N-gated Hacker NewsMar 2
🤓 Ah yes, the riveting tale of #Lagrange Interpolating Polynomials—a math nerd’s wet dream! 📈 Just what the internet needs: *another* 5000-word novel on fitting curves to dots! 🤯 Because who doesn’t love a good bedtime story about polynomial coefficients? 🙄
https://eli.thegreenplace.net/2026/notes-on-lagrange-interpolating-polynomials/ #mathnerd #mathstories #polynomials #datafitting #curvefitting #HackerNews #ngated
Notes on Lagrange Interpolating Polynomials - Eli Bendersky's website

Karsten SchmidtNov 29

Researching DIY solutions for preparing digital negatives for my salt/kallitype printing experiments. Working on a little curve fitting tool/library (of course using https://thi.ng/umbrella) for computing tone mapping curves derived from sample swatches (here 5% steps of gray/density). The screenshot shows 2nd - 5th degree polynomials. Looks like a quartic is more than good enough... (I know there are ready-made tools for this, but I'm learning more this way... :)

#ThingUmbrella #CurveFitting #Regression #DigitalNegative

N-gated Hacker NewsMay 1, 2025
👀 Oh, look! Yet another tech bro discovers the wonders of AI reviewing AI code. 🙄 Daksh wonders if he should play with his own toy while basking in the #irony of a power law #discovery, because clearly, curve-fitting is the new "Eureka!" moment. 🚀
https://www.greptile.com/blog/ai-code-reviews-conflict #techbro #AIcode #curvefitting #Eureka #HackerNews #ngated
AI Code Review: Should the Author Be The Reviewer?

Exploring the paradox of using AI to review code that was written by AI, and whether this approach makes sense.

cathillApr 15, 2025
Is machine learning merely a form of curve-fitting?
#machinelearning #ai #curvefitting #linearregression #buzzwords
SplinesFeb 6, 2025
#ReverseEngineer #ImageScans for #restoration.

This is a side view from #Vignola's #RegolaArchitettura at https://archive.org/details/gri_33125008229458/page/n39/mode/2up.

We recover the essential geometry of #primaryCurves using #curveFitting by trial and error — a human endeavor by "eye" and heuristics — not to be confused with mathematical curve fitting by regression analysis.

The heights of rectangles labeled N, P, Q, and R are 128, 80, 80, and 48, respectively. P is halfway between N and R, and Q is halfway between P and R.

The curve labeled S is the counterpart to the curve labeled O in the previous post. The purpose of these curves will be explained when we derive the #secondaryCurves from the primary curves.

For now, just note that curve O in the previous post is derived by simple proportion arithmetic. Width of N is 112 units and width of R is 28 units [https://pixelfed.social/p/Splines/793169876757012827]. Since the gap between start of curve O and the curve closest to it is 32 units at the front, the gap at the rear is 32*28/112 = 8, and 16 in the middle.

Curve S is derived in a slightly different manner because, unlike curve O where we knew the starting point, we know neither the start nor the end of curve S. Instead, we look at another clue that Vignola left for us — The 4 long leaves emanating from the rear and spreading towards the front on each bell shape. So we divide the front height of N and rear height of R into 4, giving us the start of S at 32 units from the top (miraculously in agreement with the start of curve O) in front and 12 units in the rear.

The top profile curve does not seem to "fit" Vignola's sketch. First, this is a hand sketch. Second, I tried to fit the curve more closely, but the design broke down later. Third, realize that if we fit the curve more closely to what's in the sketch, this will be the ONLY curve to have a tangent at the inflection point (switch from convex to concave) that is neither horizontal nor vertical.
Regola delli cinque ordini d' architettura : Vignola, 1507-1573 : Free Download, Borrow, and Streaming : Internet Archive

48 leaves : 44 cm (fol.)

Internet Archive
SplinesFeb 6, 2025
#ReverseEngineer #ImageScans

We now dig into the archives and resurface old sketches for #restoration. This one is from #Vignola's #RegolaArchitettura at https://archive.org/details/gri_33125008229458/page/n39/mode/2up. This lavishly illustrated book with copious notes that also flaunt his #calligraphy was written (in Italian) when America was still a British colony. The book went out of copyright a long time ago.

