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Today I somehow was wondering how could you define a 3d angle. First what I could think of was volume of a cone with side a=1. Then if it wasn't round, volume of a tetrahedron with side 1. Is any of this fisible? How is it defined, if this field exists?

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(For the following, note that I refer to quantities having a certain number of "dimensions". By this I am referring the number of separate numbers a quantity contains. For example, defining position in 3d space relative to some co-ordinate system requires three separate numbers; that is a 3-dimensional vector by definition.)

Solid angles (which abcabc123 described) are single (1-dimensional) measurable quantities that are useful for things like measuring radiation, but there also multi-dimensional quantities that can be used to describe directions, rotations and orientations in space.

Angles in a (2-dimensional) plane are actually 1-dimensional quantities; they can be described as distance around a unit circle. (That is what radians are.) Just as a circle has finite circumference, for some purposes, plane-angles have limited range. (470° and -45° are useful for describing rotations in time, but, as directions, they are equivalent to 110° and 315°.) It is also worth noting that an angle can only describe direction relative to some reference direction. (right/positive-x in Cartesian coordinates)

Directions in (3-dimensional) space can be represented as points on a unit sphere. Since the surface of a sphere is 2-dimensional and has finite area, directions in space can be represented by 2-dimensional quantities, where both dimensions have finite range. There is more than one way to describe points on a sphere, but the most common and way is with some variation of latitude(φ) and longitude(λ) (which I'm assuming you're familiar with). One likely difference is that NSEW are often replaced with +-+-. (The term spherical coordinates refers to a 3D coordinate system analogous to 2D polar coordinates, which uses latitude, longitude, and radial distance. There is more than one convection https://en.wikipedia.org/wiki/Spherical_coordinate_system#Conventions) There are other possible differences in presentation, but the basic concept is the same for any variation.

Note that, while 2D direction is relative to one reference direction, 3D directions require at least 2 reference directions. These might be the (0,0) point and one of the poles. To understand why, imagine a sphere where only one of these directions was fixed: no matter where that direction was on the sphere, (in terms of latitude and longitude you define) it would serve as an axis the sphere could rotate around, thus any point on the sphere might represent any number of directions. Adding any other reference direction other than the exact opposite direction (by defining it's latitude and longitude) would fix the sphere in place, thus any point on the sphere would correspond to a definite direction.

I've been using the terms "latitude" and "longitude", but these are often not the terms used. The following vocabulary are common for spherical direction:

Angular distance above/below/north/south of the equator/horizontal may be:

  • φ = "latitude"(in geography), "inclination", or "elevation"
  • δ = "declination" (in astronomy, must be celestial equator)

Angular distance right/left/east/west along the equator/horizontal may be:

  • λ = "longitude"(in geography)
  • θ = "azimuth"
  • α = "right ascension"(astronomy, celestial equator, measured in hours: $24hr=360°$)

I've noticed that, for some reason, longitude is sometimes listed before latitude in ordered pairs, so be sure to check the order if you see direction as an ordered pair.

There are also other ways than simple latitude and longitude to define points (or areas) on a sphere and/or directions. One way is simply to define a 3D vector parallel to that direction. This link might be a good place to start looking for other ways: https://en.wikipedia.org/wiki/Discrete_global_grid#Hierarchical_grids

Note that between any two directions there is a simple plane-angle between these two directions. When the directions correspond to points on a sphere, this is the angle between two line-segments (or rays) connecting the center of the sphere and each of those points, in the plane which contains all three points (and both line-segments/rays). The magnitude of this angle can found using the spherical law of cosines:

$$\cos c = \cos a \cos b + \sin a \sin b \cos γ,$$where a, b, and c are the sides of a spherical triangle on the surface of a unit sphere and γ is the angle of that triangle opposite c.

If a and b are replaced with the "latitude" and "longitude" differences between our two directions (Δφ and Δλ), then c will be the plane angle between those two directions (central angle between two points on a sphere):

$$c = \arccos (\cos Δφ \cos Δλ + \sin Δφ \sin Δλ \cos 90°) = \arccos (\cos Δφ \cos Δλ)$$

This is proportional to the distance between two points on the surface of a sphere of radius R:

$$Surface Distance = R\arccos (\cos Δφ \cos Δλ) $$

Rotations and Orientations in (3-dimensional) space cannot be completely represented by latitude and longitude. They are usually represented using either Euler angles, or 3D rotation matrices which are both 3-dimensional, or quaternions which are 4-dimensional. Generally, Euler angles are considered easier for humans to grasp, but quaternions are simpler and faster for computers, and avoid certain problems when multiple rotations are applied in sequence. (They avoid gimbal lock.) Note that any orientation an object (or co-ordinate system) has in 3d space can be reached by a single rotation around some fixed axis, according to Euler's Rotation Theorem. (https://en.wikipedia.org/wiki/Euler%27s_rotation_theorem, https://web.stanford.edu/~ajdunl2/so3/so3.html, https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Palais.pdf) This means that orientations can be represented as rotations relative to a reference orientation. Just like directional reference frames, a reference orientation only really needs two reference directions to define it because that's all that is needed to to fix a sphere (or any object) in space.

