Show HN: I was curious about spherical helix, ended up making this visualization
visualrambling.space417 points by damarberlari 5 hours ago
417 points by damarberlari 5 hours ago
I was wondering how I can arrange objects along a spherical helix path, and read some articles on it.
I ended up learning about parametric equations again, and make this visualization to document what I learned:
https://visualrambling.space/moving-objects-in-3d/
feel free to visit and let me know what you think!
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 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. Awesome visualizations. 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? I'm sure there's some fancy mathematical trick that would just do it. If I were only more comfortable with math... my handwavy sketch would be to compute the speed function by differentiating the formulas to get dx, dy, and dz and passing them through the Pythagorean equation, then reparameterize on a t' variable using the inverse of the speed function? Maybe? I feel like I'm speaking using words I don't understand. >I'm sure there's some fancy mathematical trick that would just do it. If I were only more comfortable with math... my handwavy sketch would be to compute the speed function by differentiating the formulas to get dx, dy, and dz and passing them through the Pythagorean equation, then reparameterize on a t' variable using the inverse of the speed function? Maybe? I feel like I'm speaking using words I don't understand. What you're looking for is called arc length parameterization. Basically, you need to compose the curve with the inverse of its arc length function. Aside form a few special curve families, closed-form solutions don't exist. 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. And for doing that there’s in general no better way than at each t, binary/interpolation search a dt that roughly corresponds to the ds that you want (start with the previous dt, it’s likely a very good approximation). In practice, you’d do that once and store the results, basically approximating the curve as a polyline of evenly-spaced points– at least assuming that the curve itself isn’t changing over time! 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. Generally this idea is called motion profiling: https://www.motioncontroltips.com/what-is-a-motion-profile/ 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. Note that the radius starts at zero, though, so something is going to have to deal with limits. A simpler path following approximation (e.g., for a game) might be to just give an iterative system path and tangent targets with respect to Z and then provide an iterative constraint on velocity with some sort of basic tweening (e.g., new = a * old + (1 - a) * target). Then just drag the thing along Z, like bead toys for toddlers. I've recently gotten interested in p5js, which is a fun and easy way to explore this kind of stuff; if anyone here is currently thinking (like I am) that you want to try this out yourself, I'd give it a try: https://p5js.org/tutorials/setting-up-your-environment/ You can play with the equations on 3D Desmos here: https://www.desmos.com/3d/t66etxi1y8 Interesting to note that the parametric equations of the spiral are linear in spherical coordinates. https://en.wikipedia.org/wiki/List_of_common_coordinate_tran... Thanks. 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". I happened to touch the right side so I thought that advanced it and so when I later clicked an extra time unintentionally, I tried to click the left side to go back (note that people who read right to left it top to bottom may have different intuitions [or be used to an insensitive web]). Unfortunately that just skipped another screen so that I missed two in a row. Not fatal because I kept going but disappointing and micro-sad. I think some subtle guidance could have removed the ambiguity and helped me focus and engage better. There are instructions on the first slide. I think adding a secondary swipe mechanic would be nice tho (although I prefer the tap), as that's how these "card stack" interfaces are usually navigated in social media apps. Oh, you're right. Small and at the bottom. I guess I tried to scroll before my eyes got there. Thanks for pointing out my error. 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? Was it derived in some way or simply hand-selected to look pleasing? 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: Just a thought: Make the velocity of the path constant. There should be some way to take a derivative an set it to a constant and solve for z. ( or really reparameterize the curve t' = f(t)) so the velocity is constant. Actually, now that I think about it, choosing z = c * t is kind of both influencing how the path is parameterized as well as the path carved out on the sphere. I am about to start my computer science degree, so this visualization got me really excited to dive into the mathematics of it all. I love to see these kind of animations, but the math behind them sends my brain spinning! Very cool to see it broken down in a digestible way like this. 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). If you the author aren't sure of such topics, I would recommend 3blue1brown yt videos on the matter Possibly better for that than for programmers (given it doesn't include code or libraries used or anything about actually manipulating 3d objects like vertices, stretching and morphing to achieve the effect shown etc) The animations are so fluid! I very recently was looking to generate some complex shapes, and stumbled onto the complex problem of "disperse N points on sphere" and all its nuances. There was a really cool / simple algorithm that was mentioned called the fibonacci-sphere that also generates a spiral around a sphere, but for the point dispersal. Here's a paper [1] on it that talks more about it. Really nice visualisation. Does anyone know of any tooling I could use to do some plotting like this in 3D space even a fraction as nice looking as OPs? I'm not a web dev but I am a dev, but would prefer something high enough level that I can focus on the "what" (to plot) rather than the "how". Grammar problem: "In 3D space, we position objects by setting its coordinates" should read either "their coordinates" or "an object". This reminds me of how the spins in MRI are manipulated in order to acquire the signal. Tracking the tips of the spins traces out similar looking paths. https://m.youtube.com/watch?v=vapJRr6gAds&t=2786s 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. If you want really great further consideration of creating geometric figures with parametric equations, Joseph Choma's book "Morphing" is an all-timer. https://www.quercusbooks.co.uk/titles/joseph-choma/morphing/... > If you like this, please consider following me on Twitter and sharing this with your friends. I do like this and will share with a couple of friends. But I no longer have a Twitter account and will definitely not rejoin. Would you consider adding an RSS or JSON feed to your website? Or make a Mastodon account, those provide RSS feeds by default. Man, that whole ‘please make an RSS or JSON feed for me’ request reads like Richard Stallman emailing himself a webpage to print out later, let them use Twitter or whatever medium they wants Nifty, but what I'm curious about is how you created the drain/fill effect on the cube. https://www.johndcook.com/blog/2023/08/12/fibonacci-lattice/ If you want a spiral that covers the sphere with evenly spaced samples, consider this approach. And the Matt Parker Stand-up Maths video: "Golf balls: How many holes in one?"
