When an AI sys­tem solves a dif­fi­cult math­e­mat­i­cal prob­lem, the usual ex­pla­na­tion is that it has be­come more in­tel­li­gent.

Perhaps it has ab­sorbed mil­lions of math­e­mat­i­cal ex­am­ples. Perhaps re­in­force­ment learn­ing has taught it bet­ter rea­son­ing strate­gies. Perhaps it is be­gin­ning to de­velop some­thing re­sem­bling gen­uine math­e­mat­i­cal in­tu­ition.

All of these ex­pla­na­tions may con­tain some truth. But they over­look a sim­pler pos­si­bil­ity:

AI has ac­cess to a vastly larger work­ing mem­ory than the hu­man brain.

Or, more pre­cisely, it has ac­cess to an enor­mous ex­ter­nal sym­bolic work­space that per­forms many of the func­tions that work­ing mem­ory per­forms in hu­mans.

This dif­fer­ence may be es­pe­cially im­por­tant in math­e­mat­ics.

A hu­man math­e­mati­cian can hold only a small num­ber of un­fa­mil­iar el­e­ments in mind si­mul­ta­ne­ously. An AI model can keep the en­tire prob­lem state­ment, hun­dreds of in­ter­me­di­ate equa­tions, sev­eral aban­doned ap­proaches, de­f­i­n­i­tions, con­straints and ear­lier con­clu­sions in­side its con­text win­dow.

We nor­mally in­ter­pret the re­sult­ing per­for­mance as ev­i­dence of su­pe­rior rea­son­ing. But some of it may in­stead re­flect the re­moval of one of the most im­por­tant bi­o­log­i­cal lim­its on hu­man rea­son­ing: our ex­tremely re­stricted work­ing-mem­ory ca­pac­ity.

Working mem­ory is the men­tal sys­tem that al­lows us to hold and ma­nip­u­late in­for­ma­tion over short pe­ri­ods.

When solv­ing an equa­tion, you must re­mem­ber what each vari­able rep­re­sents, which op­er­a­tions have al­ready been per­formed and what the cur­rent goal is. During a proof, you may need to keep track of as­sump­tions, in­ter­me­di­ate lem­mas, ex­cep­tions and mul­ti­ple pos­si­ble cases.

Human work­ing mem­ory is re­mark­ably lim­ited.

Its ex­act ca­pac­ity de­pends on the task and on how in­for­ma­tion is or­ga­nized, but the gen­eral lim­i­ta­tion is ob­vi­ous from every­day ex­pe­ri­ence. Try mul­ti­ply­ing two three-digit num­bers in your head. The un­der­ly­ing op­er­a­tions are sim­ple. The dif­fi­culty comes largely from hav­ing to pre­serve par­tial re­sults while per­form­ing ad­di­tional cal­cu­la­tions.

Writing the num­bers down trans­forms the prob­lem.

Paper does not make you more in­tel­li­gent. It ex­pands your ef­fec­tive work­ing mem­ory.

The same prin­ci­ple ap­plies at higher lev­els of math­e­mat­ics. A math­e­mati­cian uses no­ta­tion, scratch pa­per, di­a­grams and pre­vi­ously writ­ten lem­mas not merely to com­mu­ni­cate the so­lu­tion, but to make the rea­son­ing cog­ni­tively pos­si­ble.

Experts com­pen­sate through “chunking.” A novice sees a long se­quence of sym­bols. An ex­pert rec­og­nizes a fa­mil­iar struc­ture and treats it as a sin­gle con­cep­tual ob­ject. This al­lows far more in­for­ma­tion to fit in­side the same bi­o­log­i­cal work­ing-mem­ory limit.

But chunk­ing does not elim­i­nate the limit. It merely com­presses the in­for­ma­tion.

An AI model faces a very dif­fer­ent con­straint.

The im­por­tance of work­ing mem­ory for math­e­mat­ics is not merely the­o­ret­i­cal. It is vis­i­ble in the dif­fer­ences be­tween hu­man be­ings.

Working mem­ory is strongly re­lated to gen­eral in­tel­li­gence, which raises an ob­vi­ous ques­tion: does it in­de­pen­dently pre­dict math­e­mat­i­cal per­for­mance, or is it merely an­other im­per­fect mea­sure of IQ?

Several stud­ies sug­gest that it con­tributes some­thing be­yond con­ven­tional in­tel­li­gence mea­sures. Alloway and Passolunghi (2011), for ex­am­ple, ex­am­ined work­ing mem­ory, ver­bal abil­ity and math­e­mat­i­cal skills in chil­dren. They found that work­ing-mem­ory mea­sures made a dis­tinct con­tri­bu­tion to math­e­mat­i­cal per­for­mance rather than sim­ply re­pro­duc­ing the as­so­ci­a­tion be­tween math­e­mat­ics and gen­eral ver­bal abil­ity.

In a sep­a­rate six-year lon­gi­tu­di­nal study, Alloway and Alloway (2010) mea­sured chil­dren at age five and then ex­am­ined their aca­d­e­mic achieve­ment six years later. Early work­ing-mem­ory per­for­mance pre­dicted later lit­er­acy and nu­mer­acy even af­ter IQ was in­cluded in the analy­sis. Indeed, work­ing mem­ory was a stronger pre­dic­tor of the later aca­d­e­mic out­comes than the IQ mea­sure used in the study.

Blankenship and col­leagues (2015) sim­i­larly re­ported that work­ing mem­ory ex­plained unique vari­a­tion in math­e­mat­i­cal flu­ency and cal­cu­la­tion af­ter sta­tis­ti­cally con­trol­ling for IQ and age. A large meta-analy­sis by Friso-van den Bos and col­leagues (2013) also found a con­sis­tent re­la­tion­ship be­tween work­ing mem­ory and math­e­mat­ics across pri­mary-school stud­ies, al­though the strength of the re­la­tion­ship var­ied ac­cord­ing to the type of work­ing-mem­ory and math­e­mat­i­cal task be­ing mea­sured.

These find­ings should not be ex­ag­ger­ated. Working mem­ory and in­tel­li­gence over­lap sub­stan­tially, and sta­tis­ti­cal con­trol can­not per­fectly iso­late them as in­de­pen­dent psy­cho­log­i­cal mech­a­nisms. Nor does the ev­i­dence im­ply that com­mer­cially train­ing work­ing mem­ory will nec­es­sar­ily pro­duce large im­prove­ments in in­tel­li­gence or math­e­mat­ics.

The nar­rower con­clu­sion is nev­er­the­less im­por­tant: among chil­dren with sim­i­lar mea­sured in­tel­li­gence, dif­fer­ences in the abil­ity to hold, up­date and ma­nip­u­late in­for­ma­tion still pre­dict dif­fer­ences in math­e­mat­i­cal per­for­mance.

This pro­vides a cru­cial clue for un­der­stand­ing AI. If hu­man math­e­mat­i­cal per­for­mance is partly capped by a work­ing-mem­ory bot­tle­neck, then giv­ing a ma­chine an enor­mous sym­bolic work­space changes the na­ture of the con­test. The ma­chine may ap­pear more math­e­mat­i­cally in­tel­li­gent partly be­cause it is much less con­strained by a cog­ni­tive lim­i­ta­tion that sup­presses hu­man per­for­mance.

A mod­ern lan­guage model can process an enor­mous se­quence of to­kens at once. This se­quence may in­clude the orig­i­nal ques­tion, de­f­i­n­i­tions, ex­am­ples, in­ter­me­di­ate cal­cu­la­tions and the mod­el’s own ear­lier rea­son­ing.

The con­text win­dow is not iden­ti­cal to hu­man work­ing mem­ory. It is bet­ter un­der­stood as a gi­gan­tic ex­ter­nal note­book com­bined with an im­per­fect sys­tem for search­ing and us­ing what has been writ­ten in it.