Obviously, Euler's Rotation Theorum means that any rotation or orientation can be represented by defining the axis of rotation (as a direction or by a vector parallel to the axis) and the angle of the rotation. Note that if you use a latitude-longitude like method for direction this is 3 numbers, and if you use a axis-parallel 3D vector, it involves 4 numbers.

Euler angles are basically applying 3 different rotations about 3 perpendicular axes in sequence. Such a sequence of three rotations can yield the same result as any single rotation about any axis, but the order of the rotations matters. Naturally, this means there there are many theoretically possible ways to define Euler Angles. (Wikipedia claims there are six, but I wonder if there are more.) Fortunately, only a few are common.

In Euler's own definition of Euler angles, the x, y, and z axes are defined according to the normal "right-hand-rule"; that is, if +x is right and +y is up, then +z comes out of the page. The three rotations are as follows:

  • α or φ = a rotation about the z-axis moving the the +x-axis towards the +y-axis. (counter-clockwise if +z comes up out of the page) This creates a new co-ordinate system, (x',y',z') where z' = z.
  • β оr θ = a rotation about the x'-axis moving the +z'-axis away from the +y'-axis. (counter-clockwise if +x' comes up out of the page) This creates a new co-ordinate system, (x'',y'',z'') where x'' = x'.
  • γ or ψ = a rotation about the z'' axis moving the the +x''-axis towards the +y''-axis. (counter-clockwise if +z'' comes up out of the page) This creates a new co-ordinate system, (x''',y''',z''') where z' = z; this is the final result.

That's not how it's always described but I believe it's accurate. I think these "classic" Euler angles are still popular among mathematicians because they fit mathematical conventions in an orderly way.

Tait-Bryan angles, also called Cardan angles, or nautical angles are a version of Euler angles which are commonly used in aviation, seamanship, and space-flight. The three angles are usually called either heading, elevation, and bank, or yaw, pitch and roll. These can be described in terms of x, y, and z, axes but they are usually described in terms of yaw, pitch, and roll axes (which pass through the center of mass of an aircraft, ship, etc.):

  • The Yaw/Normal/Vertical/Heading/Bearing/z axis points straight down. This is the first axis rotated around. The rotation (ψ) is positive right, when facing forward, or counter-clockwise if this axis points up out of the page. (I think "bearing" is often relative to due North rather than forward, but I'm not sure if that applies here.)
  • The Pitch/Transverse/Elevation/y' axis points right when the you are facing forward. (It points starboard.) This is the second axis rotated around. The rotation (θ) is positive when the nose of the plane is going up (if you are right-side-up), or counter-clockwise when this axis points up out of the page.
  • The Roll/Longitudinal/Bank/z'' axis points straight forward. This is the final axis rotated around. The rotation (φ) is positive when you are rotating clockwise if you are facing forward, or counter-clockwise if this axis is pointing up out of the page. This means your longitudinal axis (right wing) will be going down if you are right-side-up.

You may have noticed that, if you imagine the positive side of the axis you're rotating around coming straight up (perpendicularly) out of a piece of of paper with the other two axes drawn on it, all these rotations will be counter-clockwise (like normal plane-rotations). I imagine this is a useful aid for remembering these rotations.

Note that I defined Euler and Tait-Bryan angles in terms of intrinsic rotations, where the co-ordinate system is moved by each rotation as if it is fixed to the object(s) you are rotating. It is also possible to define them in terms of extrinsic rotations, where your co-ordinate system stays fixed and only the objects move. I think the extrinsic definitions basically just have the order of the axes backwards.

Rotation Matrices are a computational tool for finding the position of points after a rotation, given their position before that rotation. If $x_i$, $y_i$, and $z_i$ represent co-ordinates before the rotation, $x_f$, $y_f$, and $z_f$ represent co-ordinates for the same point after the rotation, and R represents the rotation matrix for that rotation, then the following equation holds:

$$R\begin{bmatrix}x_i\\y_i\\z_i\end{bmatrix} = \begin{bmatrix}x_f\\y_f\\z_f\end{bmatrix}$$

That is the equation for rotations in three dimensional space, in which case the rotation matrix (R) is a 3×3 matrix. Note that order matters in matrix multiplication.

Into each of the nine slots in this matrix can be inserted an equation in terms of some other method of measuring 3D angles, such as axis and angle or Euler angles. You can find specific equations on the Wikipedia page.

The equations for converting Euler or Tait-Bryan angles into Matrix form can be derived by multiplying together the matrices for each of the 3 rotations, in the correct order for the extrinsic rotations definition of the type of angle your using. This is in accordance with the principles of linear algebra. If you don't currently understand linear algebra, check out this Youtube series by 3Blue1Brown explaining linear algebra in a mostly visual-spatial way that is very appropriate for this particular application: https://www.youtube.com/watch?v=kjBOesZCoqc

Note that rotation matrices only relate final position to initial position; they do not provide any information as to the path, so rotations greater than 360° or less than 0° have identical rotation matrices to rotations within this range. One could theoretically attach additional information to one though, if it seemed useful.

