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.
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...
You inspired me to submit one of my 2022 projects
https://observablehq.com/@jrus/spheredisksample
https://news.ycombinator.com/item?id=44963521
to fit the trend of the day. People may also enjoy
https://observablehq.com/@jrus/sphere-resample
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!
An equirectangular projection just means plotting latitude and longitude in a rectangle.
Do you mean my diagonal grid that I projected back onto the sphere? I'm not sure that has a name.
Great to see you. I look forward for your comments on geometry, multivariate calculus and rotations.
Edit: fantastic graphics. You should submit the other one as an HN post too.
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.
---
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:
https://beta.dwitter.net/d/34223
Indeed. It is one of the many well known spherical spirals / seiffert spirals.
Don't forget this post, which spawned a discussion of Rhumb lines etc. in the comments: https://news.ycombinator.com/item?id=44962767
I had missed this one ! Thanks.
It is indeed raining spherical geometry today.
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."
Jeez Erdos. This man was so prolific he was still publishing 4 years after he died :o
Many after he passed