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The solid angle at a point subtended by a circle
The Solid Angle at a Point Subtended by a Circle* bI
R.
E. ROTHE
Chemical Company Golden, Colorado
Dow
_%BSTRACT: The solid angle subtended calculated.
The solid angle is expressed
central normal
by a circle of radius as a function
to the circle and the distance
hR from
R with respect
of the inclination
to any point angle 9 from
the center of the circle.
given in terms of tabulated
elliptic
and related functions.
function of the inclination
angle is given for several values of h. Several
Cs the
The result is
A graph of the solid angle as a special
cases are
discussed.
Introduction
The solid angle subtended by a circle with respect to a point along its central normal is easily calculated by handbook mensuration formulas (1). 3s the point moves off the central normal, however, the solid angle calculation becomes more complicated. The circle’s projection becomes an ellipse whose minor axis diminishes as the angle of inclination increases. As the elliptical eccentricity increases, the solid angle generally decreases. In fact, as the point approaches the plane of the circle, the solid angle approaches either 0 or 2~ depending on whether the point lies outside or inside the circle. The results presented in this paper give the solid angle subtended by a circle with respect to any point in space. Several special cases are given which illustrate the theory. Interest in this problem originated with critical mass studies on fissionable material having hemispherical cavities. The question is raised as to what fraction of the neutrons leaking out of the interior surface would escape the hemisphere without further interaction with the fissionable material. Another critical mass application of these results involves the calculation of the geometrical interaction between elements of a uniform array of thin circular disks of fissionable material. The formulas presented in this paper are not restricted to critical mass physics; they are applicable to any problem dealing with the linear propagation of radiation from a point source. Statement
of the Problem
The point Q of Fig. 1 is a distance AR from the center of a circle of radius R and inclined from the central normal to the circle by an angle 0. The shading * This work was supported AT(29-l)-1106.
by the U.S. Atomic
Energy
Commission
under
contract
R. E. Rothe . represents the boundary to the solid angle to be calculated. All lines within the boundary cone contribute to the solid angle ; those outside the cone do not. The problem is azimuthally symmetric.
Q
FIG. 1. The point Q is a distance AR from the center of t,he circle of radius R and inclined from the cent,4
normal to the circle by an angle 0.
Lines emanating from point Q of Fig. 2 which pass through the area element dA = rd rd $ in the equatorial plane (z = 0) contribute to the solid angle to be calculated. The total solid angle subtended by the circle of
X
Y
FIG. 2. Lines leaving point Q at (R’ = AR, 8, 4 = 0) pass through the area element dA = T dr dq5 in the plane z = 0. The solid angle contribution vector p between Q and dA makes an angle [ with the normal to the plane z = 0.
radius R from the point Q is given by R Cl=
516
2n
ss0 0
cos
f p-2
rd rd+.
(1)
Journal of The Franklin
Institute
The Solid Angle at a Point Subtended by a Circle Coordinates r and I$ are in the plane z = 0; the vector element rdrd+ to the point Q ; and cost
p joins the area
= Ip*&Ip+ = R’p-lcos8,
where R’ = hR. The magnitude
(2)
of p is
p2 = r2+ R’2-
2r*R’,
(3)
where 1.8’
= cos 4 cos ($7 - 0) + sin 4 sin ($7 - 0) cos &r = cos $ sin 8,
(4)
according to the law of cosines for spherical triangles. The integral form for the solid angle is obtained by substituting into Eq. 1 to obtain Q(8,
R’) =
R
2n
ss0 0
rR’ cos 8 (r2 + RI2 - 2rR’sin 0 cos +)-* dr d$.