This dis­tinc­tion mat­ters.

Humans pos­sess a form of ac­tive in­ter­nal mem­ory. We can silently choose a num­ber, hold it in mind, trans­form it and re­place it with a new value with­out say­ing or writ­ing any­thing.

Standard lan­guage mod­els are much weaker at main­tain­ing this kind of pri­vate, con­tin­u­ously up­dated men­tal state. Their most sta­ble form of mem­ory is usu­ally the se­quence of to­kens that has al­ready been gen­er­ated.

If the model writes:

x=6

and later writes:

x+3=9,

those state­ments re­main in­side the con­text. The model can at­tend to them again when gen­er­at­ing the next step.

Its rea­son­ing is there­fore of­ten ex­ter­nal­ized. The text is not merely a re­port of a com­pleted thought process. The text is part of the mech­a­nism by which the rea­son­ing oc­curs.

Humans do some­thing sim­i­lar when us­ing scratch pa­per. The ma­jor dif­fer­ence is scale.

An un­aided hu­man may strug­gle to keep five un­fa­mil­iar con­di­tions ac­tive si­mul­ta­ne­ously. An AI can pre­serve dozens or hun­dreds of them in ex­plicit form.

This does not mean that every item in a long con­text is re­trieved per­fectly. Models can over­look rel­e­vant in­for­ma­tion, be­come dis­tracted or lose track of de­tails. Advertised con­text length is not the same as per­fectly us­able mem­ory.

Nevertheless, the dif­fer­ence in po­ten­tial ca­pac­ity is enor­mous.

The con­text-win­dow ad­van­tage is not equally use­ful in every kind of rea­son­ing.

It mat­ters es­pe­cially for math­e­mat­ics be­cause math­e­mat­i­cal rea­son­ing can be trans­lated un­usu­ally well into ex­plicit sym­bols.

Almost every rel­e­vant el­e­ment of a math­e­mat­i­cal prob­lem can be writ­ten down:

the as­sump­tions;

the as­sump­tions;

the de­f­i­n­i­tions;

the de­f­i­n­i­tions;

the known equa­tions;

the known equa­tions;

the cur­rent ob­jec­tive;

the cur­rent ob­jec­tive;

the re­sults al­ready proved;

the re­sults al­ready proved;

the cases that have been elim­i­nated;

the cases that have been elim­i­nated;

the con­di­tions un­der which each step re­mains valid.

the con­di­tions un­der which each step re­mains valid.

Once writ­ten, this in­for­ma­tion re­mains sta­ble.

If (x) is de­fined as an in­te­ger at the be­gin­ning of a proof, it re­mains an in­te­ger un­less the proof ex­plic­itly changes the de­f­i­n­i­tion. A strict in­equal­ity does not grad­u­ally be­come a non-strict in­equal­ity be­cause of changes in mood, con­text or in­ter­pre­ta­tion.

Mathematical sym­bols are de­signed to re­duce am­bi­gu­ity.

This makes math­e­mat­ics al­most per­fectly suited to an in­tel­li­gence that op­er­ates through a large tex­tual work­space.

Consider a prob­lem re­quir­ing the solver to re­mem­ber that:

n is odd;

n is odd;

p is prime;

p is prime;

x≠0

x≠0

and that one branch of the ar­gu­ment has al­ready pro­duced a con­tra­dic­tion.

A hu­man may un­der­stand the ba­sic strat­egy but di­vide by x be­fore es­tab­lish­ing that x≠0. The er­ror is not nec­es­sar­ily caused by a lack of in­tel­li­gence. It may be a fail­ure of book­keep­ing.

An AI can re­state the ac­tive con­straints at each stage:

We are work­ing un­der the as­sump­tions that (n) is odd, (p) is prime and (x≠0).

We are work­ing un­der the as­sump­tions that (n) is odd, (p) is prime and (x≠0).

The con­text be­comes a ledger of the rea­son­ing state.

Many dif­fi­cult math­e­mat­i­cal prob­lems con­tain a pro­found in­sight some­where near the be­gin­ning, but they also con­tain a large amount of less glam­orous work af­ter­ward: ex­pand­ing ex­pres­sions, check­ing cases, car­ry­ing con­di­tions through trans­for­ma­tions and mak­ing sure that the fi­nal con­clu­sion is com­pat­i­ble with every ear­lier as­sump­tion.

A ma­chine does not need to pos­sess deeper in­sight than a hu­man to gain an ad­van­tage here. It may sim­ply be bet­ter equipped to pre­serve the en­tire state of the prob­lem while com­plet­ing a long se­quence of op­er­a­tions.

Mathematics is highly com­po­si­tional.

A proof can of­ten be rep­re­sented as:

A → B → C → D

If each step is valid and the chain is pre­served ac­cu­rately, the con­clu­sion fol­lows.

A large work­ing space al­lows the model to con­struct much longer chains be­fore los­ing the thread.

This is im­por­tant be­cause the dif­fi­culty of a prob­lem does not de­pend only on the dif­fi­culty of each in­di­vid­ual step. It also de­pends on how many steps must be co­or­di­nated.

A per­son may be per­fectly ca­pa­ble of un­der­stand­ing every lo­cal in­fer­ence in a 100-step ar­gu­ment while still be­ing un­able to gen­er­ate the en­tire ar­gu­ment un­aided. The prob­lem ex­ceeds the per­son’s abil­ity to main­tain the global struc­ture.

AI can po­ten­tially com­pen­sate by writ­ing down nearly every­thing.

This may ex­plain why ad­di­tional “thinking time” of­ten im­proves model per­for­mance. More com­pu­ta­tion al­lows the sys­tem to pro­duce more in­ter­me­di­ate states, ex­am­ine al­ter­na­tive branches and pre­serve par­tial con­clu­sions.

What looks like deeper thought may some­times be broader search con­ducted in­side a much larger note­book.

Now com­pare a math­e­mat­i­cal prob­lem with a so­cial ques­tion:

Why has Maria sud­denly stopped re­ply­ing to my mes­sages?

Why has Maria sud­denly stopped re­ply­ing to my mes­sages?

A larger con­text win­dow might al­low an AI to ex­am­ine years of cor­re­spon­dence. It could iden­tify changes in tone, tim­ing and vo­cab­u­lary.

But the de­ci­sive in­for­ma­tion may still be miss­ing.

Perhaps Maria is an­gry. Perhaps she is busy. Perhaps she is ill. Perhaps she has lost her phone. Perhaps she is avoid­ing an un­re­lated prob­lem.

No amount of mem­ory can re­trieve facts that were never ob­served.

The chal­lenge is not sim­ply to pre­serve a known set of premises and de­rive their con­se­quences. It is to rea­son un­der un­cer­tainty about hid­den causes.

Informal rea­son­ing also de­pends heav­ily on con­cepts whose mean­ings are un­sta­ble.

Words such as “fair,” “successful,” “responsible,” “harmful” or “intelligent” do not pos­sess the ex­act­ness of math­e­mat­i­cal vari­ables. Their mean­ing de­pends on cul­ture, goals and con­text.

A model can re­mem­ber every sen­tence in a dis­cus­sion while still mis­un­der­stand­ing what the par­tic­i­pants mean.

The same ap­plies to po­lit­i­cal analy­sis, his­tor­i­cal in­ter­pre­ta­tion, busi­ness strat­egy and psy­cho­log­i­cal judg­ment. In these do­mains, the cen­tral prob­lem is of­ten not work­ing-mem­ory ca­pac­ity. It is iden­ti­fy­ing the cor­rect causal model when the ev­i­dence is in­com­plete and am­bigu­ous.

A larger note­book helps, but it does not solve the fun­da­men­tal prob­lem.