Straighten the image as much as you can in an image editor and crop it before bringing it into a #CAD tool.

Then, stare at the image for a while and squint occasionally until you "see" crucial features and patterns emerge, while ignoring the "noise."

Finally, try #curveFitting with the simplest of curves — straight lines, circular arcs, ellipse, and so on to get as close an approximation as possible. Remember that with hand-drawn sketches, the fit will rarely be perfect. So use some structure as a guide or #scaffolding as I laid out in https://pixelfed.social/p/Splines/792966507797633558.

In the top left of the diagram, I show the measurements that I was satisfied with after a lengthy process of trial and error because the numbers comport with my understanding of the proportions the original designers intended — many, but not all of which are documented in #Scarlata's #PracticalArchitecture with #VignolaProportions in tabular form.

For measurements that are missing, use plausible heuristics to fill in the blanks and try to justify your choices using simple rules. In this case, the bedrock rules are:

1. The entire #volute is exactly µ = 144 units wide, including #ArcZero, which extends 32 units beyond the portion of the volute that is actually used in the design.

2. The portion of the volute that is actually used in the design is 112 units wide, same as the height of the unadorned #capital.

3. Width of the #scroll bell shape as seen from the bottom is 112 units in front, 56 units in the middle and 28 units in the rear — all in #geometricSequence.
Regola delli cinque ordini d' architettura : Vignola, 1507-1573 : Free Download, Borrow, and Streaming : Internet Archive

48 leaves : 44 cm (fol.)

Internet Archive
SplinesFeb 5, 2025
The classic #IonicScroll is the most complex of all components in the #IonicOrder mainly because it is poorly documented, if at all, and even poorly understood. It is as if the classical architects deliberately concealed its enigmatic design secrets within the confines of a smooth elegant shell that could only be revealed after intense study and analysis.

I got this impression because I spent years searching for credible and actionable documentation on how to recreate this beautiful design in a #CAD tool. In the Age of Internet and Social Media, my web searches always disappointed me because the results lacked something vital in one respect or another. Over the years, I created hundreds of versions of the scroll that looked so perfect and pleasing that I thought I had cracked it, only to find some flaw or another in my work.

So, it is with caution that I present my work on the scroll in the hopes that someone will build upon this knowledge and either validate the design, or correct it and share it with me and the rest of the world.

Looking back at my progress, I'm now surprised at how remarkably simple and elegant the design is that defied familiar geometrical construction techniques I had been using until now.

As I mentioned in my introductory post, this design can be recreated by drawing simple 2-dimensional lines and circular arcs, but instead of just #primaryProfileCurves, we will use up to three additional sets of curves — #secondaryCurves, #tertiaryCurves, and #quaternaryCurves — each derived from the previous set.

I extracted the #primaryCurves after a lengthy trial-and-error process that involved #curveFitting image scans from #Vignola’s book, #RegolaArchitettura. I had to #reverseEngineer the details because the measurements have either been lost, or are locked away in some library.

Even though we start with lines and arcs, the end results are always #NURBS curves and surfaces, but everything is done by the CAD tool, and no additional math is needed.
SplinesJan 30, 2025
This shows macro-level measurements for the #IonicPedestal.

The key to #effectiveModeling is to simplify a complex shape into elementary components. Sometimes, this involves mentally flattening and reducing 3D shapes to 2D shapes, extracting elementary curves from them, and then recreating the 3D shapes from the extracted 2D curves.

This is not always easy for organic shapes (which can still be approximated by Bézier curves). I extracted the #primaryCurves for the #IonicScroll surface in https://pixelfed.social/p/Splines/789956327130679640 after a lengthy trial-and-error process that involved #curveFitting images from #Vignola’s book, #RegolaArchitettura. I had to #reverseEngineer the details because the measurements have either been lost, or are locked away in some library. Web search yields no details on these measurements.

