# STC position angles

David Berry dsb at ast.man.ac.uk
Tue Jun 21 01:02:54 PDT 2005

```Steve,

> I think that there is no way to stress the following too strongly:
> Pixel coordinates in a FITS array or table do not have a metric.
> When creating a FITS array there may be effects such as binning,
> non-adjacency, lack of orthogonality and other issues.

I would say these effects only become apparent when you project pixel
coordinates into WCS coordinates. If you forget all about WCS, and just
think within the context of an 2D array of numbers in a computer, then can
you not associate a metric with pixel coordinate? Is it not fair to say
that array element A(4,7) has 4 direct neighbours A(3,7), A(5,7), A(4,6)
and A(4,8), and 4 diagonal neigbours? This defines adjacency. Also, can
you not meaningfully choose to say that the distance between A(3,5) and
A(4,7) is sqrt( (3-4)**2 + (5-7)**2 ) pixels, and that the two axes are
orthogonal?

Of course, if you project the pixel axes into WCS, the *projected* pixel
axes could in general do anything at all, so you may not be able to use
them to define a metric in WCS.

> If FITS data have merely a "linear" WCS (and the default pixel
> coordinates are the principal example of such) then there is no
> metric.  In the absence of a well-defined reference frame there is no
> a priori justification to presume that the axes are orthonormal, or
> that the metric signature is Riemannian.

Do we need a justification? Can we not say it's a *convention* to describe
pixel coordinates as flat cartesian axes?

> For that reason I wonder whether, for the purposes of interpretation
> by machines, it might be better to encode the PositionAngle
> information as a coordinate pair along the two axes which are relevant
> to the coordinate system in question.

I agree that your suggested scheme would be far less susceptible to
mis-interpretation, but of course a lot of software would probably
immediately turn it back into an angle!

> In any case I think the notion of position angle measured from
> "X" toward "Y" should not be permitted unless it can be
> made absolutely clear that it is only permissible in cases where
> X and Y have a clearly-defined Euclidean metric.

It depends on what is meant by that phrase "have a clearly-defined
Euclidean metric". I would say that pixel coordinates can, by convention,
be described using a Euclidean metric. Is it not just a case of
specifying the convention being used? Such as, if the STC position
angle reference is "X" then a position angle of theta defines a curve
in the (x,y) coordinate system which, for points very close to the origin,
takes the form

y = x.tan( theta )

and if the reference is Y then

y = x/tan( theta )

These are mathematical conventions which define what is meant by the
position angle theta. Obviously for some coordinate systems (e.g.
spherical), the above relationships are only geodesic at points very
close to the origin, but that's not a problem.

David

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