The integral over r is performed sphere radius R vanishes leaving Q(B,X) =
Eqs. 2-4
s2n
first ; explicit dependence
(5)
upon the hemi-
COS ed4 o (1 -sin28cos2+)
s
2n (Lsin0cos+) o (l-sin2Bcos2+)
-cose(l+xy+
1_2Xsin0cos+ 1+x2
d4*
(6)
The first integral is 277; the second integral may be expressed in terms of elliptic integrals or related special functions. The solid angle subtended by a circle of radius R from a point Q at R’ = AR and 6’ in terms of tabulated functions is (1+X2-22Xsin0)n(N,ol)
2cose
~(e”)=2,-(l+,2+2,sin8)t
[ (l+h)(l+sin@
(1-h) -(l+sin8)
rI(n,(II)+ &)
G)]
(7)
Here II and K are the complete elliptic integrals of the third and first kind, respectively. The elliptic parameters n and N and the modular angle 01are related to the inclination angle 0 and to X by the relations: 2sin8 n=l+ (1+A)” (1+h2+2hsin0)’
sin2a =
4h sin
e
(l+h2+2XsinO)
=
k2.
(9) (10)
The complete elliptic integrals of the third kind, II(N, CX)and II(n, 01), of Eq. 7 diverge for any a!at N = 1 or n = 1. These values of N and n correspond to 8 = ~12. The solid angle does not become infinite there since the coefficients
Vol.287,No.F,June1969
517
R. E. Rothe of these functions vanish at that angle. To facilitate calculations of the solid angle for all angles 0 including those near ~12, the transformation set given in Eqs. 11 is used, i.e. IW,
a) = K(a) + Q~S,vP -A&v,
rI(n,cY)=
41,
(11%)
K(ol)+$&[l-R,(&,,CY)].
(11b)
Here A, is the Heuman lambda function. The parameters Ed and E, and the coefficients 6, and 6, are related to the inclination angle 19and to h by t,he relations : ‘lnE~~ =
1 -sin@ * 1 + sin B ’
(
)
(12) (13)
6,, =
1+h l+P-22hsin0
1 ( l-sine
6, = k(1+X2+2XsinB)* n 1-x
‘(1+X2+2Xsine)&,
(14)
)
l+sine 1 (
1 1 *
(15)
In Eq. 15, the plus sign is used for X < 1 and the minus sign is used for h > 1. M7hen the substitutions of Eqs. 11 are put into Eq. 7, the solid angle expression becomes
-
4h( 1 -sin
e)*(l + sin
e)-$(I + A2 +
2h sin
e)-+ K(E)
(164
e)-*(I + A2 +
2h sin
e)-* K(a)
(1610)
for the case h < 1 and We, A) = “[A&,,
-
a) + A,,(+,
4h( 1 - sin
e)*(l +
41 sin
for the case h > 1. Equations 16a and 16b, containing the Heuman lambda functions,* facilitate calculations in the vicinity of the divergence of II(n, a) and Il(N, a) since 0 < AO(.e,a) < 1 for all 0 < E6 77/Z and 0 6 01Q 7r/2. The graph of Fig. 3 presents the solid angle of a circle of radius R with respect to a point AR from the center of the circle and inclined from the central normal by an angle 8. The solid angle is shown as a function of the angle 0 for several values of the parameter h. Table I presents the solid angle data in tabular form. Note that the solid angle does not decrease montonically with inclination angle for all X > 1 as might be expected. For values of X only slightly greater than one, the solid angle increases slightly with increasing angle. It does reach zero, however, when the inclination angle finally reaches rr12 (see Fig. 3, X = 1.1). This may be understood by imagining your eye to be viewing a scene through a circular opening along the central normal from a point slightly greater than one radius. Now imagine the circle is rotated about a 1 A table of the Heuman lambda function is given in (3), pp. 622-624.
518
Journal of
The Franklin Institnte
The Xolid Angle at a Point Subtended by a Circle diameter to an inclination angle of, say, rr/4. That portion of the circle which approaches your eye opens up a greater portion of the scene to your vision than the portion removed from view by the receding half-boundary. For larger values of X (see Fig. 3, X = l-5) the solid angle diminishes monotonically with 0.
h= 3/2 I*O-
0
IO
20
30
50
40
60
70
00
90
FIG. 3. The solid angle of a circle of radius R with respect to a point hR from the center of the circle and inclined from the central normal by an angle 0 as a function of the inclination angle 0 for twelve values of the radial parameter A.
