Talk:Solid angle/Archive 1

Archive 1

Solid angle formulas

I've updated the page with a formula for the solid angle subtended by a rectangular pyramid. If you'd like to know where this somewhat messy-looking formula comes from I've posted a derivation here (pdf file): [1].

Bgerke 20:50, 1 February 2007 (UTC)

Excellent job. I always wondered where that pyramid formula came from, which was originally in the article. Thanks for generalizing it to a rectangular pyramid. -Amatulic 22:28, 1 February 2007 (UTC)

Confused Definition of Subtend

At the top of the article, it is said that "The solid angle, Ω, is the angle in three-dimensional space that an object subtends at a point." In other words, an object subtends an angle. But further down in the article under Tetrahedron, it says, "The solid angle at O that subtends the triangular surface ABC..." This implies that an angle subtends an object. I believe Wolfram uses the former definition, although in the body of their own article on solid angle, there is one instance of the reverse, I think. Could someone please rectify this confusion? I've seen the former definition being used most frequently, so this article should match this for both consistency and accuracy, if indeed that is the correct definition.Firth m (talk) 00:13, 25 February 2008 (UTC)

Yes I agree, it is the object that subtends the solid angle. So the solid angle either extends the object or is subtended by the object. Tetrahedron page updated accordingly Frank M Jackson (talk) 10:40, 25 February 2008 (UTC)

Solid angle of a triangle: inside-outside clarification and periodicity of solid angle ?

When I reverse all the 3 points in the triangle formula: x_i := -x_i, then the determinant in the enumerator also flips by -1. The denominator does not change though and therefore the resulting fraction flips by -1. I feel that solidangle=solidangle+4pi (like angle=angle+2pi) and therefore the -1 can be represented in the positive range again. This would then basically be the difference between the solid angle "inside" the triangle and outside (the remaining sphere). If I have just three points, do I have to enter them in a specific order into the formula to get inside/outside right (right-handed rule pointing towards the center or something)? Now, I test for the smaller solid angle (inside should be less than outside), since the triangle cant occupy more than 2pi. I feel however, this could be clarified in the article ... 89.53.89.210 (talk) 17:20, 6 July 2008 (UTC)

How to Calculate the sum of Solid angles of a triangulated 3D object respect to a point which is inside the object?

How to Calculate the sum of Solid angles of a triangulated 3D object respect to a point which is inside the object? —Preceding unsigned comment added by 59.160.213.4 (talk) 06:57, 14 October 2008 (UTC)

old comments

Shouldn't this be listed under geometry, not physics? LlubNek (talk) 19:37, 3 March 2009 (UTC)

Possible correction (I'm not sure): Under Tetrahedron, the definition of "Theta sub a" should perhaps be angle BOC instead of OBC and similarly for the others. Could the author or someone confirm this?Tap17 (talk) 07:14, 23 February 2008 (UTC). Yes it is a correction. Page now updated. Thanks Frank M Jackson (talk)

The solid angle is not an angle!! it's a surface area of a projection!! check Wolfram's mathworld. —Preceding unsigned comment added by 200.119.3.210 (talk) 22:36, 18 December 2007 (UTC)

"To get an angle, imagine two lines passing through the center of a unit circle. The length of the arc between the lines on the unit circle is the angle, in radians."

this definition would make the angle dependent on radius. this definition definition gives the angle units of length, absurd. this should be "length of arc divided by the radius"

Unit circle = circle with radius 1. EJ 19:29, 6 Jun 2005 (UTC)

Can someone explain the notation on the equation at the bottom of the page? Is Omega the solid angle? Do the italics represent the magnitude of the vectors? What do the brackets at the top mean?

Use of the word "identically"

The articles states:

"Over 2200 years ago Archimedes proved, without the use of calculus, that the surface area of a spherical cap mapped identically onto the area of a circle whose radius was equal to the distance from the rim of the spherical cap to the point where the cap's axis of symmetry intersects the cap."

It appears that the intent here is to say that a spherical cap maps onto a disk of a certain radius so that area is preserved. But this is certainly not what the word "identically" means, in or out of mathematics. I suggest a clearer wording here.

Also, when a mapping is mentioned, it is usually helpful to include a description of how the mapping is defined.Daqu (talk) 18:19, 10 February 2009 (UTC)

Rather than complicate the paragraph why not simplify it to say -

"Over 2200 years ago Archimedes proved, without the use of calculus, that the surface area of a spherical cap was always equal to the area of a circle whose radius was equal to the distance from the rim of the spherical cap to the point where the cap's axis of symmetry intersects the cap."Frank M Jackson (talk) 18:35, 12 February 2009 (UTC)

Sounds good. Also, could there be a simpler way to express "the point where the cap's axis of symmetry intersects the cap" ?

