(2007-08-13) Planar Angles are Signed Quantities (Carnot, Möbius)
An angle is what separates the directions of two half-lines.
A planar angle (or circular angle) measured in radians (rad),
between two straight lines originating at the center of a circle of unit radius
is the length of the circular arc between them.
Such an angle can be considered to be a signed quantity if we specify that one of the
line is the direction we choose as "reference". The angle to the other line is
then counted positively if that line is reached by turning counterclockwise.
A clockwise rotation from the line of reference corresponds to
a negative angle.
(2011-03-19) Bearings
Navigation at the surface of a sphere and orientation of a local plane.
Unless otherwise specified,
global terrestrial, maritime or aerial navigation, bearings are understood to
be angles west of north.
For example, +45° is northwest.
-90° is due east.
On the other hand, cartesian coordinates are often used for local maps
(where the curvature of the Earth is negligible).
The x-axis is due east and the y-axis is due north.
Commonly, polar coordinates in the plane use the x-axis as direction of
reference and the usual counterclockwise angular coordinate in the plane is
thus sometimes specified as the bearing (or angle)
north of east.
Brngths (Yahoo!
2007-08-12)
Solid Angles
Solid angles are to spherical patches what angles are to circular arcs.
Loosely speaking, a solid angle is the ratio of an apparent area to the square of the
distance it's observed from. That simplified definition applies only
to a constant distance, as we view the areas of
regions drawn on a sphere centered on the observer.
Euclid (c.325-265 BC)
talked about
visual pyramids and Ptolemy
(c.AD 87-165) called them visual cones...
A solid angle
(usually expressed in steradians "sr") is a measure of a cone generated by
a ray (i.e., half a straight line)
originating at the center of a sphere of unit radius with one point of the ray
moving in a closed loop at the surface of that unit sphere.
The solid angle is simply the spherical surface area enclosed by
the aforementioned loop at the surface of the unit sphere.
Just like a planar angle, a solid angle can be oriented
(i.e., assigned a sign) according to the direction in which the loop is traveled.
The universal
convention is to count a solid angle positively if its bounding loop is traveled
counterclockwise when seen from the outside of the sphere or, equivalently,
clockwise when observed from the center of the sphere.
(This way, the solid angle formed by a directtrihedron is positive.)
You may want to recall that the south face of a loop
is seen at a positive solid angle (using the usual
convention to define the "north" and "south" side of an oriented loop).
There's an S in "positive" and an N in "negative".
The loop may cross itself many times: The enclosed spherical area
is tallied algebraically as in the planar case,
described elsewhere.
Just like planar angles are defined modulo
2p,
solid angles are defined up to a multiple of 4p,
because such is the entire surface area of a unit sphere
A spat (sp)
is the solid angle subtended by the whole sphere
(4p).
Indeed, consider that a solid angle A changes to -A
when you reverse the direction of its defining loop.
However, you could also consider that the solid angle has become
(4p-A)
because the new orientation of the loop makes it enclose (as its "south side")
whatever part of the sphere was not previously enclosed by the loop as originally
oriented.
You may use this argument to convince yourself that multiples of
4p
are as irrelevant to solid angles as multiples of
2p
are irrelevant to planar angles.
The steradian is not the only unit of solid angle.
Astronomers routinely express solid angles in square degrees,
they also use square minutes or square seconds
for tiny solid angles.
Indeed, if a "rectangular" patch of sky is so small that the curvature of the celestial
sphere is negligible, then its surface is almost flat and it has an area very nearly equal
to the product of its angular width by its angular height (technically,
those concepts of "width" and "height" become precise only in the context of that flat
approximation). The result is in steradians if those angles are given
in radians. On the other hand,
if such angles are given in degrees, then the result is, by
definition, obtained in "square degrees". The square degree
is thus just a practical unit of solid angle which could be used to measure solid
angles of any size, although the aforementioned "small angle"
computation is only valid for very tiny rectangular patches of the sphere.
1 square degree =
( p / 180 ) 2 =
0.0003046... sr
A square minute is 602 = 3600 times smaller
than that.
A square second is 12960000 times smaller than a
square degree; it's roughly 2.35 10 -11sr.