08 Jul, 2026

Table of Contents

Intro Contest in short Problem in­tro

Contest in short

Problem in­tro

Why this prob­lem is auto-re­search-able

Learning Enough to Ask Better Questions

(Optional) Math for QR de­com­po­si­tion: Householder re­flec­tions

Make se­r­ial work small with the help of the blocked Householder al­go­rithm

Other chal­lenges

Codex-maxxing Kernel progress break­throughs Breakthrough ideas

Kernel progress break­throughs

Breakthrough ideas

Introducing idea di­ver­sity to es­cape the lo­cal max­ima

Implementation Hints

What I could have done bet­ter

Conclusion

References

Acknowledgements

Intro

Contest in short

GPU Mode, in col­lab with Core Automation, re­cently hosted an auto-re­search themed con­test. The prob­lem state­ment was to im­ple­ment batched square com­pact-House­holder QR fac­tor­iza­tion aka QR de­com­po­si­tion. I placed 12th out of 183 par­tic­i­pants, end­ing up with a 232x speedup over the base­line so­lu­tion. This post is about how I got there. I will go through my ap­proach, learn­ings, and bot­tle­necks I ran into dur­ing the con­test. It was my first se­ri­ous at­tempt at auto-re­search. Some peo­ple will call this “loop en­gi­neer­ing”, and hon­estly that is fine too.

Note that you don’t need to go through the math­e­mat­ics or the prob­lem it­self in de­tail to fol­low most of this blog post. I have fo­cused on my ap­proach while keep­ing the math and the prob­lem it­self sec­ondary as most peo­ple who will read this won’t have par­tic­i­pated in the con­test.

You can check out the full con­test page here: Problem Link and Leaderboard

This con­test was part of GPU Mode’s Linear Algebra Kernels in the Age of Research se­ries.

Problem in­tro

We were given a batch of square FP32 CUDA ma­tri­ces A with shape batch x n x n, and had to re­turn the same com­pact Householder QR rep­re­sen­ta­tion as torch.geqrf(A): an H ma­trix whose up­per tri­an­gle is R and whose lower tri­an­gle stores Householder vec­tors, plus a tau vec­tor of re­flec­tor co­ef­fi­cients. The checker re­built Q with torch.linalg.house­hold­er_prod­uct(H, tau), took R = triu(H), and ver­i­fied:

A≈QR,Q⊤Q≈I,Q⊤A≈R

Among cor­rect sub­mis­sions, the leader­board ranked run­time by geo­met­ric mean across shapes and con­di­tion­ing cases. The im­por­tant sizes were batched square ma­tri­ces like 512 x 512, with larger 1024, 2048, and 4096 cases too. Low-bit FP16, FP8, or NVFP4 was al­lowed in­ter­nally, but re­turned fac­tors still had to sat­isfy FP32-style QR checks.

A tiny 3 x 3 ex­am­ple is:

A=[12−5146167−68−424−41]=[6/7−69/175−58/1753/7158/1756/175−2/76/35−33/35]⏟Q[1421−140175−700035]⏟R

Here Q is or­thog­o­nal, which means its columns are unit-length and per­pen­dic­u­lar to each other, and R is up­per tri­an­gu­lar, which means every­thing be­low the di­ag­o­nal is zero. The con­test was not ask­ing us to print dense Q and R di­rectly; it asked for the com­pact Householder ver­sion that lets the checker re­con­struct Q and read R from the up­per tri­an­gle.

For the 3×3 ex­am­ple above, the very first re­flec­tor maps the first col­umn (12, 6, −4) straight onto (−14, 0, 0) in one shot. The −14 be­comes R11. How that works is in the math sec­tion.

Why this prob­lem is auto-re­search-able

GPU Mode pro­vides par­tic­i­pants with the pop­corn CLI mak­ing it agent-friendly. Agents can use this to test, bench­mark, and sub­mit to the leader­board di­rectly. The checker also pro­vided shape-wise feed­back along with the over­all geo­met­ric mean tim­ing.

Astute ob­servers will no­tice this is an apt setup for writ­ing a loop. Agents yearn for tight feed­back loops. They al­low them to hill-climb to their heart’s con­tent.

GPU Mode con­tests usu­ally give you some way to it­er­ate on ker­nels. Either you sub­mit di­rectly, or a spon­sor like Modal chips in cred­its. Here the or­ga­niz­ers ba­si­cally al­lowed un­lim­ited sub­mis­sions as long as you spaced them out. If you did­n’t, the queues got long and every­body’s runs timed out. At one point the work­space even ran out of Modal cred­its be­cause every­one had been ham­mer­ing sub­mis­sions. It’s a nice way to make learn­ing ac­ces­si­ble.

Over the course of 14 days, I made over 1500 sub­mis­sions.

Learning Enough to Ask Better Questions

I have known the ba­sics of GPU ker­nel op­ti­miza­tion (mainly in Triton with some un­der­stand­ing of CUDA) for a year, but haven’t worked in this do­main pro­fes­sion­ally. What I am try­ing to tell you is that I was an un­der­dog among the peo­ple around me on the leader­board. The per­son just above me on the leader­board (CUDA Colonel) is a prin­ci­pal en­gi­neer at NVIDIA.

Anyway aura farm­ing aside, since I know the ba­sics and had re­cently read about GatedDeltaNet, I was fresh on the gen­eral GPU ker­nel lingo.

The bet­ter you know some­thing, the bet­ter you can prompt the LLMs, be­cause you con­vert un­known un­knowns into known un­knowns.

At the same time, it’s worth not­ing that this con­test was doable with­out do­main knowl­edge - like you prob­a­bly won’t make it to the top 10, but you can get a re­spectable speedup over base­line by just re­ly­ing on your har­ness/​agent loop or what­ever.

My first steps in the con­test were to learn what QR de­com­po­si­tion is and how it can be done. There are a bunch of ways to do it - like Gram-Schmidt and Householder re­flec­tions. The con­test man­dated Householder re­flec­tions. I went back and forth with Claude and watched a few YouTube videos to build in­tu­ition. After my dis­cus­sions with Claude, it was clear that we needed to use the blocked Householder al­go­rithm as the main ar­chi­tec­ture with the trail­ing WY-update. As it turns out, GPT-5.5 also had a good idea about this. QR de­com­po­si­tion is a fairly well known prob­lem.

I found the con­cept in­ter­est­ing as ma­trix de­com­po­si­tions show up in sev­eral mod­ern op­ti­mizer vari­ants for LLM train­ing, es­pe­cially in meth­ods that use ma­trix pre­con­di­tion­ing, such as Shampoo-style op­ti­miz­ers and re­lated ap­proaches. Muon (used by Kimi) is an­other good ex­am­ple: in­stead of treat­ing a weight up­date as one gi­ant flat­tened vec­tor, it keeps the ma­trix struc­ture around and or­thog­o­nal­izes the mo­men­tum up­date, usu­ally through a few Newton-Schulz it­er­a­tions that ap­prox­i­mate the po­lar fac­tor.

(Optional) Math for QR de­com­po­si­tion: Householder re­flec­tions

I rec­om­mend skim­ming through this sec­tion if you are cu­ri­ous about the math oth­er­wise feel free to skip. The only thing to note is that there is a se­quen­tial de­pen­dency in Householder QR which makes it prob­lem­atic to do GEMM. We use blocked Householder to make it more ma­trix-mul­ti­pli­ca­tion shaped.

The con­tract

Quickly re­view­ing the con­tract: in­put is a batch of square FP32 ma­tri­ces A; out­put is the com­pact (H, tau) for­mat that torch.geqrf re­turns. The up­per tri­an­gle of H is R. Below the di­ag­o­nal, H stores the Householder vec­tors, and tau stores one scalar per col­umn. The checker re­builds Q from (H, tau) and ver­i­fies A ≈ QR.

Mirrors

Forget ma­tri­ces for a sec­ond. In a bath­room mir­ror, your re­flec­tion is ex­actly as far be­hind the glass as you are in front of it, straight through.