Have amended article as discussed. Please suggest alternative to "the point where the cap's axis of symmetry intersects the cap" Frank M Jackson (talk) 10:20, 15 February 2009 (UTC)

Finally, when speaking of spherical caps and their generalizations -- spherical zones -- there is a rather counterintuitive theorem perhaps worth mentioning as well: On a given sphere, the spherical area lying between two parallel planes that intersect the sphere is proportional to the distance between the planes. Daqu (talk) 19:33, 12 February 2009 (UTC)

Suggest that you include such a theorem only if it is pertinent to the calculation of a specific Solid Angle. Remember that the Archimedes post is an historical aside. So an aside to an aside may not be so good. Frank M Jackson (talk) 10:20, 15 February 2009 (UTC)

Good point. But in fact, it is relevant. Spherical zones, of which a spherical cap is a special case, have a simple area formula on a sphere of radius R:

A = 2π R h

where h is the distance between the two parallel planes that define the zone as the portion of the sphere (which they are required to intersect) lying between them. To calculate the solid angle of a spherical zone with respect to the sphere's center, just shrink it so that it lies on a unit sphere -- or in other words, just divide its area by R². This results in the formula (where Ω means solid angle with respect to the center of the sphere)

Ω(spherical zone of height h and radius R) = 2π h / R

A spherical zone is a very common shape. (For example, the Tropic of Cancer, the Tropic of Capricorn, everything above the Arctic Circle or below the Antarctic Circle, and the Temperate Zones are all examples of spherical zones.)Daqu (talk) 04:24, 3 April 2009 (UTC)

Amending fundamental formula

Will the anonymous person who is changing fundamental formula please check both the original citation and Mathworld and discuss in the talk page before making amendments. Please note that both are correct because

$\Omega =\iint _{S}{\frac {{\vec {r}}\cdot {\vec {n}}dS}{r^{3}}}.$

is the same as

$\Omega =\iint _{S}{\frac {{\vec {l}}\cdot {\vec {dS}}}{r^{2}}}.$

where, if you use the original formula, ${\vec {r}}$ is the vector position of an infinitesimal area of surface $\,dS$ with respect to point P and where ${\vec {n}}$ represents the unit vector normal to $\,dS$.

Or where, if you use the formula at Mathworld ${\vec {l}}$ is a unit vector from the origin i.e. ${\frac {\vec {r}}{r}}$ and ${\vec {dS}}$ is the differential area of a surface patch i.e. ${\vec {n}}dS$.

Though both are right the formula

$\Omega =\iint _{S}{\frac {{\vec {r}}\cdot {\vec {n}}dS}{r^{3}}}.$

is a better representation when trying to perform actual integrations in 3D to calculate the Solid angle.

The current amendment is clearly dimensionally incorrect.

This is why I have changed the citation and formula back to the original. Frank M Jackson (talk) 18:47, 12 December 2008 (UTC)

I missed the section about defining ${\vec {r}}$ as the position of an infinitesimal area. I would contend that using ${\vec {r}}$ to denote this is confusing to the average reader. At the very least, we should create an image that makes this clearer. My apologies for the mistake. ~Anonymous —Preceding unsigned comment added by 136.152.175.75 (talk) 00:20, 13 December 2008 (UTC)

Perhaps using $\Omega =\iint _{S}{\frac {{\vec {r}}\cdot {\hat {n}}dS}{r^{3}}}$ would make it more obvious.

Good point. Have updated page accordingly. 13:36, 13 December 2008 (UTC) —Preceding unsigned comment added by Fjackson (talk • contribs)

Error (?) in quote on Archimedes proof----------------- Have not worked on geometry at this level for my many years as a field and bench scientist. I think properly worded version of the quoted proof by Archimedes would be remarkably useful to me at the moment, but I am concerned there is an error of some sort in the wording as it makes no physical sense. Please consider the following (from below):

"Over 2200 years ago Archimedes proved, without the use of calculus, that the surface area of a spherical cap was always equal to the area of a circle whose radius was equal to the distance from the rim of the spherical cap to the point where the cap's axis of symmetry intersects the cap."Frank M Jackson (talk) 18:35, 12 February 2009 (UTC)