Fortunately, for geometrical shapes like pedestals, this is very easy. Because of its square footprint, mentally slicing it through the middle from top to bottom, it is easy to “see” the outline. Another way to think about #curveExtraction is to shine an imaginary bright light on an object from behind in a dark room to reveal its silhouette.

For the pedestal, even this silhouette or outline can be further reduced because the shape is symmetrical about the #columnAxis. With this realization, we only need to focus on one half of the outline, and methodically proceed from bottom to top, marking every kink and inflection point on the outline.

Fortunately, the other authoritative book, #Scarlata’s #PracticalArchitecture, I mentioned in my introductory post already documents #VignolaProportions in tabular form. So we can skip everything else and go directly to that.

Total height of #IonicPedestal is 864 units (108 parts, or 6*µ) of which the #PedestalBasement and #PedestalCap are each 72 units (9 parts, or µ/2) and the #Dado is 720 units (90 parts, or µ*5) tall.
Splines (@[email protected])

This sweeping shape is a timeless design that first appeared in the scrolls of the #IonicCapital as the most distinctive part of the #IonicOrder in classical Greco-Roman architecture more than 2500 years ago. Shown here with a zebra pattern on the wireframe of a CAD model to accentuate its features and attest to the smoothness of its 3-dimensional surface, the design has been refined many times since the original version over the last two millennia. The two most remarkable things about this design are that: — It can be recreated with modern CAD tools by drawing simple 2-dimensional straight lines and circular arcs exclusively. The end result is truly breathtaking and makes one wonder how architects visualized the result and put theory into practice. In the CAD model, the ultimate surface is a #NURBS surface that uses #BSplines extensively, but none of the B-splines or surfaces need to be created "by hand." One only has to draw straight lines and circular arcs with accurate measurements snapped to grids. — For a design that has survived the ages, it is lamentable how few authoritative sources that accurately describe fine details and exact reconstruction methodology remain accessible to the general public in the age of Internet. The most comprehensive is the 10-volume tome that Marcus #Vitruvius Pollio, a Roman architect and engineer, wrote for #JuliusCaesar and his successor Emperor #CaesarAugustus. [https://www.gutenberg.org/files/20239/20239-h/20239-h.htm] I frequently use two more authoritative sources: — "Regola delli cinque ordini d' architettura," or simply #RegolaArchitettura by Giacomo Barozzi da #Vignola [https://archive.org/details/gri_33125008229458/page/n3/mode/2up], and — "A Course in Theoretical and Practical Architecture," or simply #PracticalArchitecture by Francisco Salvatore #Scarlata (#Bordonaro), which documents #VignolaProportions in tabular form [https://babel.hathitrust.org/cgi/pt?id=mdp.39015031201190&view=1up&seq=5]

Pixelfed
Show threadNumeRe.org Jan 11, 2024
@stfn Yeah, #machinelearning is only #maths. Back in university that was called #statistics and #CurveFitting
Victoria Stuart 🇨🇦 🏳️‍⚧️Jul 27, 2023

Predicting Ordinary Differential Equations with Transformers
https://arxiv.org/abs/2307.12617

The application of ML / transformers to ODE is fascinating.
There's a terrific 2019-01 blog post that covers this (and more!): 👍️

Understanding Neural ODE's
https://jontysinai.github.io/jekyll/update/2019/01/18/understanding-neural-odes.html
Discussion: https://news.ycombinator.com/item?id=18978764

Transformer (machine learning model): https://en.wikipedia.org/wiki/Transformer_(machine_learning_model)

#ML #OCE #mathematics #MachineLearning #transformers #MathematicalModeling #regression #CurveFitting #differentiation

Predicting Ordinary Differential Equations with Transformers

We develop a transformer-based sequence-to-sequence model that recovers scalar ordinary differential equations (ODEs) in symbolic form from irregularly sampled and noisy observations of a single solution trajectory. We demonstrate in extensive empirical evaluations that our model performs better or on par with existing methods in terms of accurate recovery across various settings. Moreover, our method is efficiently scalable: after one-time pretraining on a large set of ODEs, we can infer the governing law of a new observed solution in a few forward passes of the model.

arXiv.org