Comment by srean

Comment by srean 3 days ago

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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...

jacobolus 3 days ago

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

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  • egwor 3 days ago

    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!

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    • jacobolus 3 days ago

      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.

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  • srean 3 days ago

    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.

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mxfh 3 days ago

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

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  • srean 2 days ago

    Indeed. It is one of the many well known spherical spirals / seiffert spirals.

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cwmoore 3 days ago

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."

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patcon 3 days ago

Jeez Erdos. This man was so prolific he was still publishing 4 years after he died :o

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