It seems to follow directly that this does not make sense physically or mathematically. In essence it is stating that the surface area of a spherical cap is equal to the surface area of the wall of a cone the base and height of which are equal to those of the spherical cap. To take a simple case consider a hemisphere (bear with my use of math notations as I am unfamiliar with the wiki text/editing). The surface of a hemisphere is 1/2 that of a sphere, or [2 π r²]. The surface area of a cone is[ π r s], where s =(square root of the sum of the squares of the radius and height of the cone). In this case, however, the space is a hemisphere and the height of the subject cone is equal to the radius of the cone. Therefore, the surface area of this cone is given by [π r √(r²+r²)], or, [√2 π r²]. So, if the area of the spherical cap were equal to the area as described in the quote above, then it would be necessary that [2 π r²] = [√2 π r²], that is, that 2 = √2. That seems to me to present a problem. Have I made a mistake? If so, can someone show me my mistake? If not, can someone direct me to a more thorough treatment of Archimedes proof, and make appropriate corrections to the wording in the relevant part of this wiki article? Additionally, I think the associated figure should be reviewed for correctness and clarity in that it is explicitly or implicitly associated with this quote and subsequent equations/discussions in the article. Beepbeep777 (talk) 15:59, 18 May 2011 (UTC) found my mistake -- suggest improving figure associated with discussion of Archimedes proof -- see at my user page Beepbeep777 (talk) 16:26, 18 May 2011 (UTC)

Solid angles in higher dimensions?

Does anyone know formulas for solid angles (aside from the surface (d-1)-volume of the d-dimensional ball) in higher dimensions? Especially appreciated would be an analogue of the tan(Omega/2) formula (the one using the triple product and the 3 dot products) in an arbitrary number of dimensions.

i dont think its likely. the cross product is only defined for the r3 euclidean vector space, so i dont expect its in that form. btw might they define it as (in r4) (surface volume)/r^3? I got scammed 10:09, 5 September 2006 (UTC)

There must be a way. I'm not a mathematician (just a physicist). The 3D version can be derived from the Levi-Civita tensor (see Levi-Civita symbol) using 3 indices. Perhaps a similar tensor can be defined in N-space? If any pair of indices are equal, the element is zero. If the indices are permuted evenly, the element is 1, and if they are permuted oddly, the element is -1. Summation of the LC Density over N-1 dimensions with N-1 vectors might give you a N-dimensional cross-product. Can anyone confirm this or play with it? --MxBuck 05:58, 2 November 2007 (UTC)

• There is a recurrence formula for "surface" and "volume", S_n and V_n, respectively, of n-D spheres; namely, S_n=2πR×V_n−2 and V_n=[R×S_n]/n. If the n-D solid angle is defined as: Ω_n=S_n/Rⁿ⁻¹, I wonder if we could determine Ω_n subtended at the center of the "sphere" in this manner. If so, we would have the following results: Ω₁=2, Ω₂=2π, Ω₃=4π, Ω₄=2п², Ω₅=8п²/3, Ω₆=п³, etc. Zymogen (talk) 14:18, 19 May 2008 (UTC)

Or if you want a single formula for the (n-1)-dimensional volume S_n of a unit sphere in n-space (which, by the way, is not called an n-dimensional sphere but rather an (n-1)-dimensional sphere), and the n-dimensional volume V_n of a unit ball (everything inside and on the unit sphere) in n-space:

   Sn  =  2 πn/2 / Γ(n/2)

and

   Vn  =   2 πn/2 / (n Γ(n/2))

where Γ(x) denotes the gamma function evaluated at x.

For a sphere of radius r in n-space, multiply S_n by r^n-1. For a ball of radius r in n-space, multiply V_n by rⁿ.Daqu (talk) 04:39, 3 April 2009 (UTC)

"The Sun and Moon both subtend a solid angle of 5.11×10-6 sr in the sky of Earth, or about 1/2,500,000th of the celestial sphere."

Could this be wrong? Using the formula provided an answer of 5.98×10-5 is given.

Another web based article appears to be correct -

"The sun and moon are both seen from Earth at a fractional area of 0.001% of the celestial hemisphere or 5.98×10-5 steradian [1]."

Proper definition of solid angle (as segment area of the unit sphere) works in any dimension (including lower dimension 2D!!!). Any arguments about coefficients are meaningless. Just go to definition of area in higher dimension. —Preceding unsigned comment added by 128.231.61.91 (talk) 18:21, 23 April 2009 (UTC)

Much appreciate if the author could kindly give a source of the formula for solid angle Omega-d. Is the formula true for any d? 174.91.78.44 (talk) 23:49, 9 October 2011 (UTC)

Pyramid - possible problem with formulae

Anonymous user suggests that there is a problem with these formulae - "Please check the validity of result for n=4,r=1 and h=1 (i.e. pyramid with square base (of circum-radius 1) and height 1 ) by 1st n 3rd formula!!!". Could main contributor please comment.Frank M Jackson (talk) 10:25, 15 November 2011 (UTC)

Qualifying formulas with "sr" and "rad"

Throughout this article the units of measure "sr" and "rad" are placed, variously, at the end of solid angle formulas. These units of measure that are dimensionless are adequately explained at the beginning of the article. To me it seems pointless "peppering" the article with "sr" and "rad". Furthermore it leads to confusion since "sr" and "rad" can be construed incorrectly as additional variables in these formulas. Unless there are any objections, I plan to remove these unnecessary qualifiers. Frank M Jackson (talk) 13:15, 3 January 2015 (UTC)

Now completed. Frank M Jackson (talk) 08:24, 9 January 2015 (UTC)

Incredibly inadequate definition

This article attempts — but fails miserably — to define its subject with the sentence:

"An object's solid angle in steradians is equal to the area of the segment of a unit sphere, centered at the angle's vertex, that the object covers."