If x⟂ is the part of x stick­ing out per­pen­dic­u­lar to the glass, re­flec­tion just sub­tracts that part twice:

xre­flected=x−2x⟂

So a Householder re­flec­tion is about find­ing the per­pen­dic­u­lar part and sub­tract­ing it twice.

Storing the mir­ror

A Householder vec­tor is the mir­ror, stored com­pactly. In code, we don’t carry around the whole mir­ror plane. We store one vec­tor v stick­ing straight out of it. The mir­ror is every­thing per­pen­dic­u­lar to v, and the re­flec­tion moves along v.

The per­pen­dic­u­lar part is just the shadow of x along v, which is v⊤xv⊤v copies of v. Plug that into the sub­trac­tion above:

ℋx=x−τv(v⊤x),τ=2v⊤v

So tau is just 2v⊤v: the fac­tor of 2 and the length of v bun­dled into one pre­com­puted num­ber. v picks the mir­ror, tau scales the up­date. (I’ll write the math­e­mat­i­cal re­flec­tor as ℋj and re­serve H for the com­pact out­put ma­trix.)

Householder re­flec­tion in 2D tau = 0.00

Why QR cares about mir­rors

QR wants to turn A into an up­per-tri­an­gu­lar ma­trix R. Column 1 should be­come some­thing like (*, 0, 0), col­umn 2 should have ze­ros be­low row 2, and so on.

A Householder mir­ror is use­ful be­cause it can do that to a col­umn in one shot. Take the first col­umn of the 3×3 ex­am­ple above: (12, 6, -4). We want to send it to the x-axis so the lower en­tries be­come zero. A re­flec­tion can only change di­rec­tion, not length, so the tar­get must also have length 14. One valid tar­get is (-14, 0, 0). After that re­flec­tion, the 6 and -4 en­tries are gone, which is ex­actly what we wanted.

How do we find the mir­ror? It sits halfway be­tween the col­umn and its tar­get, so v, the vec­tor pok­ing through the mir­ror, is just the col­umn mi­nus its tar­get:

v = (12, 6, -4) - (-14, 0, 0) = (26, 6, -4)

The Householder up­date

Then com­pute tau = 2 / (vᵀv). The re­flec­tor it­self is:

ℋ=I−τvv⊤,τ=2v⊤v

ℋx=(I−τvv⊤)x

ℋx=x−τv(v⊤x)

ℋAactive=(I−τvv⊤)Aactive

ℋAactive=Aactive−τv(v⊤Aactive)

This turns the cur­rent col­umn into (-14, 0, 0) and rewrites the other columns con­sis­tently, so the next re­flec­tor is built from the up­dated ma­trix.

We keep do­ing this once per col­umn. In rough no­ta­tion, the re­peated up­dates look like:

A(1)=A(0)−τ1v1(v1⊤A(0))

A(2)=A(1)−τ2v2(v2⊤A(1))

A(3)=A(2)−τ3v3(v3⊤A(2))

⋯

R=A(n)

Each line uses the ma­trix pro­duced by the pre­vi­ous line. Each re­flec­tor ze­roes out every­thing be­low the di­ag­o­nal of its col­umn with­out dis­turb­ing the columns al­ready fin­ished. After the last one, A has walked down to an up­per-tri­an­gu­lar R:

A=ℋ1ℋ2⋯ℋn⏟QR

Mirrors don’t change lengths or an­gles, so each ℋj is or­thog­o­nal, and so is their prod­uct Q. That’s where the or­thog­o­nal­ity the checker ver­i­fies comes from for free.

What’s the com­pact for­mat

Once col­umn j is processed, every­thing be­low its di­ag­o­nal is dead space. geqrf reuses those slots to stash the tail of v_j (the lead­ing 1 is im­plicit). On and above the di­ag­o­nal you’re look­ing at R; be­low it, the re­flec­tors; and tau rides along as a sep­a­rate vec­tor. That’s why the checker needs both H and tau to re­build Q.

Compact geqrf stor­age (n = 6) hover a col­umn

H  (upper tri­an­gle = R, be­low di­ag­o­nal = re­flec­tor tails)

tau  (one scalar per re­flec­tor)

r en­tries of R tail of vj τj

Make se­r­ial work small with the help of the blocked Householder al­go­rithm

Householder QR ze­roes out A be­low the di­ag­o­nal one col­umn at a time. Each step builds a re­flec­tor from the cur­rent col­umn and ap­plies it to every­thing on the right. The prob­lem is re­flec­tor j+1 is built from the ma­trix af­ter re­flec­tor j has al­ready hit it. So you can’t re­order the steps and you can’t fuse them. It’s se­r­ial, and the se­r­ial ma­trix-vec­tor work runs in the slow vec­tor lanes of the SM while the ten­sor cores just sit there idle.

Householder QR, one re­flec­tor at a time (n = 5) math view: ze­ros ap­pear, trail­ing block gets rewrit­ten

orig­i­nal a fi­nal en­try of R rewrit­ten by this re­flec­tor 0 zeroed (vj gets stashed here)

The clas­sic fix is the blocked al­go­rithm. You pick a nar­row panel of b columns (say 32 or 64) and do all the se­r­ial work in­side it. That’s fine, be­cause the panel is only b columns wide, so it stays cheap. Then, in­stead of ap­ply­ing the pan­el’s b re­flec­tors to the rest of the ma­trix one at a time, you com­press them into a sin­gle rank-b up­date (the “WY rep­re­sen­ta­tion”) and hit the en­tire trail­ing block in one shot with three back-to-back ma­trix mul­ti­plies. The se­r­ial work stays con­fined to the panel, and every­thing else turns into GEMMs, which is ex­actly the shape the ten­sor cores want.

Concretely, the WY rep­re­sen­ta­tion col­lapses a pan­el’s b re­flec­tors into a sin­gle rank-b up­date. Stack the pan­el’s Householder vec­tors as columns of V=[v1,v2,…,vb], build a small b×b up­per-tri­an­gu­lar T, then:

ℋ1ℋ2⋯ℋb=I−VTV⊤

and the trail­ing-block up­date be­comes three GEMM-shaped steps:

W=V⊤Atrail

Z=T⊤W

Atrail←Atrail−VZ

Blocked Householder (n = 6, panel width b = 2) con­fine the se­r­ial work, GEMM the rest

ASUS Oxiis E250G1

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Oxiis is a uni­ver­sal, fric­tion-drive mo­tor sys­tem that trans­forms any con­ven­tional bi­cy­cle into a smart e-bike. Its name fuses the Greek Oxis (agility) and Axis (pivot)—perfectly re­flect­ing how its high-per­for­mance drive core de­liv­ers ul­ti­mate ac­cel­er­a­tion and in­stant, nim­ble re­spon­sive­ness right to your ride.

Focus on es­sen­tials

Adaptive boost tech­nol­ogy

Precisely de­tects in­clines, pro­vid­ing seam­less as­sis­tance for ef­fort­less climbs.

Peak power 500W

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Wireless ca­dence sen­sor

Ditch the wires. Simple to in­stall, smart to ride.

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Design

Simple de­sign. Impressive ben­e­fits.

The ASUS Oxiis E250G1 de­signed with the user in mind: min­i­mum struc­ture, max­i­mum util­ity. Every de­tail is metic­u­lously crafted to el­e­vate your daily ex­pe­ri­ence, de­liv­er­ing seam­less in­ter­ac­tion and a func­tional mas­ter­piece of pre­mium qual­ity and beauty.

Premium alu­minum con­struc­tion

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Anti-slip tech­nol­ogy

Dynamically pres­sure au­to­mat­i­cally grips the tire for ef­fi­cient, slip-free power trans­fer.

Easy in­stal­la­tion

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Reduced risk of over­heat­ing.

