These used to be super important in early oceanic navigation. It is easier to maintain a constant bearing throughout the voyage. So that's the plan sailors would try to stick close to. These led to let loxodromic curves or rhumb lines.
This configuration is a mathematical gift that keeps giving. Look at it side on in a polar projection you get a logarithmic spiral. Look at it side on you get a wave packet. It's mathematics is so interesting that Erdos had to have a go at it [0]
On a meta note, today seems spherical geometry day on HN.
Except the helix curve shown in OP is NOT a loxodrome or rhumb line.
It has equal spacing on the surface between lines, a loxodrome can't have that property since by definition it must cross the meridians at the same angle at all times. That means it always gets denser near the poles.
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Start with the curve:
x = 10 · cos(π·t/2) · sin(0.02·π·t)
y = 10 · sin(π·t/2) · sin(0.02·π·t)
z = 10 · cos(0.02·π·t)
Convert to spherical coordinates (radius R=10):
λ(t) = π/2 · t (longitude)
φ(t) = π/2 - 0.02·π·t (latitude)
Compute derivative d(λ)/d(φ):
d(λ)/dt = π/2
d(φ)/dt = -0.02·π
d(λ)/d(φ) = (π/2)/(-0.02·π) = -25 (constant)
A true rhumb line must satisfy:
d(λ)/d(φ) = tan(α) · sec(φ)
which depends on latitude φ.
Since φ(t) changes, sec(φ) changes, so no fixed α can satisfy this.
Conclusion: the curve is not a rhumb line.
this is how one should look for varying intersection angles:
In my early teens I used to try to create something like a equirectangular projection because when drawing it, it looked cool. Obviously I had no idea that it was called this. I was trying to draw reflections of a square window onto a sphere, and then I moved on to trying to cover the sphere in a checkered pattern. This is awesome to see, thank you!
To quote the storytelling quality of Erdos's abstract:
"The simple requirement that one should move on the surface of a sphere with constant speed while maintaining a constant angular velocity with respect to a fixed diameter, leads to a path whose cylindrical coordinates turn out to be given by the Jacobian elliptic functions."
One way to fix the problem is to sample uniformly not on the latitude x longitude rectangle but the sin (latitude) x longitude rectangle.
The reason this works is because the area of a infinitesimal lat long patch on the sphere is dlong x lat x cosine (lat). Now, if we sample on the long x sin(lat) rectangle, an infinitesimal rectangle also has area dlong x dlat x d/dlat sin(lat) = dlong x dlat cos (lat).
Unfortunately, these simple fixes do not generalize to arbitrary dimensions. For that those that exploit rotational symmetry of L2 norm works best.
If you want uniformly random on the spherical surface then uniformly at random in polar coordinates will not cut it.
To appreciate why, consider strips along two constant latitudes. One along the Equator and the other very close to the pole. The uniformly random polar coordinates method will assign roughly the same number points to both. However the equatorial strip is spread over a large area but the polar strip over a tiny area. So the points will not be uniformly distributed over the surface.
What one needs to keep track of is the ratio between the infinitesimal volume in polar coordinates dphi * dtheta to the infinitesimal of the surface area. In other words the amount of dilation or contraction. Then one has apply the reciprocal to even it out.
This tracking is done by the determinant of the Jacobian.
Conformal mappings are not nearly as rich in >2 dimensions. There is a much stronger rigidity constraint and you end up limited to just Möbius transformations. The 2 dimensional case is special.
Yes of course, but I am curious about any interesting structures that functions from quaternion to quaternion may possess. I used conformal mapping as an example of an interesting structure. I could have used Cauchy Riemann as another example.
Heard the word 'nepers' after many decades. Are you by any chance an Electrical major ?
Thanks for your comment. To be fair, I had not done due diligence before asking. There's a Wikipedia pages on quaternion calculus.
Complex analysis (calculus on functions from 2D rotations to 2D rotations) is beautiful -- Once differentiability guarantees infinite differentiability. Wondering what would the analogue of that be for quaternions
In India where a social media post pissed off someone. It was sarcastic criticism - no threats, no slurs). But won't get into details, sorry. (Also arrested a few times for participating in mass protests against corruption, but that was in a large group, so wasn't all that stressful.)
I also had bad luck when traveling to the US. Got detained by the CBP - I think because I accidentally sneezed on the officer and pissed him off. (Either that or I looked like some terrorist). Had to stay in a cell for more than a day. I wasn't even questioned!
Thankfully, nothing happened after that. Was good to catch up on sleep though, since there was nothing to do.
Seems like all the more reason to put the responsibility and blame on the government. You will never eliminate “influence”, and especially the more power the government has, the more value there is in spending on “influence”. The only possible solution is to hold the government and the representatives responsible for taking actions to the detriment of their constituents. If we give them a pass because “elections can be influenced” we might as well just disband the government and allow governing by the highest bidder.
It doesn't work that way. What we have seen is that big money will always find a way to corrupt the government. In any case the priority 0 of most elected officials is to raise money for their re-election.
Eh, even the distinction between private enterprise and government is largely irrelevant. At the end of the day, forces too large to fight conspire to make peoples' lives miserable.
Given all the news about elevated levels of human Coyote interactions over the last 10 ~ 15 years, I wonder whether we are witnessing the beginnings of another domestication/speciation event -- new "dogs".
https://en.m.wikipedia.org/wiki/Rhumb_line
Mercator maps made it easier to compute what that bearing ought to be.
https://en.m.wikipedia.org/wiki/Mercator_projection
This configuration is a mathematical gift that keeps giving. Look at it side on in a polar projection you get a logarithmic spiral. Look at it side on you get a wave packet. It's mathematics is so interesting that Erdos had to have a go at it [0]
On a meta note, today seems spherical geometry day on HN.
https://news.ycombinator.com/item?id=44956297
https://news.ycombinator.com/item?id=44939456
https://news.ycombinator.com/item?id=44938622
[0] Spiraling the Earth with C. G. J. Jacobi. Paul Erdös
https://pubs.aip.org/aapt/ajp/article-abstract/68/10/888/105...
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