Show HN: I was curious about spherical helix, ended up making this visualization

Here's a summary of the themes from the Hacker News discussion:

Appreciation for Visualizations and Educational Value

Many users expressed admiration for the presentation's aesthetic appeal and its effectiveness in explaining complex mathematical concepts. The visual nature of the content was frequently highlighted as a significant advantage over traditional learning methods.

• "It's pretty and really easy to digest." - fleebee
• "This is excellent. I'm always looking for good things to show my students on coordinate systems and geometry, and this joins the list. Thank you for diving down the rabbit hole and bringing this back for everyone." - DuaneMclemore
• "This is beautiful animation. This is a great example of a visual lesson that leaves a chalkboard in the dust (ha)." - 1970-01-01
• "This lays it out so plainly." - chamomeal
• "quite beautiful" - nikolayasdf123
• "Beautifully done, thank you for sharing. :-)" - markusw
• "This is fantastic. What a terrific combination of the creative presentation and the clear exposition of information. You've hit on a very nice aesthetic and a stunningly clear articulation of the underlying mechanics." - thopkinson
• "This is so cool! Thanks for sharing, stuff like this is why I love HN." - ramathornn
• "Brilliant, I learned something today." - nedsma

Expanding on Mathematical Concepts and Applications

Users recognized the core mathematical principles demonstrated and suggested avenues for further exploration, connecting the visualizations to broader mathematical fields and historical applications.

• "It's a pretty basic primer to the subject, but good for kids learning maths. Could do with some callbacks to maths concepts like the circle equation ( x = r cos (t) and y = r sin (t) ). Possible topics to branch further into would be polar coordinates and linear algebra basics (vectors, transformations, transformations in 3d space)." - RugnirViking
• "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." - srean
• "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]" - srean
• "For me personally it's simpler to think about it as having an f(theta, r) = r (cos(thetha), sin(theta)), interpreting theta as a compass direction and r as a distance to walk along a great circle. So g(t) = polar_to_R3(f(t k, t l)). Changing the relative sizes of k and l changes the tightness of the helix." - fluoridation
• "The solar system spiraling through the universe." - ezconnect
• "I harmoniously love the mathematics of the spherical helix. The way the x and y coordinates scale proportionally to t, and the z coordinate changes with the sine of t. It's a very elegant way to represent the path of a point (or object) moving in 3D space." - rishabh_p

Interest in Animation Techniques and Implementation

A significant portion of the discussion revolved around how the animations were created, with many users expressing curiosity about the tools and specific parameters used. This also led to suggestions for creating similar visualizations.

• "I was also hoping to see more about how the animations were done here. They look great. I love the shifting camera perspectives." - jhaile
• "I think 3blue1brown might have an animation library, the same one he uses in his videos that might help with that" - qwertytyyuu
• "He created a visualization library called Manim and it's great." - megaloblasto
• "Best thing I have seen on HN in ages. Also interesting for a CNC geek." - maxbaines
• "I had a similar thought about 3D printing - particularly extruding mathematically defined shapes in vase mode." - ejp
• "Could you share how you made it? I want to make something similar for Rotation Matrices" - mostlyk
• "The animations are so fluid!" - reece

Discussions on Parameterization and Constant Speed

Several users delved into the nuances of parameterizing curves, particularly in relation to generating smooth motion and maintaining a constant speed along a path. This led to technical discussions about integration, reparameterization, and numerical methods.

• "The part that I was expecting to see but didn't: how can you move at a constant speed? For the original purpose of positioning objects along a path, it doesn't matter. But when moving, you can see it's moving much more slowly at the beginning and end (mostly determined by the radius). What if I want it to travel at a constant rate? Or even apply an easing function to the speed?" - ssilver
• "For constant speed you need a so-called “Euclidean parameterization” where the t value is proportional to s, the Euclidean distance traveled (and thus no matter the value of t, if you add some dt it always works out to the same ds). This is super commonly needed when animating motion along all sorts of curves, as you might guess. Unfortunately, there’s usually no closed-form solution for it, so we have to do it numerically." - Sharlin
• "I was wondering about the “correctness” of the z-axis movement for the spherical helix. You could pick lots of different functions, including simple linear motion (z = c * t). This would obviously affect the thickness and consistency of the “peels”. The equation used creates a visually appealing result but I’m wondering what a good goal would be in terms of consistency in the distance between the spirals, or evenness in area divided, or something like that. How was this particular function selected?" - pimlottc
• "I think this particular function was selected because it happened to be convenient to program and the visual effect was pleasant enough. The actual "correct" thing to do would probably be to have the point maintain constant speed in 3D space like a real boat sailing on a globe, right? But that's a rather bigger lift:" - crdrost
• "This pattern of stepping with different ds along a path has a lot of applications in control theory. Often we change the ds/dt ratio based on the known acceleration profile of a motor to minimize jerk and reach our destination as fast as possible." - Kennnan
• "The instinct to sort of slow t is right, as the governing functions are maintaining angular velocity with respect to t but scaling radius also with respect to t. It’s sort of like an Archimedean spiral. So, yeah, if you parameterize velocity and make that constant, you’re in better shape." - chaboud
• "Okay, I have some followup questions. Are the points equally spaced? I.e. the cube's |∆p| is constant? I see you scale z by the sin. What happens of you don't?" - Tyr42

User Experience and Browser Compatibility

A subset of the discussion focused on the user experience, particularly regarding navigation on mobile devices and browser compatibility.

• "I think the solar system spiraling through the universe. Together, these functions create a spherical helix... That's all! This strikes me as backwards reasoning. You are showing "these functions" -> spherical helix But I actually want spherical helix -> "these functions" 1. What if I want to make some other shape? I'm lost. 2. I have learned nothing about the spherical helix." - tantalor
• "My useful feedback is that navigation violated my expectations. I was in mobile FWIW. I didn't know what to do so I started to scroll. My touch to the screen kicked into the next pane so I'm like 'oh, okay'." - erikerikson
• "When I first opened it, its basically a bunch of static pages that made absolutely no sense. My first question was 'why is this garbage being #1 on HN?' Then I realized that, unlike the early web with banners of "best viewed in Netscape navigator", this was an unstated "best viewed in google chrome". Alas. At least please check and validate if the site works in Firefox, or notify appropriately. Because this demonstrably does not." - mystraline
• "I use Firefox (dev edition, v142.0b9) and it works great." - rocmcd
• "as lefthanded person it is quite cumbersome to tap right side of screen. Go to solution is to navigate using swiping which is ambidextrous." - aacid

Technical Discrepancies and Potential Errors

A few users pointed out potential inaccuracies or areas for clarification, such as typos in equations or misinterpretations of the visualization's intent.

• "I believe there is a typo here: y = 10 * sin(πt/2) * sin(0.02 * πt) On the previous two slides the end is sin(0.2 * πt)" - jtbayly
• "This is very cool, but somewhat confusing to the eye, because you are actually demonstrating the movement of a point along a path, while visualising it with a cube whose orientation doesn't change when it feels like it should. The point that is moving is in the centre of the cube. But the cube's orientation is fixed in global space. So the cube's orientation relative to the path of the spiral/helix is not quite the same as its orientation relative to the path of the straight line." - exasperaited
• "I feel a little let down. There is a huge leap from the basics of 3d plotting & spheres to the crazy pattern you tease and then show at the end. I understand it as someone who kind of knows this stuff already, but I think its way too big of a leap for someone who doesn't have the background." - dgrin91