You're probably looking for something like Sudbery 1977,
https://dougsweetser.github.io/Q/Stuff/pdfs/Quaternionic-ana...
(published 1979, doi: 10.1017/S0305004100055638)
Fantastic. Thanks for the reference.
Readit Newssrean commented on Visualizing quaternions: An explorable video series (2018) eater.net/quaternions... · Posted by u/uncircle
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.
srean commented on Visualizing quaternions: An explorable video series (2018) eater.net/quaternions... · Posted by u/uncircle
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.
See: https://en.wikipedia.org/wiki/Liouville's_theorem_(conformal...
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.
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
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.
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...
srean commented on A simple way to generate random points on a sphere johndcook.com/blog/2025/0... · Posted by u/piinbinary
I think it can be done that way yeah but in order to yield a uniform-density of points on the surface of the sphere there's some pre-correction (maybe a sqrt or something? I can't remember) that's needed before feeding the 'uv' values to the trig functions to make 3D positions. Otherwise points will 'bunch up' and be more dense at the poles I think.
Indeed.
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.
srean commented on A simple way to generate random points on a sphere johndcook.com/blog/2025/0... · Posted by u/piinbinary
srean commented on A simple way to generate random points on a sphere johndcook.com/blog/2025/0... · Posted by u/piinbinary
Please forgive me my naivete, but won't generating two random polar coordinates do? I'm bad at math, so I might as well be very very wrong here, but I'd like to know.
Edit: see @srean's excellent explanation why that won't do.
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.
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