Download SHORT COMMUNICATION Differing angles on angle

INSTITUTE OF PHYSICS PUBLISHING
METROLOGIA
Metrologia 42 (2005) L23–L26
doi:10.1088/0026-1394/42/4/L02
SHORT COMMUNICATION
Differing angles on angle
W H Emerson
Le Trel, 47140 Auradou, France
Received 14 March 2005
Published 10 May 2005
Online at stacks.iop.org/Met/42/L23
Abstract
Values of plane angles are expressed with a choice of several units.
Historically the quantity needed a unit because it was, and still is, used as a
base quantity. ISO/TC 12 defines it as a derived, dimensionless quantity, and
the International System of Units (SI) gives it the ‘dimensionless unit’
radian, which now means no more than ‘one’. This paper discusses the
confusion that arises from the dual uses of these terms.
The author argues that solid angle is logically a two-dimensional angle
derived from plane angle, and has units naturally derived from units of plane
angle. ISO/TC 12, however, defines the quantity independently of plane
angle as a dimensionless quantity to which, nevertheless, the SI assigns
a ‘unit’.
1. Plane angle
The word ‘angle’, when used to designate a quantity, means
different things to different people. Essentially it is a
phenomenon at a point; it has no linear dimension, and
historically it has been regarded as a difference of direction as
viewed from a point. Babylonian astronomers needed a way of
expressing differences between the directions of ‘points’ in the
heavens as viewed from Earth. They invented a unit by which
such a difference, an angle, could be given a value: the degree.
They made a degree one 360th of a full turn around the sky,
or one 180th of the greatest amount by which the directions of
two stars could differ.
Navigators needed a way of expressing changes of heading
of their ships. They divided their compass cards to indicate
32 evenly-spaced directions, called points of the compass, the
total amounting to a change that would bring the ship back to
its original heading. A point is thus equal to 11 41 degrees, and
nowadays navigators express changes of heading in degrees.
They formerly expressed the altitude of a star above the horizon
in units of degrees, minutes and seconds of arc; now they
state it as a decimal number of degrees. The degree, the
Babylonian astronomers’ unit, is still the universally accepted
unit of angle for anyone who measures angles, whether or not
he calls himself a metrologist, or who uses the results of such
measurements.
All the dictionaries that I have consulted [1] define angle
in terms of the differences of direction of two intersecting
lines. The larger, internationally respected dictionaries define
0026-1394/05/040023+04$30.00
it as the amount of turning of one of those lines that would
be necessary to make it coincident with or parallel to the
other. The quantity thus defined has all the hallmarks of a base
quantity in any system of quantities; it is not derived from any
other quantity and is indeed difficult to define without some
circularity; and, as with the other ancient base quantities of
our modern systems, nobody really needs to ponder the term’s
meaning.
Some two centuries ago mathematicians started using a
new unit of angle. Instead of dividing a complete reversal of
direction by 180, which gives the unit ‘degree’, they divided it
by the number π and called the unit ‘radian’. It is defined as
the angle subtended at the centre of a circle by an arc whose
length equals that of the circle’s radius. That is, of course, a
definition of a particular unit of angle, not of the quantity angle,
which is not associated by definition with circular geometry.
The new unit, more difficult to realize than degree, was not and
is not used to express the results of measurements of angles;
it was adopted because it simplified certain mathematical
expressions. For example, it is impossible to define the
centripetal acceleration of a point on a rotating shaft, radius r,
without stating its angular velocity ω using a relative angle,
that is angle relative to a reference angle, a unit. If that unit
2
r, where ωdeg is the
is degree, the acceleration is (π/180)2 ωdeg
shaft’s angular velocity with the changing angle expressed as
a number of degrees. With radian as the unit of angle the
angular velocity is, say, ωrad and the centripetal acceleration
2
r. By putting π into a definition of
becomes simply ωrad
a unit of angle the constant π may be eliminated from the
© 2005 BIPM and IOP Publishing Ltd
Printed in the UK
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Short Communication
definitions of many derived quantities that are functions of
angle. Mathematicians, physicists and engineers who derive
and manipulate expressions for quantities that describe cyclic
phenomena use radian as their unit of angle as a matter of
course, and when they give an angle a value, the numerical
part of the value is nearly always a multiple or a fraction of π.