Special
Cases
When using either Eq. 7 or Eqs. 16 to calculate the solid angle at some inclination angle 0 and at some radial parameter A, the solid angle at the same angle but at the radial parameter l/X can be obtained easily with little additional effort. This fact becomes clear as the several terms and factors in those two equations are examined under the reciprocal transformation: h -+ l/A. Table II lists the response of several quantities to this transformation. Since the quantity 6, changes sign under the reciprocal transformation, the form of Eqs. 16 is modified in the terms dealing with the Heuman lambda functions as well as in the coefficient to the complete elliptic integral of the first kind. Equations 16 become C2 e,; = 277- ‘rr[A&, ( 1 -(34h(Z)
Vol. 287, No. 6, June 1969
CL)-A&$7,
a)]
(1 +X2 + 2h
sin e)-* K(a)
Wa)
519
R. E. Rothe TABLE I So&id Angle of a Circle of radius R with Respect to a point hR from the Center of the Circle and Inclined from the central Normal by an angle tP
h=O
ii=&
X=%
h=+
0
R
e
R
e
a
All
6.283
0.0 12.06 33.64 47.48 62.37 73.54 90.00
4.759 4.795 4.983 5.311 5.535 5.820 6.283
0.0 11.00 22.30 30.30 42.00 55.40 62.70 74.90 81.18 90.00
3.473 3.501 3.597 3.707 3-940 4.355 4.657 5.317 5.995 6.283
h = 1.0
x = 1.02
0s~
en
en
73.01 79.97 83.61 86.47 88.11 90.00
0.0 16.99 31.30 48.78 69.84 78.58 86.38 90.00
h = 0.9804
2.587 2.873 3.131 3.527 4.081 6.283
1.840 1.865 1.927 2.068 2.385 2.603 2.898 3.142
l9 o-0 10.28 20.85 30.30 39.06 51.32 58.02 63.94 74.96 84.44 90.00
n
e
2.798 2.820 2.887 2.995 3.142 3.456 3.702 3.986 4.731 5.653 6.283
0.0 8.56 15.84 20.32 31.73 38.68 46.55 55.44 65.63 78.72 82.71 90.00
h = 1.1
2.316 2.416 2.423 2.246 1.914 o*ooo
9.90 20.07 31.38 41.96 55.09 63.92 71.91 79.20 84.60 88.37 90.00
X=+$ R 2.513 2.523 2.558 2.586 2.710 2.800 2.961 3.214 3.658 4.707 5.195 6.283
x = 1.5
e
en 0.0
73.01 79.97 83.61 86.47 88.11 90.00
X=2
1.634 1.638 1.655 1.689 1.732 1.799 1.838 1.830 1.691 1.258 0.482 0.000
o-o 10.28 20.85 30.30 39.06 51.32 58.02 63.94 74.96 84.44 90.00
~2 1.055 1.055 1.050 1.043 1.024 0.971 0.916 0.843 0.604 0.254 0.000
d
s2
0.0 52.65 61.56 71.21 79.68 84.75 90.00
2.080 2.484 2.698 3.077 3.738 4.582 6.283
--
h = 2.0
e
n
o-0 11.00 22.30 30.30 42.00 55.40 62.70 90.00
0.663 0.655 0.644 0.627 O-578 0.500 0.432 0.000
1 The angle B is expressed in degrees. TABLE
Respome
II
of Various Quantities in the Solid Angle Eqs. 7 and 16 to the Reciprocal Transformation X + l/h Quantity
k2=sin2a Coefficient of lI(N, a) Coefficient of Il(n, a) Coefficient of K(a) in Eq. 7 Sin&* Sin &N &z 8N
Coefficient of K(a) in Eqs. 16 Coefficient by A, functions
520
Response Unchanged Unchanged Unchanged Unchanged Multiply by - 1 Unchanged Unchanged Unchanged Multiply by - 1 Unchanged Multiply by l/h See Eqs. 17
Journal of The Franklin
Institute
The Xolid Angle at a Point Xubtended by a Circle for the case h < 1 and
-($4h(S)
(1+X2+2Xsin8)-*K(a)
(17b)
for the case h > 1. Another special case is the solid angle for a circle with respect to a point along its central normal. In this case, the inclination angle 0 = 0. This solid angle may be calculated from mensuration formulae to obtain fi(O,h) = 257[1-h(l
+P)-*I,
(16)
which agrees with the degenerate forms of Eqs. 7 and 16 when 0 = 0. When h = 1 Eqs. 7 and 16 reduce to the special case where the point Q is on the surface of the hemisphere having the circle of radius R as its great circle. In this case
fi(8, 1) =
297-J(2)
(1 - sin 19)*[i&%)
B(N, a) + K(a) ]
(19)
or !qe, 1) = ?r[l +A&-CY,OL)]--J(2)
G C
1
*K(a).
