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June 27

Arccos(1.1)

I was recently looking at a problem which involved taking the inverse cosine of a calculated number which, in one case, turned out to be greater than 1. A quick examination revealed that the way I'd sketched out the problem made no physical sense but it occurred to me that this might just mean "has no solution in the real numbers".

Is there a definition for arccos(1.1) which lies outside the reals?

2A01:E34:EF5E:4640:BD32:A0DC:FFF7:43C0 (talk) 11:38, 27 June 2022 (UTC)[reply]

Using

\cos x={\frac {e^{ix}+e^{-ix}}{2}},

we have that

\cos(ix)=\cos(-ix)={\frac {e^{x}+e^{-x}}{2}}=\cosh x.

So it makes some sense to use

\arccos x=i\operatorname {arcosh} x,

although there is no obvious reason to prefer this choice over

\arccos x=-i\operatorname {arcosh} x.

--Lambiam 12:33, 27 June 2022 (UTC)[reply]

You should use, of course, whichever corresponds to arccos on the usual domain, which will depend on which branch of the complex logarithm you choose.

The result, depending on which branch of the function you use, is approximately ±0.44356825 i + n * 2

π

, where n is an integer. IpseCustos (talk) 14:50, 27 June 2022 (UTC)[reply]

Thanks a lot both of you. 37.166.79.23 (talk) 18:23, 27 June 2022 (UTC)[reply]

What's a Mobius strip with a 1-dimensional centerline called?

Or the roll equivalent of a great circle? A great circle of course being the geodesic where you pitch at constant rate in 3D Euclidean space but don't yaw or roll and the "straight Mobius" or "zero radius helix" being the geodesic of rolling at constant rate with no pitch or yaw. Sagittarian Milky Way (talk) 23:09, 27 June 2022 (UTC)[reply]

The Möbius strip is a 2-dimensional metrizable manifold, but from the way the question is phrased it seems that you think of the strip as being embedded in 3-dimensional Euclidean space, inheriting its metric. These embeddings are not unique; some give rise to an Euclidean metric on the strip, while others don't. Are there Möbius strips whose centre line is not 1-dimensional? It seems to me that the centre line of anything, being a line, is 1-dimensional. --Lambiam 11:22, 28 June 2022 (UTC)[reply]

Unglue the ends and pull the rectangle taut without untwisting it. Sagittarian Milky Way (talk) 12:08, 28 June 2022 (UTC)[reply]

And then what? Is that different from twisting one end of a plain strip a half turn while holding it taut? Topologically, it is a disk, twisted or not. --Lambiam 17:42, 28 June 2022 (UTC)[reply]

Same thing, it becomes not a spiral, not a typical helix but a another kind of twister with zero diameter if it was infinitely thin. Sagittarian Milky Way (talk) 22:18, 28 June 2022 (UTC)[reply]

It is no longer flat (in the inherited metric), so the twisting also causes a deformation. We can consider the spiraling surface parametrized by two parameters,

r\in [-1,1]

and

z\in \mathbb {R} ,

embedded in

\mathbb {R} ^{3}

by

(x,y,z)=(\eta \,r\cos \lambda z,\eta \,r\sin \lambda z,z),

in which

\eta

and

\lambda

are constants. If

\eta =0,

this degenerates into simply a straight line, indistinguishable from

(x,y,z)=(0,0,z).

--Lambiam 22:52, 28 June 2022 (UTC)[reply]

What does geometry call it if it's not infinitely thin? If a line segment on the z=seconds since t=0 plane is spinning on the z-axis and the midpoint is always x=0 y=0 z=time then the centerline of the surface that touched the segment would be the z-axis thus flat despite the surface not being flat. Sagittarian Milky Way (talk) 01:26, 29 June 2022 (UTC)[reply]

SMW, it's really hard to figure out what your actual question is. Are you asking for the name of this surface? I don't know that it has a standard name. --Trovatore (talk) 01:39, 29 June 2022 (UTC)[reply]

Yes what do you call this surface (doesn't have to be in that location or orientation) Sagittarian Milky Way (talk) 02:12, 29 June 2022 (UTC)[reply]

Helicoid. More precisely, a helicoid stretches out from the z-axis to infinity, as when we allow the

r

parameter above to range over

\mathbb {R} .

--Lambiam 07:35, 29 June 2022 (UTC)[reply]

If

r

is constrained, you get an (idealized) Archimedes screw. --Lambiam 09:03, 29 June 2022 (UTC)[reply]

Fusillioid? Sagittarian Milky Way (talk) 11:47, 29 June 2022 (UTC)[reply]