3 ms·

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 woul

by srean 1y ago

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 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 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=44956297 https://news.ycombinator.com/item?id=44939456 https://news.ycombinator.com/item?id=44939456 https://news.ycombinator.com/item?id=44938622 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/68/10/888/1055391/Spiraling-the-Earth-with-C-G-J-Jacobi https://pubs.aip.org/aapt/ajp/article-abstract/68/10/888/105...

jacobolus 1y ago
You inspired me to submit one of my 2022 projects https://observablehq.com/@jrus/spheredisksample https://observablehq.com/@jrus/spheredisksample https://news.ycombinator.com/item?id=44963521 https://news.ycombinator.com/item?id=44963521 to fit the trend of the day. People may also enjoy https://observablehq.com/@jrus/sphere-resample https://observablehq.com/@jrus/sphere-resample
- srean 1y 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.
- egwor 1y 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!
  - jacobolus 1y 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.
taco_emoji 1y ago
Don't forget this post, which spawned a discussion of Rhumb lines etc. in the comments: https://news.ycombinator.com/item?id=44962767 https://news.ycombinator.com/item?id=44962767
- srean 1y ago
  I had missed this one ! Thanks. It is indeed raining spherical geometry today.
mxfh 1y 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 https://beta.dwitter.net/d/34223
- srean 1y ago
  Indeed. It is one of the many well known spherical spirals / seiffert spirals.
cwmoore 1y 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."
patcon 1y ago
Jeez Erdos. This man was so prolific he was still publishing 4 years after he died :o
- srean 1y ago
  Many after he passed

by srean 1y ago

jacobolus 1y ago

You inspired me to submit one of my 2022 projects https://observablehq.com/@jrus/spheredisksample https://observablehq.com/@jrus/spheredisksample https://news.ycombinator.com/item?id=44963521 https://news.ycombinator.com/item?id=44963521 to fit the trend of the day. People may also enjoy https://observablehq.com/@jrus/sphere-resample https://observablehq.com/@jrus/sphere-resample

srean 1y 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.
egwor 1y 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!
- jacobolus 1y 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.

taco_emoji 1y ago

Don't forget this post, which spawned a discussion of Rhumb lines etc. in the comments: https://news.ycombinator.com/item?id=44962767 https://news.ycombinator.com/item?id=44962767

srean 1y ago
I had missed this one ! Thanks. It is indeed raining spherical geometry today.

mxfh 1y 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 https://beta.dwitter.net/d/34223

srean 1y ago
Indeed. It is one of the many well known spherical spirals / seiffert spirals.

cwmoore 1y 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."

patcon 1y ago

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

srean 1y ago
Many after he passed