One final special case is B = 7~12.Here the point Q is coplanar with the circle. By inspection the solid angle for all h < 1 at 0 = x/2 is 2n; similarly, the solid angle for all h > 1 at f3 = n/2 is 0. Equations 7 and 16 do approach these limiting values as the angle 0 approaches 7~12. There is a singular point, however, at h = 1, 8 = n/2 since fi (n/2,1) = n whereas s2(,/2,h < 1) = 2n and fi(r/2, h > 1) = 0. This discontinuity may be understood physically by realizing that if h = 1, 4 = 0, and 0 < &T by an infinitesimal amount, the remaining wall completely excludes the quadrant x > R, z > 0 (see Fig. 2). On the other hand, if for any 4, 8 = +T and h < 1 by an infinitesimal amount that quarter of the full 4~ solid geometry is accessible. Acknowledgement The author wishes to thank H. E. Clark for calculating some of the data for Fig. 3. References (1) “Handbook of Physics and Chemistry”, 45th ed., Robert C. Weast, Chemical Rubber Co., 2310 Superior Ave., N.E., Cleveland, Ohio, p. A-168, 1964. (2) C. Heuman, J. Math. Whys., Vol. 20, pp. 127-206, 1941. of Mathematical Functions”, (3) M. Abramowitz and I. Stegun (eds.), “Handbook U.S. Government Print.ing Office, Washington, D.C. 20402, National Bureau of Standards Applied Math. Ser. No. 55, pp. 587-626, 1964.
Vol.287,No.&June
1969
521
R.
E. ROTHE
Chemical Company Golden, Colorado
Dow
_%BSTRACT: The solid angle subtended calculated.
The solid angle is expressed
central normal
by a circle of radius as a function
to the circle and the distance
hR from
R with respect
of the inclination
to any point angle 9 from
the center of the circle.
given in terms of tabulated
elliptic
and related functions.
function of the inclination
angle is given for several values of h. Several
Cs the
The result is
A graph of the solid angle as a special
cases are
discussed.
Introduction
The solid angle subtended by a circle with respect to a point along its central normal is easily calculated by handbook mensuration formulas (1). 3s the point moves off the central normal, however, the solid angle calculation becomes more complicated. The circle’s projection becomes an ellipse whose minor axis diminishes as the angle of inclination increases. As the elliptical eccentricity increases, the solid angle generally decreases. In fact, as the point approaches the plane of the circle, the solid angle approaches either 0 or 2~ depending on whether the point lies outside or inside the circle. The results presented in this paper give the solid angle subtended by a circle with respect to any point in space. Several special cases are given which illustrate the theory. Interest in this problem originated with critical mass studies on fissionable material having hemispherical cavities. The question is raised as to what fraction of the neutrons leaking out of the interior surface would escape the hemisphere without further interaction with the fissionable material. Another critical mass application of these results involves the calculation of the geometrical interaction between elements of a uniform array of thin circular disks of fissionable material. The formulas presented in this paper are not restricted to critical mass physics; they are applicable to any problem dealing with the linear propagation of radiation from a point source. Statement
of the Problem
The point Q of Fig. 1 is a distance AR from the center of a circle of radius R and inclined from the central normal to the circle by an angle 0. The shading * This work was supported AT(29-l)-1106.
by the U.S. Atomic
Energy
Commission
under
contract
R. E. Rothe . represents the boundary to the solid angle to be calculated. All lines within the boundary cone contribute to the solid angle ; those outside the cone do not. The problem is azimuthally symmetric.