Quaternions are a special type of (4-dimensional) number, which extend the concept of complex numbers. They are written in the form a+bi+cj+dk, where i, j, and k are "quaternion units" that form the basis for 3 imaginary number lines in addition to to the line of real numbers. One important feature of quaternion units is that the commutative property of property of multiplication does not quite apply to them. ij = -ji, and the same relation is true for other such combinations. This allows them to satisfy the following equation which defines quaternion units:

$$\mathbf i^2 =\mathbf j^2 =\mathbf k^2 = \mathbf i\mathbf j\mathbf k = -1$$

If you treat the coefficients a, b, c, and d as measurements in four perpendicular directions and find the "magnitude" of this "4D vector", you get a value called the norm of the quaternion, which has important mathematical properties:

$$\Vert a+b\mathbf i+c\mathbf j+d\mathbf k\Vert = \sqrt{a^2 + b^2 + c^2 + d^2}$$

Quaternions whose norm is $1$ are called unit quaternions. I turns out that every orientation (but not every rotation because some rotations are indefinitely more than 360° less than 0°) in 3-dimension space is uniquely described by by a certain unit quaternion. That Youtuber I mentioned (3Blue1Brown) has some content on quaternions and there are lots of other sources too. I must admit that I don't currently quite understand how unit quaternions relate to orientations, but I think it has something to do with the following:

Consider that the surface of a unit sphere is a finite but unbounded 2-dimensional non-Euclidean plane. Every direction in space is uniquely defined by a particular point on the surface of such a sphere, and every point the surface on such a sphere can be defined by either a unit vector describing its 3D position relative to the center of the sphere or by "latitude" and "longitude" given at least 2 reference directions. "Latitude" and "longitude" are plane-angle quantities, who's orientation and directional senses "wrap around" and become redundant if you go to high or to low. This can be seen as the reason that the surface of sphere is finite but unbounded.

If we extend this concept into four dimensions, we are considering the surface of 3-sphere (a hypersphere in 4D (Euclidean) space.) Its surface is a finite but unbounded non-Euclidean space (3D). Clearly, (since we are in 4D space) each point on the "surface" of this 3-sphere can be defined by a unit 4D vector describing that point's 4D position relative to the "center" of the 3-sphere. It also turns out that, much like a normal sphere, every point on this surface can be defined by three (rather than 2) angular values. These are sometimes called (ψ,θ,φ), just like Euler angles. When you consider that any orientation in space can be uniquely defined by some set of 3 Euler angles, it seems natural that every orientation in space MIGHT be uniquely described by some point on this 3D "surface". If that's true (it must be because quaternions work), then clearly every orientation is also uniquely described by a unit quaternion, since unit quaternions all correspond to unit 4D vectors, which correspond to points on our unit 3-sphere.

Thats certainly not a proof, but I THINK it's a good intuition.

For completeness, I feel I should mention the angle between two planes in 3D space, which Axel Hansen mentioned, and the angle between a line and a plane in 3D space, which is also the angle between that line and its orthognal projection onto the plane, and is often calculated using the equation for the dot product of a unit vector parallel to the line and one perpendicular to the plane. I'm sure you can find more information on these elsewhere, and this answer is already extremely long so I won't try to go into more detail here (at least not yet).

There might other things that are 3D analogs to plane-angles (eg. the Wikipedia page on Rotation Matrix mentions "polar decomposition" and other "decompositions"), but that's everything I know about and could think of.

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Well one property of the 2-dimensional angle is that the length of a circle's arc $l$ is proportional to the angle of the arc $\theta$ for a given radius $r$, i.e. if we put $r=1$ then $$l=k \theta$$ where $k$ is a proportionality constant (radians have been defined such that $k=1$ when $\theta$ is in those units, and $k=\pi/180$ if you work in degrees).

To define your 3-dimensional angle, you could choose it to have a similar property. In this case, you would have the area of a spherical cap $A$ proportional to the 3d angle of the cap $\Omega$ for a given radius, i.e. for $r=1$ $$A=\kappa \Omega$$ And such angle has a name, it is called a solid angle and the units in which $\kappa =1$ is called a steradian.

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A 3d angle is an angle between 2 planes in space. For example a valley gutter. You essentially need to know the coordinates of 4 points, 2 of which are common for both planes.

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I would not define a solid angle with circular functions, but in terms of squared areas kept in proportion, because a point on a sphere can be located with two right triangles, one which emanates from the origin, and the other being a cross section.

The image below shows that you can drop a perpendicular from the point on the sphere's surface to the horizontal plane, and then to a vertical plane (which is omitted), while the cyan triangle sits in the frontal one.

The slanted red triangle would be the 3D analog of an hypothenuse. You would then be able to relate the squared area of the cyan triangle to the squared area of the red one to get the separation between the three lines in three dimensional space.

rational solid angle

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