STC position angles

David Berry dsb at ast.man.ac.uk
Wed Jun 22 00:00:45 PDT 2005

```Steve,

> On Tue 2005-06-21T09:02:54 +0100, David Berry hath writ:
> > 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?
>
> You can choose to say that, but then you may display it, and nothing
> requires that your display have square pixels.

Screen (X,Y) and pixel (X,Y) are two distinct coordinate systems. The
mapping from one to the other could in general be anything at all - it
just depends how you choose to display the image. So there's no reason to
suppose that a position angle measured in pixel (X,Y) should be the same
when measured in screen (X,Y). When a value is given for a position angle
it's essential that the coordinate system to which it refers is also
given.

> > Do we need a justification? Can we not say it's a *convention* to describe
> > pixel coordinates as flat cartesian axes?
>
> I would be a lot more comfortable if the document contained explicitly
> that along with a bunch of caveats regarding the inapplicability of
> the concept of angle in some circumstances.

Agreed.

> > 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 )
>
> In cases where the array or table represents phase space, or has a
> Minkowski metric, that angle has extremely little meaning.
> Indeed, in a Minkowski space you really want to be using sinh, cosh,
> and tanh, and you need the metric to define the speed of light.
> In such cases it is far better to express an ellipse or vector as a
> covariance matrix or coordinate pair.

I suppose it's a case of making things comfortable in the most common
cases, without making difficult cases impossible. For most 2D coordinate
systems associated with images of the sky, the above angle has a meaning.
For others, it can be converted into a vector coordinate pair using the
convention:

x = cos( theta )
y = sin( theta )

(swapped if reference=Y).

David

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