Battery

Removable mod­u­lar bat­tery

Easy to charge and swap, of­fer­ing ul­ti­mate free­dom for your jour­ney.

100W USB-C® PD

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158 Wh

bat­tery ca­pac­ity

Flight-safe for carry on lug­gage (Requires air­line ap­proval pri­or­ity to fly­ing)

Compatibility

Universal de­sign, per­fect fit

ASUS Oxiis E250G1 works with a wide range of bike frame types, in­clud­ing city, road, gravel, and fold­ing bikes, as well as hy­brid or hard­tail moun­tain bikes.

Accommodates

Find your fit

To guar­an­tee op­ti­mal per­for­mance, seam­less in­stal­la­tion, and max­i­mum safety, please dou­ble-check that your bike’s spe­cific tire width, wheel size, and seat post mea­sure­ments fully meet our cri­te­ria be­fore your first setup.

Tire width

Supports tire widths up to 60 mm.

Tire size

Compatible with tire sizes from 16 to 29 inches, plus 700C.

Seat post

Fits 25.4 – 34.9 mm posts (spacers in­cluded).

Installation Guide

Easy in­stal­la­tion

No com­plex tools are needed to in­stall ASUS Oxiis E250G1.

App

ASUS Oxiis app

Three modes. One seam­less ex­pe­ri­ence.

Switch modes ef­fort­lessly us­ing the in­te­grated but­ton or ded­i­cated app.

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Sport mode: Instant boost for steep climbs and high-speed per­for­mance.

Spec

Product Specifications

Motor Power

250W (Rated) / 500W (Peak)

Range

50 km (31 miles) in eco mode

Battery

158.4 Wh / 36 V

Weight

3.7 kg (including bat­tery)

Dimensions

400 x 84 x 128 mm (L x W x H)

Max Speed

32 km/​h (Limited to 25 km/​h in spe­cific re­gions)

Charging Time

2 hours (100W PD Charger re­quired)

Waterproof Rating

IPX4

App Compatibility

Android & iOS

Get lo­cal ASUS Zenbook 14 avail­abil­ity alerts - and more from ASUS/ROG

allen@bargi:~/​notes$ cat work­ing-with-ai.md

For most of my ca­reer, code gave me cer­tainty. A pro­gram did what its in­struc­tions told it to do. If the same in­put pro­duced a dif­fer­ent re­sult, we called it a bug.

People were never like that. As a leader, I can ex­plain a task and get ex­actly what I asked for. I can also get some­thing bet­ter be­cause a col­league un­der­stood the in­tent be­hind the re­quest. Sometimes the re­sult shows that I was not as clear as I thought.

Working with AI feels closer to the sec­ond ex­pe­ri­ence.

AI runs on soft­ware, but work­ing with it is not fully pre­dictable. The same re­quest can pro­duce a dif­fer­ent an­swer. It can make a use­ful con­nec­tion, miss an ob­vi­ous point, or sur­prise me with an ap­proach I had not con­sid­ered.

This is frus­trat­ing when I treat AI like a com­piler. It be­comes more use­ful when I treat the in­ter­ac­tion as a form of col­lab­o­ra­tion.

That does not make AI a per­son. It has no lived ex­pe­ri­ence, ac­count­abil­ity, or hu­man judg­ment. The com­par­i­son is about how we work. Good lead­ers do more than is­sue in­struc­tions. They share con­text, ex­plain the de­sired out­come, set bound­aries, and re­spond to what comes back.

The same habits im­prove my work with AI. A good prompt helps, but a shared work­ing con­text helps more. Examples, cor­rec­tions, and reusable in­struc­tions re­duce mis­un­der­stand­ings. Over time, the sys­tem be­comes bet­ter aligned with how I think and what I need from it.

The in­vest­ment is not in pre­tend­ing that AI is hu­man. It is in be­com­ing bet­ter at ex­press­ing in­tent.

We spent years learn­ing how to tell com­put­ers ex­actly what to do. Now we also need to ex­plain why the work mat­ters, what a good re­sult looks like, and where judg­ment is needed.

For me, that is the shift. AI is mak­ing soft­ware work less like is­su­ing com­mands to a ma­chine and more like lead­ing through a con­ver­sa­tion. The tech­nol­ogy is new. The lead­er­ship skills are not.

This note led to a thought­ful dis­cus­sion on Hacker News. I rec­om­mend read­ing through all the com­ments; the agree­ment, crit­i­cism, and dif­fer­ent ex­pe­ri­ences add more to the idea than I could fit here.

Contact: Olivia Walther, owalther@acc.org ,

WASHINGTON (Aug 11, 2026) -

The size of a per­son’s mid­sec­tion is a bet­ter in­di­ca­tor of car­dio­vas­cu­lar dis­ease risk when com­pared to their body mass in­dex (BMI) alone, ac­cord­ing to a study pub­lished in JACC, the flag­ship jour­nal of the American College of Cardiology. BMI has tra­di­tion­ally been used to de­ter­mine over­weight or obe­sity, com­mon risk fac­tors for heart dis­ease, but this study shows that not ac­count­ing for waist cir­cum­fer­ence (WC) or waist-to-hip ra­tio (WHR) can lead to mis­clas­si­fi­ca­tion of car­dio­vas­cu­lar dis­ease risk.

“Indeed, it ap­pears that WC and WHR re­clas­sify risk de­fined by tra­di­tional BMI thresh­olds,” said Michael J. Blaha, MD, MPH, se­nior au­thor of the study and di­rec­tor of clin­i­cal re­search at Johns Hopkins Ciccarone Center for the Prevention of Cardiovascular Disease. “We saw in­di­vid­u­als with clin­i­cally de­ter­mined nor­mal weight who had el­e­vated cen­tral adi­pos­ity and high WHR, as­so­ci­at­ing them with higher risk across most out­comes.”

BMI is cal­cu­lated by di­vid­ing weight in kilo­grams by height in me­ters squared and is a com­mon prac­tice for di­ag­nos­ing obe­sity. However, BMI can­not ac­count for dis­tri­b­u­tion of body fat. Studies have shown that vis­ceral fat, which is fat that sur­rounds the in­ter­nal or­gans in the ab­dom­i­nal area, is as­so­ci­ated with chronic dis­eases like heart dis­ease and di­a­betes, while sub­cu­ta­neous fat, which is lo­cated di­rectly un­der the skin, is not as strongly as­so­ci­ated.

Despite ev­i­dence link­ing cen­tral adi­pos­ity, the ac­cu­mu­la­tion of both vis­ceral and sub­cu­ta­neous fat in the ab­dom­i­nal area, to ad­verse car­dio­vas­cu­lar health out­comes, BMI is still the most used met­ric to de­ter­mine over­weight and obe­sity and fu­ture car­dio­vas­cu­lar risk.

This study ex­am­ines whether adding WC and WHR to BMI bet­ter pre­dicts fu­ture car­dio­vas­cu­lar risk. Researchers from the Cross Cohort Collaboration looked at over 260,000 peo­ple over an av­er­age of 20 years who had ei­ther WC or WHR data and at least one of nine out­comes: time to first fa­tal and non-fa­tal my­ocar­dial in­farc­tion, fa­tal and non-fa­tal stroke, heart fail­ure, atrial fib­ril­la­tion, to­tal coro­nary heart dis­ease (CHD), to­tal car­dio­vas­cu­lar dis­ease (CVD), CHD mor­tal­ity, CVD mor­tal­ity and/​or all-cause mor­tal­ity.

In in­di­vid­u­als with nor­mal weight as de­ter­mined by BMI, 5% had high WC and 18% had high WHR; among those with over­weight, 39% had high WC and 40% had high WHR. Among those with obe­sity, 9% had low WC and 45% had low WHR.

