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Final Answers
© 2000-2016   Gérard P. Michon, Ph.D.

    Planar Angles
&  Solid Angles

Every man of genius sees the world 
at a different
  angle  from his fellows,
and there is his tragedy.

H. Havelock Ellis  (1859-1939)  

Related articles on this site:

Related Links (Outside this Site)

Angle  &  Solid Angle,  by  Eric W. Weisstein.
On the Concept of the "Visual Angle" in an Optic Array and its History
by  James J. Gibson (1904-1979)  Cornell University  (October 1972).

Wikipedia:   Solid Angle


Angles, Angular Measures, Solid Angles...

(2007-08-13)   Planar Angles are  Signed  Quantities   (Carnot, Möbius)
An angle is what separates the directions of two half-lines.

planar 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.

Lazare Carnot (1753-1823)   |   August Möbius (1790-1868)

(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...

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  direct  trihedron 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.

square degree   =   ( p / 180 ) 2   =   0.0003046... sr

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 -11 sr.

The whole celestial sphere (twice the sky) corresponds to a solid angle of

1 spat   =   4p [sr]   =   41252.96... square degrees

 Chevalier de Borda (2005-07-21)   Units for Angles and Solid Angles
Their special status should be restored among SI units.

The CGPM is the international body responsible for enacting the definitions of the SI units, which are now used throughout the scientific world. 

 Come back later, we're
 still working on this one...

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 vectors  u, v and w.

      tg (  W  )    =     det (u,v,w)       
Vinculum Vinculum
2 u.v ||w||  +  v.w ||u||  +  w.u ||v||  +  ||u||.||v||.||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  ab  and  c :

a 2  =  x 2 + z 2             b 2  =  y 2 + z 2             c 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       
Vinculum Vinculum Vinculum
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)
Vinculum Vinculum
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)     
Vinculum Vinculum
4 ab + z2
    sin (  W  )    =     x y     
Vinculum Vinculum
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
Vinculum Vinculum
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/sinW/4  =  1 + 1/tgW/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   sin W/4

The leading coefficient  (i.e., the square bracket)  is:

(x2-y2 )2 -  x2 y2 / tgW/4)

 Come back later, we're
 still working on this one...

Solid angle of rectangular pyramid (Planetmath)
Solid angle at the apex of a right pyramid whose base is a rhombus ?
Adriaan Van Oosterom   and   Jan Strackee
"The Solid Angle of a Plane Triangle"  IEEE Trans. Biom. Eng. 30, 2, pp.125-126 (Feb. 1983)

(2008-05-28)   Solid Angles Subtended by Simple Shapes
The solid angle corresponding special angular configurations.

Band (between two parallel lines) :

 Spherical Lune  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).

 Full Moon

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.

 Rectangular field of view

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  2pnot  p).

 Basic Triangular Patch

Triangles :

W   =    j  -  Arctg ( cos q   tg j )

 Come back later, we're
 still working on this one...

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  d  from 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:

W   =   2p - 8 y   =   4 (p/2 - 2 y)   =   4 [ p/2 - Arccos ( a2 / 4d 2 ) ]  QED

(2011-02-21)   Right Ascension  (a)  and Declination  (d)
The  precession  of the Earth's axis makes celestial coordinates vary.

 Come back later, we're
 still working on this one...

"Sixty Symbols" Video:   Right Ascension and Declination (a,d)   by  Mike Merrifield.

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