Indeed, the presence of π in the expression of a value is usually
seen as sufficient to indicate that the unit is radian, and the unit
is not then stated.
However, not stating the unit can lead to anomalous
statements. For example, the sine of the angle θ can be
represented by a summed infinite series:
sin θ = ϑ −
1 3
ϑ
3!
+
1 5
ϑ
5!
− ···,
(1)
where
θ
.
(2)
rad
Many texts, however, show that equation with the symbol
θ on both sides of the equation, that is with θ replacing ϑ
by omission of the unit radian, and without mentioning that
omission, thus:
ϑ=
sin θ = θ −
1 3
θ
3!
+
1 5
θ
5!
− ···.
(3)
That transformed equation can be satisfied dimensionally only
if θ is made a dimensionless quantity, without a unit. By the
definition of radian, ϑ is the ratio of the length of an arc of a
circle subtending an angle at the circle’s centre, to that of the
radius of the circle. That ratio is, of course, a number that is
proportional to the angle at the centre, but most ordinary people
would not regard it as the same thing, as being that angle. They
would argue that an arc subtending an angle cannot at the same
time be that angle, even if the arc is normalized by dividing
its length by that of its radius. An angle is a phenomenon
at a point, but clearly an arc of a circle is not ‘at’ a point,
particularly a point that is outside itself.
If an angle is an arc of a circle made dimensionless by
dividing it by its radius of curvature ρ, then the absolute values
of the lengths of the arc and its radius of curvature are irrelevant.
An elemental angle may be written as dθ = ds/ρ where ds is
an elemental length of a curved line in the neighbourhood of a
point where the radius of curvature is ρ and may be a function
of s. If angles, like elements of arc length, are additive, then
logically the angle corresponding to the length s is
s
ds
θ=
.
(4)
0 ρ
That seems to be a legitimate representation of the quantity, but
an angle is supposed to be ‘at’ a point, and there is no single
point that can be assigned to the quantity expressed thus as an
integral.
The substitution of the symbol for angle for that of a pure
number, and the adoption of the name angle for that number,
changes the meaning of the term. It is no longer a difference of
direction or an amount of turning from one direction to another,
defined independently of all other quantities; it becomes a ratio
of two lengths, like a trigonometric function. Moreover, if θ
is no longer a symbol for an angle as the term is traditionally
understood, what is the sine of θ if the argument is the ratio of
two lengths? Formerly, in a right-angled triangle of sides a, b
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and c (c being the hypotenuse), the ratio a/c was the sine of the
angle opposite the side of length a, by definition. Now it is said
to be the sine of the ratio of the length of an arc, bounded by
the lines b and c and centred on their point of convergence, to
the arc’s radius. We are asked to accept that those are identical,
trigonometric functions, yet sin θ now defined by equation (3),
with θ as a number between −2π and +2π, rather than as a
trigonometric function.
I have suggested elsewhere [1] that the quantity ϑ, defined
as the ratio of an arc of a circle to its radius, is akin to the
trigonometric functions sine, cosine and tangent, a function of
an angle θ, though, unlike the others, it is a linear function.
It, too, deserves a name that is not ‘angle’, like the other
trigonometric functions of angle.
When, in 1960, the International System of Units (SI) [2],
was given that name by the Conférence Générale des Poids
et Mesures (CGPM) there were six base units for six base
quantities (now seven). Those base quantities did not and do
not include plane angle. That quantity was not defined (the
SI does not define kinds of quantities or cite definitions; it
only defines their units), but the name radian was adopted for
its unit. Nor, exceptionally, was that unit defined. Unlike all
other SI units (except steradian, for solid angle) it was listed
neither as a base unit nor a derived unit. It, with steradian, was
called a ‘supplementary’ unit. Radian is, of course, defined in
almost any dictionary, but the Comité International des Poids et
Mesures (CIPM) did not adopt the lexicographers’ universally
agreed definition, nor any other.
