Wikipedia:Reference desk/Archives/Mathematics/2022 November 3

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November 3

Is there a mathematical function that is also the latitude of a great circle at each longitude?

Or at least all diagonal great circles with ascending nodes (northward equator crossings) at the longitude that makes the math simplest? Or if not the latitude in radians maybe the latitude in hectograds (0.9 degrees per grad) or 1=the northernmost latitude in the great circle? Similar to how a sine curve touches but does not breach 1 and minus 1 when the math is in simplest form ignoring the plus a constant and scaling stuff. It's not a sine curve except for the degenerate one that hugs the equator though. Sagittarian Milky Way (talk) 12:12, 3 November 2022 (UTC)

I suppose you are seeking a function that maps longitudes to latitudes for points lying on a great circle. What is a "diagonal great circle", and what are "ascending nodes"?  --Lambiam 14:29, 3 November 2022 (UTC)

(Orbital node. Diagonal surely means oblique; so not the equator (which would make the problem trivial)) catslash (talk) 15:49, 3 November 2022 (UTC)

Yes. Great circles on a perfect sphere. Doesn't need to work for ones perpendicular to or on the equator. I think it can be less verbose if the great circles ascend into positive latitudes at 0 degrees east? (2pi radians at 0 degrees west) Sagittarian Milky Way (talk) 19:12, 3 November 2022 (UTC)

We model the Earth surface as the unit sphere and use the geocentric coordinate system with the equator at $z=0,$ the North pole at $(0,0,1),$ Null Island (latitude and longitude both equal to $0^{\circ }$) at $(1,0,0),$ and the equatorial point with longitude $90^{\circ }$E (somewhere in the Indian Ocean) at $(0,1,0).$ The great circle crossing the equator at Null Island with inclination $\iota$ is formed by the intersection of the sphere with the plane defined by $z\cos \iota =y\sin \iota ,$ so $y:z=\cos \iota :\sin \iota .$ The meridian of longitude $\lambda$ is formed by the intersection of the sphere with the plane defined by $y\cos \lambda =x\sin \lambda ,$ so $x:y=\cos \lambda :\sin \lambda .$ Combining these gives us $x:y:z=\cos \iota \cos \lambda :\cos \iota \sin \lambda :\sin \iota \sin \lambda .$ Using $x^{2}+y^{2}+z^{2}=1$ and putting $Q=\cos ^{2}\!\iota +\sin ^{2}\!\iota \sin ^{2}\!\lambda =\cos ^{2}\!\iota \cos ^{2}\!\lambda +\sin ^{2}\!\lambda ,$ we obtain $z=Q^{{-}{\frac {1}{2}}}\sin \iota \sin \lambda .$ The latitude is given by $\varphi =\arcsin z=\arcsin(Q^{{-}{\frac {1}{2}}}\sin \iota \sin \lambda ).$ As a one-liner:

$\varphi =\arcsin {\frac {\sin \iota \sin \lambda }{\sqrt {\cos ^{2}\!\iota +\sin ^{2}\!\iota \sin ^{2}\!\lambda }}}.$

The maximal latitude is reached at longitude $\lambda ={\tfrac {1}{2}}\pi ,$ where $\varphi =\iota .$  --Lambiam 21:23, 3 November 2022 (UTC)

arcsin[sin1.57sin6/[(sin1.57)^2(sin6)^2+(cos1.57)^2]^.5] seems to work. Sagittarian Milky Way (talk) 06:13, 4 November 2022 (UTC)

And arcsin[sin70 degrees*sin242 degrees/[(sin70 degrees)^2(sin242 degrees)^2+(cos70 degrees)^2]^.5] gives what it should so apparently sin²foo does mean my first guess of (sin(foo))². And for only 1 extra character arcsin[sinιsinλ/sqrt[(sinι)^2(sinλ)^2+(cosι)^2]] is easier to read. Sagittarian Milky Way (talk) 04:33, 5 November 2022 (UTC)

The notational convention by which $\sin ^{2}\!\alpha$ means $(\sin \alpha )^{2}$ is standard; see e.g. our article Pythagorean trigonometric identity and in textbooks here, here and here. What is easier to read may be a matter of what one is used to.  --Lambiam 07:48, 5 November 2022 (UTC)

I probably wasn't in school when they that taught that or I forgot. The original pretty-printed line is easiest to read but for something that can be typed on the most devices (even 1970s computers and typewriters probably), closes all parentheses and still causes Google search suggestions to answer from anywhere on the Internet perhaps arcsin[sin1sin2/sqrt[(sin1)^2(sin2)^2+(cos1)^2]] is a bit easier than arcsin[sin1sin2/[(sin1)^2(sin2)^2+(cos1)^2]^.5]? (braces instead of outer brackets would be a bit better but Google calculator doesn't understand) Sagittarian Milky Way (talk) 09:46, 5 November 2022 (UTC)

Minkowski distance formula

In our collection, we have a book on abstract math theory published in 1963. In it, there is a section on Minkowski space that brings up Minkowski distance. The formula it uses is Σ|a-b|. It states that in this simple form, this is Manhattan distance. Then, the next chapter expands Minkowski distance to be (Σ|a-b|^p)^1/p for some power p. That matches the formula in Minkowski distance except tht it doesn't begin with a limit from p to infinity. It does not continue to add the limit in the textbook, so I am wondering if this is an error in the textbook or is it that the accepted formula has changed since 1963? 97.82.165.112 (talk) 13:52, 3 November 2022 (UTC)

In the usual case the parameter p is not infinite. The first distance formula involving a limit is for the Chebyshev distance.  --Lambiam 14:36, 3 November 2022 (UTC)

Thank you. Re-reading the Wikipedia article, it looks like the limit is not part of the definition, so it does match the old textbook. 97.82.165.112 (talk) 16:02, 3 November 2022 (UTC)