Q
FIG. 1. The point Q is a distance AR from the center of t,he circle of radius R and inclined from the cent,4
normal to the circle by an angle 0.
Lines emanating from point Q of Fig. 2 which pass through the area element dA = rd rd $ in the equatorial plane (z = 0) contribute to the solid angle to be calculated. The total solid angle subtended by the circle of
X
Y
FIG. 2. Lines leaving point Q at (R’ = AR, 8, 4 = 0) pass through the area element dA = T dr dq5 in the plane z = 0. The solid angle contribution vector p between Q and dA makes an angle [ with the normal to the plane z = 0.
radius R from the point Q is given by R Cl=
516
2n
ss0 0
cos
f p-2
rd rd+.
(1)
Journal of The Franklin
Institute
The Solid Angle at a Point Subtended by a Circle Coordinates r and I$ are in the plane z = 0; the vector element rdrd+ to the point Q ; and cost
p joins the area
= Ip*&Ip+ = R’p-lcos8,
where R’ = hR. The magnitude
(2)
of p is
p2 = r2+ R’2-
2r*R’,
(3)
where 1.8’
= cos 4 cos ($7 - 0) + sin 4 sin ($7 - 0) cos &r = cos $ sin 8,
(4)
according to the law of cosines for spherical triangles. The integral form for the solid angle is obtained by substituting into Eq. 1 to obtain Q(8,
R’) =
R
2n
ss0 0
rR’ cos 8 (r2 + RI2 - 2rR’sin 0 cos +)-* dr d$.
The integral over r is performed sphere radius R vanishes leaving Q(B,X) =
Eqs. 2-4
s2n
first ; explicit dependence
(5)
upon the hemi-
COS ed4 o (1 -sin28cos2+)
s
2n (Lsin0cos+) o (l-sin2Bcos2+)
-cose(l+xy+
1_2Xsin0cos+ 1+x2
d4*
(6)
The first integral is 277; the second integral may be expressed in terms of elliptic integrals or related special functions. The solid angle subtended by a circle of radius R from a point Q at R’ = AR and 6’ in terms of tabulated functions is (1+X2-22Xsin0)n(N,ol)
2cose
~(e”)=2,-(l+,2+2,sin8)t
[ (l+h)(l+sin@
(1-h) -(l+sin8)
rI(n,(II)+ &)
G)]
(7)
Here II and K are the complete elliptic integrals of the third and first kind, respectively. The elliptic parameters n and N and the modular angle 01are related to the inclination angle 0 and to X by the relations: 2sin8 n=l+ (1+A)” (1+h2+2hsin0)’
sin2a =
4h sin
e
(l+h2+2XsinO)
=
k2.
(9) (10)
The complete elliptic integrals of the third kind, II(N, CX)and II(n, 01), of Eq. 7 diverge for any a!at N = 1 or n = 1. These values of N and n correspond to 8 = ~12. The solid angle does not become infinite there since the coefficients
Vol.287,No.F,June1969
517
R. E. Rothe of these functions vanish at that angle. To facilitate calculations of the solid angle for all angles 0 including those near ~12, the transformation set given in Eqs. 11 is used, i.e. IW,
a) = K(a) + Q~S,vP -A&v,
rI(n,cY)=
41,
(11%)
K(ol)+$&[l-R,(&,,CY)].
