I would like to pursue the issue of the unit of measure of an angle, since
it also relates to other "ratio'-like quantities that may be used as
measures. The gist of this note is that I would not like the 'dimension' of
an angle to be considered as null, or 1, or dimensionless, but as something
that means 'angular measure'. For an ontology that is intended to represent
meanings, I am very leery of oversimplifications that work fine in
restricted contexts but may prove confusing in missed contexts. (01)
The last note from Ingvar Johansson had this portion of a discussion: (02)
[John Sowa] > > I also like that analysis. But it has to be extended to
angles,
> > since we have to support multiple functions that map angles to
> > numbers: degree and radian.
>
[IJ] > I agree, and in a sense so do also the metrologists that (as I said
in
> an earlier mail) I criticize. In my opinion, one should say that radian is
> a unit of the derived dimension length/length, but the SI system and VIM
> says that it is a dimensionless unit or a unit of dimension-one.
> However, everyone agrees that angles can be measured by (or mapped on)
scales
> whose magnitudes are 'x degree' or 'x radian'.
>
> I think, by the way, that it is misleading to say that "angles are
> mapped to numbers"; angles are mapped to magnitudes of a scale.
>
Although an angle in radians can be expressed as a ratio of linear
measures, the linear measures themselves do not measure arbitrary straight
lines, but are quite specific regions of some imaginary circle. I think it
is a misleading oversimplification, when taking ratios of things that are
not themselves pure numbers, to ignore the meanings of the measures that are
being divided. A similar issue has arisen in the past about how to express
things like "weight percent" which, if one ignores the objects that are
represented by the numerator and denominator, can appear to be a
dimensionless number (grams/grams). Such ratios have an actual conceptual
"dimension" though the SI and VIM committees may have found it possible to
ignore the meanings in the case of radians, knowing that the dimensions will
likely be interpreted properly in applications. One way to recognize the
problem is to note that if one wants to represent a weight ratio, it is
possible to use micrograms per gram or grams per gram, and the "dimensions"
will appear to cancel out in either case, leaving a "dimensionless" number,
though the resulting numbers differ greatly depending on what units are
chosen for the numerator.
I would suggest that we promiscuously include all quantifiable "units"
that carry meaning in any application, and not take as "dimensionless" any
measures that are in fact distinguishable in their intended meaning. A
weight ratio does *not* have the same dimension as an angle, though one can
oversimplify either to some dimensionless number. (03)
In this view, a 'radian' is a unit of measure, as is a 'degree-of-angle',
and if the dimension is represented separately from the unit of measure, the
dimension in either case would be 'angular measure'. The dimension of a
weight ratio is the ordered pair of objects or types of objects whose
weights are being divided (weight ratios might better be treated in a
different way, but if they were treated as measures with a unit, that would
be my preference for the unit). (04)
It may be possible to consider certain ratios as the 'base unit' as in
the case of a radian, where the subtended arc length and radius are the
defining measures being divided. In the case of weight ratio,
grams-of-X/grams-of-Y might be the base unit for each X/Y pair. Measures
that are related to other ratio measures by some constant number, such as
angle degrees or micrograms/gram, would then be related to the base unit as
"prefix"-unit is to other base units, where "prefix" may be micro, kilo,
etc. or a special non-SI prefix. (05)
Pat (06)
Patrick Cassidy
MICRA, Inc.
908-561-3416
cell: 908-565-4053
cassidy@xxxxxxxxx (07)
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