Those in­di­vid­u­als with nor­mal weight or over­weight and clin­i­cally de­fined high WC or WHR were as­so­ci­ated with a 15% - 50% greater risk for most of the nine stud­ied out­comes. Those with obe­sity and low WC were not found to be as­so­ci­ated with a sig­nif­i­cantly dif­fer­ent risk of out­comes com­pared with those who had nor­mal weight and low WC, ex­cept for all-cause mor­tal­ity, for which risk was sig­nif­i­cantly lower.

“Our find­ings em­pha­size the crit­i­cal role of iden­ti­fy­ing el­e­vated cen­tral adi­pos­ity, even in in­di­vid­u­als with a nor­mal BMI or with a BMI in the over­weight range. Relying solely on BMI may re­sult in mis­clas­si­fi­ca­tion of car­dio­vas­cu­lar risk across a wide range of car­dio­vas­cu­lar out­comes,” Zeina A. Dardari, PhD, MS, lead au­thor of the study, said. “We en­cour­age clin­i­cians to con­sider cen­tral adi­pos­ity dis­tri­b­u­tion across the en­tire BMI spec­trum when eval­u­at­ing car­dio­vas­cu­lar risk in pri­mary pre­ven­tion set­tings.”

Limitations of the study in­clude that it did not have mea­sures of phys­i­cal ac­tiv­ity, diet or ge­netic obe­sity risk, which have all been shown to play a role in the de­vel­op­ment of CVD. It also in­cluded only one as­sess­ment of WC and WHR, which could limit un­der­stand­ing of how changes in cen­tral fat ac­cu­mu­la­tion over time in­flu­ences CVD risk.

“It is time to aban­don a sole fo­cus on body mass in­dex,” said Harlan M. Krumholz, MD, SM, MACC, FAHA, Editor-in-Chief of JACC and the Harold H. Hines, Jr Professor at the Yale School of Medicine. “This enor­mously im­por­tant study, based on data from hun­dreds of thou­sands of par­tic­i­pants in large-scale co­hort stud­ies, au­thor­i­ta­tively shows that waist cir­cum­fer­ence and waist-to-hip ra­tio pro­vide crit­i­cal in­for­ma­tion about car­dio­vas­cu­lar risk, even among peo­ple with a BMI con­sid­ered nor­mal. Where fat is dis­trib­uted mat­ters, and these sim­ple mea­sures should be part of rou­tine car­dio­vas­cu­lar risk as­sess­ment.”

For an em­bar­goed copy of the study “Risk Reclassification Beyond BMI by Waist Circumference and Waist-to-Hip Ratio Across Nine Cardiovascular Outcomes: Results from the Cross-Cohort Collaboration,” con­tact JACC Media Relations Manager Olivia Walther at owalther@acc.org.

The American College of Cardiology (ACC) is a global leader ded­i­cated to trans­form­ing car­dio­vas­cu­lar care and im­prov­ing heart health for all. For more than 75 years, the ACC has em­pow­ered a com­mu­nity of over 60,000 car­dio­vas­cu­lar pro­fes­sion­als across more than 140 coun­tries with cut­ting-edge ed­u­ca­tion and ad­vo­cacy, rig­or­ous pro­fes­sional cre­den­tials, and trusted clin­i­cal guid­ance. From its world-class JACC Journals and NCDR reg­istries to its Accreditation Services, global net­work of Chapters and Sections, and CardioSmart pa­tient ini­tia­tives, the College is com­mit­ted to cre­at­ing a world where sci­ence, knowl­edge and in­no­va­tion op­ti­mize pa­tient care and out­comes. Learn more at www.ACC.org or con­nect on so­cial me­dia at @ACCinTouch.

The ACC’s JACC Journals rank among the top car­dio­vas­cu­lar jour­nals in the world for sci­en­tific im­pact. The flag­ship jour­nal, the Journal of the American College of Cardiology (JACC) — and spe­cialty jour­nals con­sist­ing of JACC: Advances, JACC: Asia, JACC: Basic to Translational Science, JACC: CardioOncology, JACC: Cardiovascular Imaging, JACC: Cardiovascular Interventions, JACC: Case Reports, JACC: Clinical Electrophysiology and JACC: Heart Failure — pride them­selves on pub­lish­ing the top peer-re­viewed re­search on all as­pects of car­dio­vas­cu­lar dis­ease. Learn more at JACC.org.

###

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how it works

A drum­head clamped at its rim can only vi­brate in cer­tain shapes, at cer­tain fre­quen­cies. Those shapes and fre­quen­cies are the so­lu­tions of

−∇²u = λu  inside the shape,  u = 0 on the edge

Each so­lu­tion u is a mode, a stand­ing wave, and each λ gives a fre­quency pro­por­tional to √λ. This is an eigen­value prob­lem, and for al­most every shape it has no for­mula. So Eigendrum solves it nu­mer­i­cally: it cov­ers your shape with a mesh of tri­an­gles, builds the fi­nite el­e­ment stiff­ness and mass ma­tri­ces, and finds the small­est eigen­val­ues of Kφ = λMφ.

why you can trust the num­bers

A few shapes have spec­tra that can be writ­ten down ex­actly, and the solver is tested against them on every change. A cir­cle’s fre­quen­cies are the ze­ros of Bessel func­tions; a rec­tan­gle’s are π²(m²/​a² + n²/​b²). The solver re­pro­duces both to bet­ter than a tenth of a per­cent, and be­cause a con­form­ing fi­nite el­e­ment method min­imises en­ergy over a re­stricted space, its an­swers are guar­an­teed slight over­es­ti­mates, never un­der. The mea­sured er­ror is in “the num­bers”.

where you strike it mat­ters

Striking a spot dri­ves each mode in pro­por­tion to how much that mode moves there. Hit a line where a mode stands still and you can­not ex­cite it at all. That was not pro­grammed in; it falls out of pro­ject­ing the mal­let onto the modes.

So a strike is never one mode: it is every mode at once, in a mix­ture set by where your mal­let landed. The rules along the mode list are that mix­ture, and the modes marked with a square were the ones your mal­let could not reach. Pressing a row in­stead plays that sin­gle mode alone - some­thing no mal­let can do, and the only way to hear what one fre­quency of a shape ac­tu­ally sounds like.

drums from equa­tions

Besides trac­ing an out­line you can write one. r(t) gives the ra­dius as t sweeps one full turn, so 1 + 0.3cos(5t) is a five-lobed flower; a para­met­ric x(t), y(t) pair reaches the closed curves po­lar can­not, like a nephroid or an egg. This is not a short­cut for draw­ing. It reaches shapes no hand traces ac­cu­rately - eleven even lobes, a super­el­lipse part­way be­tween a cir­cle and a square - and it makes a shape some­thing you vary: change one num­ber and hear what moved.

A writ­ten shape trav­els as its own text. The link for a for­mula holds the for­mula, so it is some­thing you can read and re­type rather than a few hun­dred char­ac­ters of en­coded out­line, and edit­ing it in the ad­dress bar works. Anything too thin to mesh hon­estly is re­fused rather than an­swered, be­cause a sliver would still re­turn num­bers and they would be wrong.

can one hear the shape of a drum?

Mark Kac asked ex­actly that in 1966. In 1992 Carolyn Gordon, David Webb and Scott Wolpert an­swered no, by build­ing two dif­fer­ent shapes with iden­ti­cal spec­tra. Both are in the form list as Kac drum I and II. Each is made from the same seven tri­an­gles, re­arranged. They en­close the same area and the same perime­ter, and every fre­quency matches. Switch be­tween them and lis­ten: the out­lines are plainly dif­fer­ent and the sound is not.

what’s physics, and what’s just a slider

The fre­quency ra­tios, the mode shapes, and the pitch of the fun­da­men­tal are all baked into the out­line - you can’t touch them. What you can move is the wave speed (tension and den­sity): the pitch slider names the note a cir­cle of this area would sound, and your ac­tual shape lands above or be­low that on its own. Since every shape gets scaled to the same area first, that off­set is gen­uinely about the shape - roughly six semi­tones of spread across the built-in pre­sets, with the cir­cle al­ways low­est thanks to Faber-Krahn’s in­equal­ity. Fade time is a real slider too, since that’s ma­te­r­ial and air, not some­thing the maths pins down.