The whole celestial sphere (twice the sky) corresponds to a solid angle of
1 spat = 4p [sr] =
41252.96... square degrees
(2005-07-21) Units for Angles and Solid Angles
Their special status should be restored among SI units.
The most common unit of angle we use (outside of the scientific realm
where the radian reigns supreme) is the degree
(°) inherited from the Mesopotamians. There are 360°
in a full turn; a right angle is 90°.
The ancient Mesopotamian sexagesimal system
(positional numeration in base 60)
is still used for fractions of a degree:
60 minutes to a degree, 60 seconds to a minute:
1° = 60'
and 1' = 60''
The CGPM is the international body responsible for enacting the definitions
of the SI units, which are now used throughout the scientific world.
Simply put, a planar angle is an
axial scalar in the plane,
whereas a solid angle is an axial scalar in
3-dimensional space.
(2013-04-04) Trihedron Formula (van Oosterom & Strackee, 1983)
Solid angle W
formed by the trihedron (u,v,w).
The words formed and subtended describe dual
views of angular quantities (either planar angles or solid angles).
Just like a planar angle is formed by two rays and
subtended by a circular arc, so too is a solid angle
formed by a cone and subtented by a spherical patch.
In this section, we're concerned with the former viewpoint and
ponder the solid angle bounded
by a particular cone composed of three planar surfaces intersecting along three rays
characterized by the three nonzero vectorsu, v and w.
Notice that both sides of the above equation
are equal to unity when the trihedron is orthonormal
(with positive winding)
as W = p/2.
(2013-04-30) Solid Angle Subtended by a Rhombus
Consider an horizontal rhombus having (perpendicular)
diagonals of lengths 2x and 2y,
viewed from an altitude z directly above its center.
The sides of the relevant right rhombic pyramid are
a, b and c :
a 2 = x 2 + z 2b 2 = y 2 + z 2c 2 = x 2 + y 2
Paying no attention to signs or orientations, the
trihedron formula
yields directly the solid angle subtented by
one quarter of the rhombus, using:
u = (x, 0, z)
v = (0, y, z)
w = (0, 0, z)
The solid angle W
subtended by the whole rhombus is thus given by:
tg (
W
) =
x y z
=
x y
8
z3 + z2 a +
z2 b + a b z
(a + z) (b + z)
Using
x 2 = a 2 - z 2
and
y 2 = b 2 - z 2
the square of that boils down to:
tg2
(
W
) =
(a - z) (b - z)
8
(a + z) (b + z)
If we call t2 that quantity, then
cos (W/4) is
( 1-t2 ) / ( 1+t2 ). So:
cos (
W
) =
z(a+b)
4
ab + z2
sin (
W
) =
xy
4
ab + z2
Also :
You may want to check that
cos2 + sin2 = 1 using the above formulas.
Notice that a thin degenerate pyramid
(W = 0)
is obtained with
a = z or
b = z.
The solid angle at the apex of a flat pyramid (z = 0)
is indeed 2p.
Let's solve the above relation for z... We have:
sin2 (
W
) =
x2 y2
4
(x2+z2 ) (y2+z2 ) + z4 +
2 z2 (x2+z2 )½ (y2+z2 )½
4 z4 (x2+z2 ) (y2+z2 )
=
[
x2 y2
-
(x2+z2 ) (y2+z2 )
- z4
] 2
sin2 W/4
Let Z = z2, S = x2+y2,
P = x2y2 and
U = 1/sin2 W/4
= 1 + 1/tg2 W/4
We have
4 Z4 +
4 S Z3 +
4 P Z2
=
[ (U-1) P -
S Z - 2 Z2 ] 2 with
(U-1) P -
S Z - 2 Z2
> 0
The coefficients of Z4 and Z3 cancel out,
leaving a quadratic relation:
[ S2- 4P - 4(U-1)P ] Z2
- 2 (U-1) S P Z
+
(U-1)2 P2 = 0
The reduced discriminant 4 U (U-1)2 P3
is the square of
2 x3 y3
tg2 W/4
sinW/4
The leading coefficient (i.e., the square bracket) is:
(2008-05-28) Solid Angles Subtended by Simple Shapes
The solid angle corresponding special angular configurations.
Band (between two parallel lines) :
The dihedral angle (q)
formed by two half-planes is proportional to the solid angle
(W) between them
for an observer located at any point O on their axis of intersection.