The word "covers" could mean almost anything. And the phrase "segment of a sphere" is a technical term — that even has its own Wikipedia article — and is totally inappropriate here.

It would greatly benefit Wikipedia if people who do not know what they are talking about — or who know what they are talking about but have no idea how to explain it clearly — refrain from editing Wikipedia articles. I say this out of exasperation after encountering probably hundreds of articles with both mistakes and extremely unclear explanations.Daqu (talk) 20:48, 2 February 2015 (UTC)

If you want to improve the quality of this article, don't rant. Simply make a positive contribution by suggesting an alternative definition. I am not sure that such a definition will fit into one sentence. Nevertheless, in order to open up a discussion on improving the definition and in fear of failing miserably, I suggest:

"Given a reference point P and a 3-dimensional surface S, then the solid angle $\Omega$ subtended by S at point P is defined as the surface area of the projection of S onto a unit sphere centered at P". Frank M Jackson (talk) 15:52, 3 February 2015 (UTC)

I think the kind of solid angle that can occur in a surface at a cusp (of the kind seen on an onion dome) needs a subtler definition of solid angle "at a point". Or even a simpler example: the solid angle at a point on the ridge of a surface of the region common to two intersecting spherical balls. These examples go beyond the usual examples "subtended by S a a point".Daqu (talk) 20:43, 4 February 2015 (UTC)

Links to other languages

Does anyone know why the article does not show any links to other languages (at least currently and on my computer)? According to https://www.wikidata.org/wiki/Q208476#sitelinks-wikipedia the article is already listed in the database. What else needs to be done? --92.224.201.116 (talk) 12:22, 27 January 2016 (UTC)

explanation

brackets = triple scalar product. italics = magnitude of vector. Omega= solid angle.

152.78.98.1 15:53, 7 April 2006 (UTC) Somehow, this formula seems to be wrong. Let's ignore the minor detail that we get positive or negative solid angles, depending on the orientation of the vertices. The problem is as follows:

Let's take a triangle whose center is very close to the observer. The angle omega then should be close to 4pi/2 = 2pi, and omega/2 should be close to pi. Now, taking the arctangent of the fraction will give something in the range -pi/2..pi/2. So, this cannot work - or can it?

I want a diagram of solid angle cause I want to know what are its uses and why I am studying it Sqadro (talk) 02:41, 23 May 2016 (UTC)

Definition is wrong.

An "object" is generally considered to be a real 3-dimensional solid. Assuming that is what is meant here, then an object does NOT "subtend" an angle. A real object subtends many angles. Even if we consider concave solids, orientation matters. Consider the case of a cube. The largest distance subtended by it's shadow (projection) onto a sphere depends on its orientation. Other items, that is: virtually all real objects, will share this characteristic. The largest angle subtended depends on their orientation with respect to the center of the sphere. If this article is trying to say that a tetrahedron, no matter its orientation, has a constant area of projection, then that should be articulated, if not then the entire discussion of tetrahedra is useless and confusing here. Even if the area of projection is constant, I see no value in discussing this special case.173.184.25.50 (talk) 05:02, 8 January 2018 (UTC)

I disagree with your premise. In 3D, an object can be any n-dimensional solid. The observer in a 3D world will see either a point, curve or lamina. The curve will contain boundary points and the lamina will contain both boundary points and curves. If the object is an n-dimensional solid (n ≥ 3) it will appear as a lamina to the observer and the appearance of this lamina will depend on the projection of the n-dimensional solid in 3D and how this 3D projection is orientated with respect to the observer.

Consider the inside of a tetrahedron and place an observer at one of its vertices. The observer will be able to see the base opposite this vertex as a triangular lamina. At the reference point, an angle will be subtended by the lamina and it can be measured as a solid angle. The observer will also see 3 edges of this base. At the reference point, an angle will be subtended by each of the 3 edges and each can be measured as a planar angle. As a solid angle, it will have zero measure. Finally the observer will see the 3 vertices of the base. At the reference point an angle will be subtended by each of the 3 points and each will have a zero measure as a linear angle (it has no definition). Also as both a planar and solid angle it will have zero measure. Frank M. Jackson (talk) 18:26, 19 January 2018 (UTC)

pyramid angles vs lengths

Is there any convention to use alpha,beta for lengths and a,b for angles, while using d for a length. It makes more sense to me to use Greek letters for angles and Latin letters for lengths. Chris2crawford (talk) 10:13, 25 May 2019 (UTC)