In 1969, the CIPM interpreted the decision of the CGPM,
in 1960, which classed the units radian and steradian as
supplementary units, as allowing the freedom of treating those
units as base units [2]. In 1980, the CIPM (following a
resolution adopted by ISO/TC 12 in 1978) decided that they
must be treated as ‘dimensionless, derived units’. No other
dimensionless quantities were considered to have need of
units. In 1995, the CGPM, acknowledging that the status
of the supplementary units in relation to the base units and
derived units of the System ‘[gave] rise to debate’, decided
to do away with the name ‘supplementary’ and to make those
units ‘dimensionless, derived units’ [derived from existing base
units]. That ‘debate’ was supposedly between the proponents
of the historical, classical view of angle as effectively a base
quantity, and generally accepted as such by metrologists, and
those of the practice of mathematical analysts of defining angle
as a ratio of lengths. The analysts prevailed and, supposedly,
the debate was officially at an end. The proposals of the
CIPM are not open to general debate in the manner of a draft
International Standard.
The definition of angle now adopted by ISO/TC 12 and
the CGPM leaves it dimensionless and thus without need of
a unit. Yet the name ‘angle’ has always been associated
with units. Furthermore, the quantity angle is used to define
derived quantities such as ‘angular momentum’ and their units,
in which a unit of angle commonly appears. The CGPM’s
decision of 1995 means that an angular velocity that was
formerly stated with rad s−1 as its unit has now, logically, the
unit s−1 . It has been necessary to invent a ‘dimensionless unit’
with a borrowed name, with no value other than unity, so that
a unit of angular velocity may still be called rad s−1 . The
unit degree is no longer permitted to be that defined by the
Metrologia, 42 (2005) L23–L26
Short Communication
Babylonians and universally still in use; it is now defined by
ISO/TC 12 [3] as π/180.
A derivation of a quantity normally either assigns a name
to a relationship between two base quantities of the system
(e.g. an area or a velocity), or to a dimensionless relationship
such as a Reynolds number, or it expresses an observed law
of nature (e.g. a force). It contains no arbitrary element
unless, rarely, it be a numerical coefficient. By contrast the
definition of a unit of a quantity is always arbitrary, normally
by consensus.
There is nothing arbitrary about the classical concept of
angle, the quantity; but that is not true of the mathematical
analysts’ concept. In the geometry of the circle the length of
an arc may be divided by any of several characteristic lengths,
the circle’s radius, diameter, circumference or any constant
fraction or multiple of any of them, to give a ratio that is
always proportional to certain angles subtended by the arc.
Those ratios are proportional not only to the angle subtended
by the arc at the centre, but also to angles subtended anywhere
on the circumference. In the mathematicians’ definition of
‘angle’ the radius is chosen arbitrarily as the characteristic
dimension, and ISO/TC 12’s definition makes the centre of
the circle the point to which the quantity is supposed to have
relevance. There is no doubt that mathematicians, scientists
and engineers find important uses for the quantity they call
‘angle’, but it is not the concept of that name that has been
recognized for some millennia. That historical concept is still
recognized throughout the world, by surveyors, navigators,
astronomers, architects, industrial technicians, ordinary people
and all who measure angles or use the results; but not by the
mathematicians and scientists who are represented in ISO/TG
12 or the CIPM.
Equation (6) defines the quantity solid angle.
An element of solid angle may be visualized by
considering a rectangular, elemental area da on the surface
of a sphere centred on the point. It subtends1 the element
dΩ of solid angle at the centre. Let c be the length of the
circumference of a great circle on the sphere, p be a unit of
plane angle and n the number of such units in a full turn. The
rectangle’s length on a great circle through the point where
the chosen axis intersects the sphere is d(cθ/np); its width is
sin θ d(cφ/np); its area is, using equation (5),
da = c2 dθ sin θ
dφ np 2
= c2
dΩ
,
np 2
(7)
whence
Ω=
a(np)2
.
c2
(8)
If p = a complete turn (tr), n = 1 and
a
tr 2 .