https://www.youtube.com/watch?v=dNTnk1VFoJY Very cool & pretty, but 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. I think the crazy plot at the end is not intentionally constructed to be exactly that way, just an example of what you can get if you vary the parameters of cosines and sines in parametric equation setups similar to the ones shown earlier (instead of seeking to align them so they wrap a sphere exactly). 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. I LOVE when people geek out about the most simple mathematical things*, especially discovering the animation power of the trigonometric functions... or any of mathematics underlying modern interactive stuff. It's one thing to know what they do, it's another to understand the power of that tool. * I wrote a similar article around making "blobs" a while back: https://www.hailpixel.com/articles/generative-art-simple-mat... > 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. I can't tell why, but I'm getting terrible performance on Linux/Brave. The CPU spikes immediately on visit. I'm on a powerful machine (32 GiB RAM, Intel i9 w/ 24 cores). > I can't tell why Let's find out where this sentence goes! > Linux Would look into that. > Brave. Definitely would look into that. > 32 GiB RAM, Intel i9 w/ 24 cores This part is not relevant How is that not relevant? Because any non-broken combination of browser and software platform can run this on any PC made this century. It runs fine on my terrible Atom Chromebook, even. Wish the bundle.js wasn't minified so I can look at the code and see what's going on! This is super nice to view, could you share how you made it? I want to make something similar for Rotation Matrices I think 3blue1brown might have an animation library, the same one he uses in his videos that might help with that I was also hoping to see more about how the animations were done here. They look great. I love the shifting camera perspectives. That is beautiful animation. This is a great example of a visual lesson that leaves a chalkboard in the dust (ha). Cool! My first instinct was to scroll. I suggest you treat the scroll down as a signal to move 1 slide further Cool! My first instinct was to scroll. I suggest you use the scroll down as a signal to move 1 slide further Really love this project, I only have small little UX nitpick:
as lefthanded person it is quite cumbersome to tap right side of screen. Go to solution is to navigate using swiping which is ambidexterous. As a left-handed person I have no problem holding the phone in my right hand when I need to use UI on that side of the screen. Though I do prefer phones on the small side. This is why I love small phones - I’m able to reach and tap with left thumb quite easily on my iPhone SE. This was amazing! I also checked the understanding neural network. Looking forward to the next part in that one. Keep up the good work! so beautiful. thank you for this. I am trying to develop intuition for sine cosine etc. and it did fire some neurons in my brain not sure i can put that into words Best thing I have seen on HN in ages. Also interesting for a CNC geek. I had a similar thought about 3D printing - particularly extruding mathematically defined shapes in vase mode. In both of these cases, mathematically generating the points to visit (in gcode) is needed, and we don't care about constant speed - the firmware handles the instruction->motion part. I am thinking about non-planar printing mostly, but this could also apply to CNC finishing passes. I love the slight aliasing on the visuals (iOS Safari). It makes it obvious that it’s rendered on the fly. 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) 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. For me, it doesn't work entirely in Librewolf 142 (shows the text, and some tags moving around an otherwise black screen), but does work in Firefox Nightly. Worked fine for me using Firefox 142 on Linux. Did you have any errors in the console? Really nice animations! This is the type of thing that was really hard for me to grasp in school. This lays it out so plainly. 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. 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? Without scaling the rate of change of the given curve would not be constant. I was expecting linear progression on z and some nasty square root for amplitude of x,y. It's cool that he basically just used another parametric circle drawn on coordinates z and amplitude of x,y oscillations. 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. Your mission, should you choose to accept it ;-) it's impressive enough to crash my browser (twise) before I got to see it
the full digital resource conflict jam and judder! Very distracting to try and read text in the middle of a constantly spinning visualisation
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with outputs, const degrees = Math.PI / 180;
const bearing = 5 * degrees; // or it might be 85 degrees? Not sure off the top of my head
const k = Math.tan(bearing);
const v = 0.001 // some velocity, adjust as needed
const phi = (t) => v*t/Math.sqrt(1 + k*k) // the sqrt is not strictly needed
const theta = (t) => k*Math.ln(Math.tan(phi(t)/2)) // this is the annoying one haha
I doubt that they did the ln(tan(phi/2)) thing though, but it's what you get when you integrate the k d{phi} = sin{phi} d{theta} equation that you have here. const x = (t) => Math.sin(phi(t)) * Math.cos(theta(t))
const y = (t) => Math.sin(phi(t)) * Math.sin(theta(t))
const z = (t) => Math.cos(phi(t))
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