(11b)
Here A, is the Heuman lambda function. The parameters Ed and E, and the coefficients 6, and 6, are related to the inclination angle 19and to h by t,he relations : ‘lnE~~ =
1 -sin@ * 1 + sin B ’
(
)
(12) (13)
6,, =
1+h l+P-22hsin0
1 ( l-sine
6, = k(1+X2+2XsinB)* n 1-x
‘(1+X2+2Xsine)&,
(14)
)
l+sine 1 (
1 1 *
(15)
In Eq. 15, the plus sign is used for X < 1 and the minus sign is used for h > 1. M7hen the substitutions of Eqs. 11 are put into Eq. 7, the solid angle expression becomes
-
4h( 1 -sin
e)*(l + sin
e)-$(I + A2 +
2h sin
e)-+ K(E)
(164
e)-*(I + A2 +
2h sin
e)-* K(a)
(1610)
for the case h < 1 and We, A) = “[A&,,
-
a) + A,,(+,
4h( 1 - sin
e)*(l +
41 sin
for the case h > 1. Equations 16a and 16b, containing the Heuman lambda functions,* facilitate calculations in the vicinity of the divergence of II(n, a) and Il(N, a) since 0 < AO(.e,a) < 1 for all 0 < E6 77/Z and 0 6 01Q 7r/2. The graph of Fig. 3 presents the solid angle of a circle of radius R with respect to a point AR from the center of the circle and inclined from the central normal by an angle 8. The solid angle is shown as a function of the angle 0 for several values of the parameter h. Table I presents the solid angle data in tabular form. Note that the solid angle does not decrease montonically with inclination angle for all X > 1 as might be expected. For values of X only slightly greater than one, the solid angle increases slightly with increasing angle. It does reach zero, however, when the inclination angle finally reaches rr12 (see Fig. 3, X = 1.1). This may be understood by imagining your eye to be viewing a scene through a circular opening along the central normal from a point slightly greater than one radius. Now imagine the circle is rotated about a 1 A table of the Heuman lambda function is given in (3), pp. 622-624.
518
Journal of
The Franklin Institnte
The Xolid Angle at a Point Subtended by a Circle diameter to an inclination angle of, say, rr/4. That portion of the circle which approaches your eye opens up a greater portion of the scene to your vision than the portion removed from view by the receding half-boundary. For larger values of X (see Fig. 3, X = l-5) the solid angle diminishes monotonically with 0.
h= 3/2 I*O-
0
IO
20
30
50
40
60
70
00
90
FIG. 3. The solid angle of a circle of radius R with respect to a point hR from the center of the circle and inclined from the central normal by an angle 0 as a function of the inclination angle 0 for twelve values of the radial parameter A.
Special
Cases
When using either Eq. 7 or Eqs. 16 to calculate the solid angle at some inclination angle 0 and at some radial parameter A, the solid angle at the same angle but at the radial parameter l/X can be obtained easily with little additional effort. This fact becomes clear as the several terms and factors in those two equations are examined under the reciprocal transformation: h -+ l/A. Table II lists the response of several quantities to this transformation. Since the quantity 6, changes sign under the reciprocal transformation, the form of Eqs. 16 is modified in the terms dealing with the Heuman lambda functions as well as in the coefficient to the complete elliptic integral of the first kind. Equations 16 become C2 e,; = 277- ‘rr[A&, ( 1 -(34h(Z)
Vol. 287, No. 6, June 1969
CL)-A&$7,
a)]
(1 +X2 + 2h
sin e)-* K(a)
Wa)
519
R. E. Rothe TABLE I So&id Angle of a Circle of radius R with Respect to a point hR from the Center of the Circle and Inclined from the central Normal by an angle tP
h=O
ii=&
X=%
h=+
0
R
e
R
e
a
All
6.283
0.0 12.06 33.64 47.48 62.37 73.54 90.00
4.759 4.795 4.983 5.311 5.535 5.820 6.283
0.0 11.00 22.30 30.30 42.00 55.40 62.70 74.90 81.18 90.00