The mal­let works the same way. Its width is a slider; its con­tact time is fixed at a few mil­lisec­onds, be­cause no real beater is in­stant, and one that was would slam every mode equally hard. Both change how much of a mode a strike can reach - nei­ther one can shift where a mode ac­tu­ally sits. Damping is Rayleigh damp­ing, so loss rises with the square of fre­quency: the high over­tones die away first, which is why a drum dark­ens as it rings.

where it lives, and how to reach me

Eigendrum is hosted at eigen­drum.com. That is the ad­dress to link to and to cite; the older base­lashraf81.github.io/​eigen­drum is a mir­ror that now redi­rects there.

For ad­ver­tis­ing or part­ner­ship en­quiries, write to u2679054@uel.ac.uk. For any­thing wrong with the maths or the in­ter­face, an is­sue on the repos­i­tory is bet­ter, be­cause then the fix is pub­lic.

colophon

No build step and no ap­pli­ca­tion back­end: the mesh, the solve and the au­dio all run on your own ma­chine. The de­ployed site uses Vercel Analytics and Google Analytics; it car­ries no ad­ver­tis­ing net­work and no con­sent ban­ner. Support to­ward the do­main and host­ing is vol­un­tary, via the link above. The shape you draw lives in the ad­dress bar af­ter the #, which browsers never send to a server, and an­a­lyt­ics is con­fig­ured not to record it. Details in the pri­vacy no­tice. Set in Jost* by in­de­struc­tible type*. After Kac, Can One Hear the Shape of a Drum? (1966); Gordon, Webb and Wolpert (1992); and Driscoll, Eigenmodes of Isospectral Drums (1997), whose co­or­di­nates the two Kac drums use.

Source, in­clud­ing the solver and the tests that check it against the closed-form spec­tra: github.com/​Base­lAshraf81/​eigen­drum

Free to use, with no ac­count and noth­ing to in­stall. If you would like to put some­thing to­wards it, or would rather it were not ad-sup­ported: ko-fi.com/​base­lashraf

Super El Niño con­tin­ues to strengthen rapidly across the trop­i­cal Pacific, with new ocean and at­mos­pheric data show­ing an­other ac­cel­er­a­tion in its de­vel­op­ment. The lat­est long-range fore­casts have raised the pro­jected peak again, push­ing the 2026 event deeper into his­toric ter­ri­tory later this year.

Behind this rapid growth is an un­usu­ally strong com­bi­na­tion of west­erly winds and sub­sur­face ocean heat. Recent data shows record-level west­erly wind anom­alies across the equa­to­r­ial Pacific, while a pow­er­ful Kelvin Wave con­tin­ues to spread east­ward be­neath the sur­face, pro­vid­ing ad­di­tional warm wa­ter that pow­ers the Super El Niño.

In this spe­cial­ized fore­cast ar­ti­cle, I break down the lat­est ocean and at­mos­pheric data be­hind the rapidly grow­ing Super El Niño, in­clud­ing the record west­erly wind anom­alies and chang­ing long-range fore­casts. I also look at how the at­mos­phere is al­ready re­spond­ing, and what the lat­est Fall and Winter 2026/2027 pre­dic­tions show for the United States, Canada, and Europe.

Super El Niño Dynamics: How Extreme Events Reshape Global Circulation

In the past few months, we have been track­ing the growth of a Super El Niño event that is al­ready a ma­jor global weather dri­ver for 2026/2027. This Super El Niño is grow­ing rapidly, and with the fore­casts con­stantly ad­just­ing its peak strength higher, we have to mon­i­tor its de­vel­op­ment on a reg­u­lar ba­sis.

Currently, we have just en­tered a very strong El Niño phase, so be­low you can see the usual changes it brings to the at­mos­pheric cir­cu­la­tion. The up­ward and down­ward at­mos­pheric mo­tion and cir­cu­la­tion in the trop­i­cal re­gions is called a Walker Cell, and is es­pe­cially sen­si­tive to strong ENSO events. Image by ESA.

In sim­ple terms, the El Niño causes a pres­sure drop in the cen­tral and east­ern trop­i­cal Pacific and a high-pres­sure zone over the west­ern Pacific. This cir­cu­la­tion change af­fects the global at­mos­phere and sig­nif­i­cantly in­flu­ences pres­sure pat­terns, winds, rain­fall, and the over­all sea­sonal weather sys­tem.

While a mod­er­ate event pro­duces a mild wave pat­tern, a Super El Niño trig­gers a far more ag­gres­sive shift in the jet stream. The tran­si­tion from a Moderate to an Extreme El Niño leads to a deeper Pacific trough and a stronger Canadian ridge, form­ing an at­mos­pheric high­way.

This fa­vors milder con­di­tions in the north while dri­ving an ac­tive storm track across the south­ern United States. The im­age above is from a study (linked be­low) that com­pared pres­sure anom­alies at 5km (3.1miles) dur­ing Winter for mod­er­ate and strong El Niño events.

Below is my com­bined analy­sis im­age of the last 4 Super El Niño events, show­ing ocean tem­per­a­ture anom­alies. You can see a strong warm anom­aly across the cen­tral and east­ern trop­i­cal Pacific. This close prox­im­ity to North America means di­rect and strong im­pacts on the sea­sonal weather in the United States and Canada in the Northern Hemisphere.

El Niño events hap­pen every few years, but Super events are rare, and usu­ally oc­cur once per decade or less.

But how does an El Niño even reach a “Super” sta­tus? To keep it sim­ple, a west­erly wind burst can pile up warm wa­ter in the west­ern Pacific. This cre­ates a warm oceanic layer at depth known as a Kelvin Wave, rais­ing ocean heat con­tent in the west­ern Pacific, spread­ing east, and ris­ing to the sur­face.

The lat­est analy­sis data now shows the ex­act same process un­fold­ing rapidly, but with an en­ergy sig­na­ture that ex­ceeds most (if not all) pre­vi­ous su­per events.

Kelvin Wave: Subsurface Heat Drives Rapid El Niño Growth

The lat­est NOAA CRW ocean analy­sis be­low shows the ENSO area cov­ered in sub­stan­tial warm anom­alies. You can see the peak warmth in the east­ern parts reach­ing more than 5 de­grees above nor­mal over a large area, peak­ing at over 6 de­grees anom­aly. This is an ex­cep­tion­ally strong anom­aly for this stage of de­vel­op­ment, in­di­cat­ing un­usu­ally rapid El Niño growth.

The analy­sis be­low also shows the 30-day ocean tem­per­a­ture anom­aly change. It re­veals a broad warm­ing trend as the El Niño is emerg­ing, in­creas­ing by more than 1.5 de­grees over a very large area. Such a trend marks ac­cel­er­ated growth, with at least 3 months of fur­ther de­vel­op­ment ahead.

The un­usu­ally rapid growth can be seen in the lat­est analy­sis graph be­low, for the main ENSO re­gion. This is based on the rel­a­tive ENSO in­dex, which nor­mal­izes the data across past decades, mak­ing all El Niño events di­rectly com­pa­ra­ble. I pro­duced this plot us­ing BOM weekly data.

As you can see, there has been ac­cel­er­ated El Niño growth and strength­en­ing since Spring. The 2026 event has al­ready sur­passed the last Super El Niño event (2015 – 2016) in speed and strength. It is al­ready not far from the peak strength of the last Super event, af­ter hav­ing one of the fastest de­vel­op­ment cy­cles in decades.

