Talk:Angle
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How about defining angle as the ?linear? codependance/independance of two curves, 0 being identical and 1 being orthogonal? I think the whole difficulty is defining angle in such a way as to make complex angles a logical extension of real angles, while keeping the definition extremely simple and clear. Kevin 2003.03.14
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Angles in Complex Hilbert spaces
Here are some thoughts about angles in complex Hilbert spaces. I moved them from the main page because they don't qualify as encyclopedic knowledge. One could also use the absolute value of the dot product I suppose. --AxelBoldt
For complex Hilbert spaces, the formula (*) can be recycled to obtain a complex angle, but it is not entirely clear that this corresponds to a real-world notion of angle. An alternative is to use
(**) R(u·v)=cosθ ||u|| ||v||
where R denotes the real part. Definition (**) also special cases to (*) for real Hilbert spaces, so that may be a reasonable choice.
Definition?
The definition of angle on the main page seems rather vague. Perhaps a better definition would be: the fraction of the arc of a circle with a center at the origin of the angle.
That way degrees can be clearly defined as:
(s/c)×360
and radians as:
(s/c)×2π = s/r
where s = arc length, c = circumphrence, and r = radius.
Grad?
Scientific calculators often have a 3rd measure of angle for trig functions, abbreviated to "grad" , with 400 per circle -- what are they?
Dimensionless?
[quote] Note that angles are dimensionless, since they are defined as the ratio of lengths. [/quote] Let us not confuse too many things here with this statement. Pi is defined by the ratio of two lengths. Pi can then be used to define the dimension of direction on a unit circle. Certainly, units are associated with things that have dimension. Angles have units, and they require the correct unit for a specific equation (Perhaps it is helpful to think of Radians as dimension-less and all other forms of angular units as dimensions.). Also, angles are a dimension, in polar and spherical coordinates, they have the dimension of direction. Angles depict two different directions.
- (William M. Connolley 07:39, 2 Sep 2004 (UTC)) Angles are dimensionless but they may have units (degrees). Use of an angular coordinate as a dimension in one application isn't the same question. Direction is not a dimension.
- User: Nobody_EDN 2004.10.22
Then why do the first three dimensions, length, width, and breadth, have three different directions??? 'Different directions' is the basic property of dimension. ---- Please tell me what definition of dimension you are using where an angle is ruled out. (They have units and they can define a point in space. Just as width, and breadth can, using length as the third dimension for either.)
- (William M. Connolley 20:09, 23 Oct 2004 (UTC)) You can sign your comments by using 4 tildes ~~~~ like that. Now, on: I'm using "dimension" is the sese of dimensional analysis. Since angles are length/length (or at least thats one good way to define them) they have dimensions of L/L = no dimensions.
Pi is a ratio of two lengths.
- (William M. Connolley 20:09, 23 Oct 2004 (UTC)) Yes, exactly. Pi is a pure number, and an angle. Hence, angles are dimensionless. Another way of seeing it.
Angles are defined by the vertex of part of a disk, cut once or more through the center. Angles are measured in fractions of Pi, or fractions of the full circle. Angle measurements are chosen by taking a full circle and cutting it up in pieces by a set constant. That constant is 2Pi for radians, 360 for degrees, 400 for gradients (grads), 800 percent for percent grade (Such that 45 degrees is a 100% grade.), and 32 for compass points.
Perhaps defining Pi, Circle, Circumference and Radius first would help in defining the Angle. Use a demonstration of the wrapping function.
A Circle, drawn with Radius of one unit, on a number line, centered at minus one, has zero located on the outer rim of the Circle. If the number line is then wrapped around the circle, the numbers one, two, three, four, five, and six will all be wrapped around the edge before we get back to the beginning of the circle where number zero is.
The number Pi is the ratio of Circumference over the Diameter, Or C/D. The Circumference over the Radius C/R is 2Pi. So the wrapping function has at 180 degrees, or the opposite side of the circle from zero, the number Pi. 2Pi coincides back again with the number zero. (A picture, or several, at this point would help.)
An angle is the position on the wrapped number line in Radians. Radians are in fractions and multiples of Pi. Pi is C/D and therefore length over length and is though of as dimension-less. Although they still can contain a dimension of direction at times, most mathematical and engineering equations eliminate any dimension from calculations. 2Pi times Radial length equals length of circumference..... 2Pi times Radius time rpm become velocity.....
Degrees are a different way of unitizing Radians. Degrees break up a circle into 360 degrees. Therefore, 2pi = 360 degrees. Once converted to Degrees the angle then carries the dimension of Degrees and all mathematical equations they are used in must deal with that dimension/unit, when doing dimensional/unit checks.
Other ways are points, grads, and percent. They carry the same dimension/unit requirement with them.
Supplied for creative incentive to improve the definition and description of an angle.
User: Nobody_EDN 2004.09.01
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