Surely angles are a special case of a relationship between two
quantities in different dimensions (in this case two spatial dimensions). (01)
Mike (02)
Patrick Cassidy wrote:
> 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.
>
> The last note from Ingvar Johansson had this portion of a discussion:
>
> [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.
>
> 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).
>
> 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.
>
> Pat
>
> Patrick Cassidy
> MICRA, Inc.
> 908-561-3416
> cell: 908-565-4053
> cassidy@xxxxxxxxx
>
>
>
>
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>
> (03)
--
Mike Bennett
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