3.473 3.501 3.597 3.707 3-940 4.355 4.657 5.317 5.995 6.283
h = 1.0
x = 1.02
0s~
en
en
73.01 79.97 83.61 86.47 88.11 90.00
0.0 16.99 31.30 48.78 69.84 78.58 86.38 90.00
h = 0.9804
2.587 2.873 3.131 3.527 4.081 6.283
1.840 1.865 1.927 2.068 2.385 2.603 2.898 3.142
l9 o-0 10.28 20.85 30.30 39.06 51.32 58.02 63.94 74.96 84.44 90.00
n
e
2.798 2.820 2.887 2.995 3.142 3.456 3.702 3.986 4.731 5.653 6.283
0.0 8.56 15.84 20.32 31.73 38.68 46.55 55.44 65.63 78.72 82.71 90.00
h = 1.1
2.316 2.416 2.423 2.246 1.914 o*ooo
9.90 20.07 31.38 41.96 55.09 63.92 71.91 79.20 84.60 88.37 90.00
X=+$ R 2.513 2.523 2.558 2.586 2.710 2.800 2.961 3.214 3.658 4.707 5.195 6.283
x = 1.5
e
en 0.0
73.01 79.97 83.61 86.47 88.11 90.00
X=2
1.634 1.638 1.655 1.689 1.732 1.799 1.838 1.830 1.691 1.258 0.482 0.000
o-o 10.28 20.85 30.30 39.06 51.32 58.02 63.94 74.96 84.44 90.00
~2 1.055 1.055 1.050 1.043 1.024 0.971 0.916 0.843 0.604 0.254 0.000
d
s2
0.0 52.65 61.56 71.21 79.68 84.75 90.00
2.080 2.484 2.698 3.077 3.738 4.582 6.283
--
h = 2.0
e
n
o-0 11.00 22.30 30.30 42.00 55.40 62.70 90.00
0.663 0.655 0.644 0.627 O-578 0.500 0.432 0.000
1 The angle B is expressed in degrees. TABLE
Respome
II
of Various Quantities in the Solid Angle Eqs. 7 and 16 to the Reciprocal Transformation X + l/h Quantity
k2=sin2a Coefficient of lI(N, a) Coefficient of Il(n, a) Coefficient of K(a) in Eq. 7 Sin&* Sin &N &z 8N
Coefficient of K(a) in Eqs. 16 Coefficient by A, functions
520
Response Unchanged Unchanged Unchanged Unchanged Multiply by - 1 Unchanged Unchanged Unchanged Multiply by - 1 Unchanged Multiply by l/h See Eqs. 17
Journal of The Franklin
Institute
The Xolid Angle at a Point Xubtended by a Circle for the case h < 1 and
-($4h(S)
(1+X2+2Xsin8)-*K(a)
(17b)
for the case h > 1. Another special case is the solid angle for a circle with respect to a point along its central normal. In this case, the inclination angle 0 = 0. This solid angle may be calculated from mensuration formulae to obtain fi(O,h) = 257[1-h(l
+P)-*I,
(16)
which agrees with the degenerate forms of Eqs. 7 and 16 when 0 = 0. When h = 1 Eqs. 7 and 16 reduce to the special case where the point Q is on the surface of the hemisphere having the circle of radius R as its great circle. In this case
fi(8, 1) =
297-J(2)
(1 - sin 19)*[i&%)
B(N, a) + K(a) ]
(19)
or !qe, 1) = ?r[l +A&-CY,OL)]--J(2)
G C
1
*K(a).
One final special case is B = 7~12.Here the point Q is coplanar with the circle. By inspection the solid angle for all h < 1 at 0 = x/2 is 2n; similarly, the solid angle for all h > 1 at f3 = n/2 is 0. Equations 7 and 16 do approach these limiting values as the angle 0 approaches 7~12. There is a singular point, however, at h = 1, 8 = n/2 since fi (n/2,1) = n whereas s2(,/2,h < 1) = 2n and fi(r/2, h > 1) = 0. This discontinuity may be understood physically by realizing that if h = 1, 4 = 0, and 0 < &T by an infinitesimal amount, the remaining wall completely excludes the quadrant x > R, z > 0 (see Fig. 2). On the other hand, if for any 4, 8 = +T and h < 1 by an infinitesimal amount that quarter of the full 4~ solid geometry is accessible. Acknowledgement The author wishes to thank H. E. Clark for calculating some of the data for Fig. 3. References (1) “Handbook of Physics and Chemistry”, 45th ed., Robert C. Weast, Chemical Rubber Co., 2310 Superior Ave., N.E., Cleveland, Ohio, p. A-168, 1964. (2) C. Heuman, J. Math. Whys., Vol. 20, pp. 127-206, 1941. of Mathematical Functions”, (3) M. Abramowitz and I. Stegun (eds.), “Handbook U.S. Government Print.ing Office, Washington, D.C. 20402, National Bureau of Standards Applied Math. Ser. No. 55, pp. 587-626, 1964.
Vol.287,No.&June
1969
521