(c)2
(9)
3602 a
deg2 .
(c)2
(10)
(2π )2 a
rad2 ,
(c)2
(11)
Ω=
If p = deg, n = 360 and
Ω=
If p = rad, n = 2π and
Ω=
or, if r is the radius of the sphere,
2. Solid angle
Ω=
The SI concept of solid angle (whose undefined unit was
the other former ‘supplementary’ unit) is equally arbitrary in
definition. It is defined by ISO/TC 12 [3] as the ratio of two
areas on or characteristic of a sphere: the ratio of an area on the
sphere’s surface to that of a square of side equal to the sphere’s
radius. The SI’s solid angle is not derived from plane angle,
yet a solid angle at a point, as naturally conceived, is clearly a
two-dimensional angle, not defined by one of many ratios of
spherical areas that are proportional to it.
A section through a solid angle at a point, and containing a
line through the point that we may call its axis, is a plane angle
bounded by a pair of half lines both terminating at the point.
One of those lines is at an angle θ to the axis. The plane of
the section is at an angle φ to a plane of reference through the
same axis. The angle θ is a function of φ.
An element of the solid angle Ω at the point is
dΩ = dθ sin θ dφ
0
the integration with respect to φ being through a complete
rotation.
Metrologia, 42 (2005) L23–L26
(12)
None of the relations (8) to (12) defines solid angle. As with
plane angles, a solid angle is a phenomenon at a point, not on
the surface of a sphere. Equation (8) and those following it
define the relations between units of solid angle and units of
plane angle.
If sp is the length of an arc, of a great circle on the sphere,
that subtends a unit p of plain angle at the centre, equation (8)
may be written as
a
Ω=
(13)
p2 .
sp2
In words:
In any system of units where p is a base unit of plane
angle, the coherent unit of solid angle [p ] is the solid
angle subtended at the centre of a sphere by an area
ap on the surface equal to sp2 , the square of the length
sp of an arc of a great circle that subtends a unit p of
plane angle at the centre, times p 2 , the square of that
unit of plane angle.
(5)
(the contribution associated with dφ increases with sin θ),
whence
θ
Ω=
sin θ dθ dφ = (1 − cos θ) dφ
(6)
a
rad2 .
r2
1
For brevity I use here the word ‘subtend’ extended to two dimensions. An
area on a sphere ‘subtends’ a solid angle at the centre equal to the solid angle
at the apex of a cone, having its apex at the centre, that cuts that area on the
sphere.
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3. Conclusion
The two concepts of plane angle, one treating it as a base
quantity, the other defining it as a dimensionless ratio of two
lengths, are incompatible. The former, historical concept fits
consistently into a system of quantities that accepts it as a
base quantity. The second is of a quantity that should not be
confused with the first; and both suffer from a dual use of
terminology.
Solid angle and its units are currently defined without
reference to and independently of plane angle. The SI ‘unit’
of solid angle bears the name steradian, which contains the
name radian but has no defined relationship with that unit of
plane angle. Solid angle was even allowed formerly to be
considered a base quantity and steradian a base unit. Such
practices overlook the fundamental relationship between the
quantities. Solid angle is a quantity derived, not from areas on
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spheres and their radii, but from plane angle, as in equation (6).
It is a two-dimensional angle with units derived from units of
plane angle. If radian is a unit of plane angle, a base unit as it is
commonly regarded, the coherent unit of solid angle is radian2 .
The numerical part of the value of a rationally defined, solid
angle expressed with radian2 as unit is the quantity called a
‘solid angle’ in the SI, without a unit but whose value may
be followed by the word ‘steradian’. The addendum means
no more than ‘this number is a value of a solid angle as ISO
defines it’.
References
[1] Emerson W H 2002 Metrologia 39 105–9
[2] 1998 The International System of Units 7th edn (Sèvres: Bureau
International des Poids et Mesures)
[3] 1993 Quantities and Units: ISO Standards Handbook 3rd edn
(Geneva: International Organisation for Standardisation)
Metrologia, 42 (2005) L23–L26