implement.
>
>  - A "great circle" BOX could represent a simple region that can describe
>the coverage of a rectangular detector with WCS TAN projection.
>
>Considering that STC/s is part of the ADQL geometry specs, whose polygons
>are apparently restricted to great circles, one could argue that BOX
>should be interpreted in the "RA-Dec" sense for simplicity, and use the
>STC/s polygon for the other case.
>
>Alternatively, one could define the BOX to be a shortcut to the STC/s
>polygon, and implement RA-Dec constraints by explicitely spelling them out
>in the query.
>
>I don't feel strongly either way but perhaps the latter is a bit more
>appealing to me, because this way a simple and common geometry constraint
>(TAN coverage of, say, a CCD) could be described in ADQL without
>supporting the more advanced STC/s constructs that might be a little more
>challenging for some implementations in terms of parsing and query
>optimization.
>
>Cheers, T.
>
>P.S. Of course, one could also just have both boxes in ADQL with separate
>keywords.
>
>
>On Fri, 23 Oct 2009, Tom McGlynn wrote:
>
>> Alberto Micol wrote:
>>> On 23 Oct 2009, at 21:19, Arnold Rots wrote:
>>>
>>>
>>>> 4.5.1.5 Box
>>>> A Box is a special case of a Polygon, defined purely for  convenience. It
>>>> is
>>>> specified by a center position and size (in both coordinates)  defining a
>>>> cross
>>>> centered on the center position and with arms extending, parallel to  the
>>>> coordinate axes at the center position, for half the respective  sizes on
>>>> either side.
>>>> The box?s sides are line segments or great circles intersecting the  arms
>>>> of the
>>>> cross in its end points at right angles with the arms.
>>>>
>>>
>>> My trouble is with the sentence that  the arms extend "parallel to the
>>> coordinate axes".
>>> "Parallel" to the equator cannot be a great circle unless it is the
>>> equator itself. Hence:
>>> Does that mean that the I should measure the size of the "horizontal"  arm
>>> along
>>> the small circle parallel to the equator?
>>> If this is correct, then a size of 180 deg is an hemisphere if and  only if
>>> the centre is placed
>>> on the equator.
>>>
>>> I appreciate some help, thanks!
>>>
>>>
>> Hi Alberto,
>>
>> I understood this to mean that the horizontal arm goes along great circle
>> which has an apex (highest latitude, or lowest
>> if the point is in the southern hemisphere) at the point.  So the great
>> circle is 'parallel' to the equator but only
>> instantaneously at that point  However I wouldn't mind one of the experts
>> chiming in here.
>>
>>> Then, regarding the usefulness of a BOX made of great circle arcs:
>>> that is useful because to find if a point is inside or outside such BOX
>>> it is just matter to compute the scalar product of the vector  representing
>>> the point
>>> and the 4 vectors representing the half-spaces of the 4 box sides.
>>>
>>> Of course this means that it will no longer be possible to use (ra,  dec)
>>> as we are used to,
>>> as in:  ra BETWEEN this AND that AND dec BETWEEN d0 AND d1
>>> and instead we have to go to a vectorial representation of the sky
>>> coordinates.
>>>
>>>
>>
>> My problem is that as far as I can see, the problem that the astronomers want
>> to answer will be phrased
>> in terms of limits on RA and Dec so that even though it might be
>> mathematically handy it's not necessarily relevant
>> to the problems we want to solve.   In any case there's nothing special about
>> the box here.  It's true for any
>> (convex?) polygon isn't it?
>>
>> Tom
>>
>> P.S.  I think I've gotten the equations for the vertices of a box (assuming
>> my interpretation above is correct).  The derivation
>> was pretty easy once I abandoned trying to do things using pure geometry and
>> attacked it using the centers of the great circles and
>> analytic geometry.  I'll try to post it somewhere tomorrow.
>>
>>
>>>
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Francois Ochsenbein    ------   Observatoire Astronomique de Strasbourg
   11, rue de l'Universite 67000 STRASBOURG  Phone: +33-(0)368 85 24 29
Email: francois at astro.u-strasbg.fr (France)    Fax: +33-(0)368 85 24 17
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