The coefficient of proportionality is simply obtained from
any special case... In particular,
if q = p/2,
the faces are perpendicular and the area of the spherical lune
between them is a quarter of the whole sphere
(W = p).
Therefore, in general:
W = 2 q
This can be interpreted as the solid angle between two parallel lines
whose largest angular separation is seen to be
q (by an observer at point O).
Disk :
The part of a solid which lays between two parallel planes that intersect it
is called a frustum.
The height (h) of the frustum is the
distance between those planes.
For a sphere of radius R,
the lateral (spherical) surface of a frustum
has an area equal to 2pRh
(remarkably, for a sphere, this
area doesn't depend on the position of the cutting planes;
it's just a function of the distance between them).
With h = 2R,
we retrieve the surface area of the entire sphere, namely
4pR2.
In particular, with R = 1, a spherical cap
of angular radius r has a height
h = 1-cos r and a
surface area equal to the solid angle seen from O, namely:
W =
2p ( 1 - cos r )
=
4p sin2 (r/2)
Note that the latter expression is required
for accurate floating-point computations with small values of
r.
You may check that a solid angle of 2p
is obtained for the entire sky (a hemisphere,
r = p/2)
and 4p for the whole sphere
(r = p).
The mean angular diameter of the full moon is
2r = 0.52° (it varies with time around
that average, by about 0.009°).
This translates into a solid angle of
0.0000647 sr, which means that the whole night sky
covers a solid angle roughly
one hundred thousand times greater than the full moon.
Symmetrical Rectangular Patch :
The solid angle of a rectangular field of view (symmetrical about
the axis of observation) of angular width
2a and
height 2b is:
W =
4 arcsin ( sin(a) sin(b) )
For small angles, the patch is nearly flat and the surface of the rectangle is
nearly the product of its angular width by its angular height
W » (2a)(2b).
The above is, of course, the origin of the units of solid angle
commonly used by astronomers: square degree, square minute, etc.
The conversion factors between those units are always obtained in
the limit of very small angles.
For example, a square degree, expressed in steradians is simply
(p/180)2.
The above exact formula shows that a square patch of sky 1° by 1°
is only about 0.9999746 of a square degree!
The larger the angular size, the greater the discrepancy...
Note that a symmetrical square field, 1 radian on
a side, is about 0.9193954 of a "square radian"
(which is just an unused—and confusing—alternative name for a
steradian ).
For such a patch, the true solid angle exceeds the naive value by less
than 9%. For the whole sky (hemisphere) however,
ignoring curvature like that would result in a discrepancy of 57%
(the true solid angle is 2p,
not p2 ).
Triangles :
W =
j -
Arctg ( cos q tg j )
Regular Polygons :
At the apex O of a regular n-gonal pyramid
(i.e., a straight pyramid whose
base is a regular polygon with n sides) the solid angle subtended by the
base consists of 2n triangular solid angles of the type just discussed,
where j is
p/n and
q is the angular radius of the circle
circumscribed to the base. This adds up to:
W
=
2p -
2n Arctg ( cos q tg p/n )
For large values of n, as the tangent or the arctangent
of a small quantity is nearly equal to it, we retrieve the disk formula:
W
=
2p (1 - cos q)
In the case n = 4, the formula can be nicely recast, as presented next.
Solid Angle Subtended by a Square :
For a square of side a viewed
from a point on its axis of rotational symmetry at a distance
dfrom the middles of its sides, we have:
W
=
4 Arcsin ( a 2/ 4d 2 )
Proof : For n = 4, the above says:
W =
2p -
8 Arctg ( cos q )
If y is the angle whose tangent
is t = cos q, Let's show that:
cos 2y =
( 1 - t2 ) / ( 1 + t2 ) = a 2 / 4d 2
The first equality is a trigonometric identity. We only have to prove the second one:
If R is the distance from the observer (O) to a corner of the square, the definition of t
makes R.t the distance from O to the center of the square.
Using the Pythagorean theorem twice, we have:
d 2 = R2 t 2 + a 2 / 4
R2 = d 2 + a 2 / 4
Thus, a2/2 = (1-t2 ) R 2 and
2d 2 = (1+t2 ) R 2. The advertised ratio follows,
which brings about the desired result via:
