Solid angle

Solid angle in a sphere with radius R

{\ displaystyle W}

The solid angle is the three-dimensional equivalent of the (for the plane defined) angle . It describes the proportion of the total space that z. B. lies inside a given cone or pyramid jacket .

definition

The solid angle is defined as the area of a part of a spherical surface divided by the square of the radius of the sphere: ${\ displaystyle \ Omega}$ ${\ displaystyle A}$ ${\ displaystyle F}$ ${\ displaystyle r}$

{\ displaystyle \ Omega = {\ frac {A} {r ^ {2}}}}

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When considering the unit sphere ( ) is the same as the corresponding solid angle. So the full solid angle is equal to the surface of the unit sphere, namely . ${\ displaystyle r = 1}$ ${\ displaystyle A}$ ${\ displaystyle 4 \ pi}$

The partial area can be of any shape. Vectorially written as a surface integral is

{\ displaystyle \ Omega = \ iint _ {F} {\ frac {{\ hat {\ vec {n}}} \ cdot \ mathrm {d} {\ vec {A}}} {r ^ {2}}} }

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It is the unit vector from the origin, the differential surface element and its distance from the origin. ${\ displaystyle {\ hat {\ vec {n}}}}$ ${\ displaystyle \ mathrm {d} {\ vec {A}}}$ ${\ displaystyle r}$

Contrary to what the picture might suggest, the shape of the area is not important. Every outline shape on the spherical surface with the same area defines a solid angle of the same size. If you put a half-line (also called a ray) through each point of the outline with the center of the sphere as the starting point, you get a geometric figure that illustrates the solid angle. (This is comparable to the representation for an angle in the plane: two half-lines with a common starting point.)

units

Although the solid angle is a quantity of the dimension number , it is usually given in the unit steradian (sr) for clarity ; this corresponds to the radian measure with the unit radian (rad) for a plane angle. A solid angle of 1 sr encloses an area of 1 m ² on a sphere with a radius of 1 m . Since a whole spherical surface has the area , the corresponding full solid angle is ${\ displaystyle 4 \ pi r ^ {2}}$

{\ displaystyle \ Omega = 4 \ pi \ \ mathrm {sr} \ approx 12 {,} 56637 \ \ mathrm {sr}}

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Occasionally, solid angles are also given in square degrees , (°) ². 1 (°) ² is equal to (around 0.00030462) sr. ${\ displaystyle \ left ({\ tfrac {2 \ pi} {360}} \ right) ^ {2}}$

The use of an auxiliary unit of measurement for a quantity of the dimension number has the advantage, as in many areas, in particular also in the case of solid angles, that the unit used shows which physical quantity is meant. In contrast to the luminous flux (lm), the luminous intensity (cd = lm / sr) shows its dependence on the solid angle through the occurrence of the steradian in the unit. The light intensity thus denotes a luminous flux that is dependent on the solid angle.

Canonical solid angle

If you choose a circle as the outline shape on the spherical surface, you get the canonical solid angle. The solid angle then forms the jacket of a straight circular cone with the center of the sphere in the tip.

The full, flat opening angle of the cone can be converted into the solid angle: ${\ displaystyle \ omega}$

{\ displaystyle \ Omega = 2 \ pi \ left (1- \ cos {\ frac {\ omega} {2}} \ right) = 4 \ pi \ sin ^ {2} {\ frac {\ omega} {4} }}

Opening angle in degrees ${\ displaystyle \ omega}$	0	1	2	5	10	15th	30th	45	57.2958
Opening angle in radians ${\ displaystyle \ omega}$	0.0000	0.0175	0.0349	0.0873	0.1745	0.2618	0.5236	0.7854	1.0000
Solid angle in square degrees ${\ displaystyle \ Omega}$	0.00	0.79	3.14	19.63	78.49	176.46	702.83	1570.10	2525.04
Solid angle in steradian ${\ displaystyle \ Omega}$	0.0000	0.0002	0.0010	0.0060	0.0239	0.0538	0.2141	0.4783	0.7692

Opening angle in degrees ${\ displaystyle \ omega}$	60	65.5411	75	90	120	150	180	270	360
Opening angle in radians ${\ displaystyle \ omega}$	1.0472	1.1439	1.3090	1.5708	2.0944	2.6180	3.1416	4.7124	6.2832
Solid angle in square degrees ${\ displaystyle \ Omega}$	2763.42	3282.81	4262.39	6041.36	10313.24	15287.95	20626.48	35211.60	41,252.96
Solid angle in steradian ${\ displaystyle \ Omega}$	0.8418	1.0000	1.2984	1.8403	3.1416	4.6570	6.2832	10.7261	12.5664

Spherical triangle

The solid angle of a spherical triangle is steradian, depending on its interior angles . ${\ displaystyle (\ alpha + \ beta + \ gamma - \ pi)}$

Solid angle and spherical coordinates

A solid angle from a Cartesian polar coordinate segment

In a spherical coordinate system , the solid angle can be defined very clearly, since there is no radial variable. Two meridian angles φ ₁ , φ ₂ and two latitude angles γ ₁ , γ ₂ determine a surface element on a spherical surface. The corresponding solid angle is:

{\ displaystyle \ Omega = \ int \ limits _ {\ varphi _ {1}} ^ {\ varphi _ {2}} \ int \ limits _ {\ gamma _ {1}} ^ {\ gamma _ {2}} \ sin \ gamma \, \ mathrm {d} \ gamma \, \ mathrm {d} \ varphi}

Oosterom and Strackee Formula

Three vectors starting from a point P , and define a general triangle. For the spanned solid angle with the vertex P applies: ${\ displaystyle {\ vec {r}} _ {1}}$ ${\ displaystyle {\ vec {r}} _ {2}}$ ${\ displaystyle {\ vec {r}} _ {3}}$ ${\ displaystyle \ Omega}$

{\ displaystyle \ tan \ left ({\ frac {1} {2}} \ Omega \ right) = {\ frac {\ mathrm {det} \ left ({\ vec {r}} _ {1}, {\ vec {r}} _ {2}, {\ vec {r}} _ {3} \ right)} {\ | {\ vec {r}} _ {1} \ | \ | {\ vec {r}} _ {2} \ | \ | {\ vec {r}} _ {3} \ | + \ langle {\ vec {r}} _ {1}, {\ vec {r}} _ {2} \ rangle \ | {\ vec {r}} _ {3} \ | + \ langle {\ vec {r}} _ {1}, {\ vec {r}} _ {3} \ rangle \ | {\ vec {r} } _ {2} \ | + \ langle {\ vec {r}} _ {2}, {\ vec {r}} _ {3} \ rangle \ | {\ vec {r}} _ {1} \ | }}}

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This representation was given and proven in 1983 by Oosterom and Strackee.

Solid angle of a pyramid

To the solid angle of a pyramid

The special case of the solid angle with a rectangular and flat outline corresponds to the geometric shape of a pyramid, the origin being exactly perpendicular to the center of the flat rectangle, see Fig. Illustration. This solid angle occurs z. B. when calculating the étendue of optical systems with rectangular apertures.

It can be calculated very easily using the Oosterom and Strackee formula. With the pyramid bases and as well as the height h we get: ${\ displaystyle w_ {x}}$ ${\ displaystyle w_ {y}}$

{\ displaystyle \ Omega = 4 \ arctan {\ frac {w_ {x} \ cdot w_ {y}} {2h \ cdot {\ sqrt {4h ^ {2} + w_ {x} ^ {2} + w_ {y } ^ {2}}}}}}

If the two opening angles 2 φ _x and 2 φ _y are used for the calculation (where tan φ _x = w _x / 2 h and tan φ _y = w _y / 2 h ), after some trigonometric transformations it follows:

{\ displaystyle \ Omega = 4 \ arcsin \ left (\ sin \ varphi _ {x} \ sin \ varphi _ {y} \ right)}

example 1: A rectangular aperture in front of a point light source limits the light beam to the angles 45 ° ( ) and 20 ° ( ). The solid angle is 0.27 sr. ${\ displaystyle \ varphi _ {x} = 22 {,} 5 ^ {\ circ}}$ ${\ displaystyle \ varphi _ {y} = 10 ^ {\ circ}}$
Example 2: If it is a square aperture and both angles are 20 °, the solid angle is 0.12 sr. The canonical solid angle of a 20 ° circular aperture is 0.10 sr.

Notation

∢

The symbol »das« ( TeX , Unicode U + 2222 spherical angle , no HTML entity ) is available for the formula set , which can be found in the Unicode block Mathematical Operators . The sign corresponds to the Anglo-American customs, in the European set of formulas a confusingly similar sign is common for the flat angle . \sphericalangle

Web links

Commons : Solid angle - collection of images, videos and audio files

Individual evidence

^ A. Van Oosterom, J. Strackee: The Solid Angle of a Plane Triangle . In: Biomedical Engineering, IEEE Transactions on . BME-30, No. 2 , 1983, p. 125-126 , doi : 10.1109 / TBME.1983.325207 .

[1] A. Van Oosterom, J. Strackee: The Solid Angle of a Plane Triangle . In: Biomedical Engineering, IEEE Transactions on . BME-30, No. 2 , 1983, p. 125-126 , doi : 10.1109 / TBME.1983.325207 .