Looking at the medium-range fore­cast be­low, the day-10 fore­cast shows con­tin­ued ex­pan­sion of the +5 de­gree (+9°F) anom­aly area, in­di­cat­ing no real break in the cur­rent growth. The 2026 event is de­vel­op­ing at an ex­cep­tional rate, with the lat­est data putting its cur­rent tra­jec­tory among the strongest in the his­tor­i­cal record.

Exceptional sur­face anom­alies are only half the story. The true power that is dri­ving this rapid warm­ing and Super El Niño de­vel­op­ment sits deep be­low the ocean sur­face.

Below you can see the sub­sur­face tem­per­a­ture anom­aly across the trop­i­cal Pacific in the top 250m (800ft) of the ocean. This re­veals the core (engine) of the 2026/2027 Super El Niño event: a pow­er­ful down­welling Kelvin Wave, with peak anom­alies over 9 de­grees (16°F) above nor­mal. It is push­ing east­ward and ris­ing to­ward the sur­face.

In sim­ple terms, the ocean sur­face anom­alies are just the sur­face foot­print of this mas­sive sub­sur­face warm core. As this warm wa­ter steadily sur­faces, it pro­vides a con­tin­u­ous sup­ply of ther­mal en­ergy to keep the El Niño strong and healthy well into the Winter sea­son.

I pro­duced a video be­low that shows the de­vel­op­ment of sub­sur­face tem­per­a­ture anom­alies in the past week un­der the ENSO re­gion. It shows clear move­ment and growth of this large Kelvin Wave and its even­tual rise as a Super El Niño in the east­ern parts.

These sub­sur­face Kelvin waves are dri­ven by the west­erly wind bursts across the trop­i­cal Pacific, push­ing the warmer sub­sur­face ocean wa­ters to the east, where they rise to the sur­face. And there are more west­erly winds com­ing to the Pacific, boost­ing the Super El Niño even higher to­wards Winter.

Westerly Wind Bursts: Record Pacific Anomalies Drive El Niño Growth

Below is the zonal wind rank­ing for June and July 2026, which I cal­cu­lated us­ing ERA5 data. It nicely shows how the west­erly wind anom­aly in the past two months com­pares to the past 86 years. You can see that a broad re­gion of the west­ern and cen­tral equa­to­r­ial Pacific recorded its strongest low-level west­erly wind anom­alies on record, with sur­round­ing ar­eas rank­ing in the top five.

This is sci­en­tif­i­cally sig­nif­i­cant be­cause achiev­ing an ab­solute record across an 86-year dataset for a two-month av­er­age re­quires sus­tained, broad trade wind col­lapse. It shows that the at­mos­phere re­ally is sup­port­ive for one of the strongest El Niño events to de­velop.

The re­sult is vis­i­ble in the ocean heat con­tent be­low, which looks at the ocean down to 300m (1000ft) depth. It per­fectly shows an ex­pand­ing warm sub­sur­face anom­aly across the trop­i­cal Pacific and ENSO re­gions from Spring to now, dri­ven by the west­erly wind bursts that push the warm Kelvin Wave east­ward.

But the story does­n’t end here, since con­tin­ued west­erly winds will help sus­tain and fur­ther strengthen the Super El Niño anom­alies at and be­low the ocean sur­face.

Below is the lat­est analy­sis and fore­cast of the winds across the trop­ics. You can al­ready see the strong west­erly wind burst anom­alies (warm hues) in the analy­sis part, dri­ving the El Niño warm­ing. But the fore­cast now also shows even stronger west­erly anom­alies across the Pacific in August, which will help fur­ther grow and strengthen the 2026/2027 Super El Niño event.

The ex­tended-range ECMWF fore­cast main­tains con­tin­u­ous west­erly wind anom­alies across the west­ern and cen­tral Pacific through late September. In en­sem­ble fore­cast­ing, a sig­nal this per­sis­tent be­yond 10 days in­di­cates an ex­cep­tion­ally ro­bust ocean-at­mos­phere cou­pling that will con­tin­u­ously sup­press trade winds and force trop­i­cal Pacific warm­ing.

The in­di­vid­ual west­erly wind burst events oc­cur on a daily or weekly scale, mak­ing it hard for sea­sonal fore­casts to sim­u­late them prop­erly. And since they are the key to El Niño growth and strength, this means the true ex­tent of the 2026 Super El Niño event was hid­den un­til re­cent weeks.

Latest El Niño Forecast: New Runs Push Further Into Record Territory

Because sea­sonal mod­els can­not prop­erly sim­u­late in­di­vid­ual fu­ture west­erly wind events, they of­ten un­der­pre­dict the ini­tial rapid growth of an El Niño. This is un­til the winds ac­tu­ally oc­cur, launch an oceanic Kelvin Wave, and al­low the mod­els to phys­i­cally see the re­sult­ing El Niño warm­ing.

This has cre­ated a very strong vi­sual fore­cast trend, with each new fore­cast show­ing a stronger peak El Niño anom­aly.

Below is a com­par­i­son I pro­duced us­ing the last seven ECMWF fore­casts, re­leased since early February. You can clearly see that each new run shows a stronger event, with the last three runs push­ing it into record-strong ter­ri­tory, above all the strongest Super El Niño events.

The same trend is also vis­i­ble on the NMME multi-model fore­cast, also trend­ing with a stronger Super El Niño with each con­sec­u­tive run since February at least. The anom­aly val­ues for the main ENSO re­gion now peak close to +4 de­grees, mak­ing this a record-strong event if ver­i­fied.

The term Super El Niño is com­monly used for events in which sea sur­face tem­per­a­ture anom­alies in the main ENSO re­gion reach or ex­ceed +2 de­grees above the long-term av­er­age. All fore­casts now in­di­cate that this event will reach far above that, ex­ceed­ing even the ex­treme +3 thresh­old (unofficial) in a lot of sce­nar­ios.

Below is the lat­est ECMWF fore­cast av­er­age for the November-December pe­riod, re­veal­ing a sig­nif­i­cant Super El Niño event. Peak anom­alies reach +7 in the east­ern parts, out­side of the main re­gion, but over­all, we are ob­serv­ing a his­toric event un­fold­ing.

El Niño events al­ways peak later in the year, with the lat­est fore­casts trend­ing to­ward a max anom­aly around November-December as seen above. The multi-model fore­cast be­low for November shows a very strong event, with the Super El Niño and re­lated anom­alies cov­er­ing over 10% of the global ocean sur­face.

But the strong anom­alies do not end in the ocean. The at­mos­phere is al­ready re­spond­ing to this El Niño event, with stronger im­pacts com­ing in Fall and Winter 2026/2027, as in­di­cated by the lat­est long-range data.

Atmospheric Forcing: Super El Niño Establishes a Global Standing Wave

To de­tect the vis­i­ble at­mos­pheric im­pact of the Super El Niño, we need to find its cir­cu­la­tion in the Walker cell, the trop­i­cal ris­ing and sink­ing of air.

Below is the lat­est 30-day analy­sis of the Velocity Potential pa­ra­me­ter from GDAS data, which shows broad ar­eas of ris­ing and sink­ing air in the at­mos­phere. You can see a large ris­ing air anom­aly (teal) di­rectly over the cen­tral and east­ern Pacific, forced by the lower pres­sure and rain­fall of the El Niño, and strong sink­ing (brown) to­wards the west.

These ar­eas of ris­ing and sink­ing air (Walker cell) are usu­ally dy­namic, mov­ing around the globe with dif­fer­ent at­mos­pheric waves and dri­vers. But when a Super El Niño ap­pears, the strong ocean heat and en­ergy over­pow­ers these mov­ing weather dri­vers.

It can force the at­mos­phere to lock down, cre­at­ing what sci­en­tists call an at­mos­pheric stand­ing wave. You can see this in the ECMWF ex­tended fore­cast be­low, which shows the main ar­eas of the Walker cell al­most fixed/​sta­tion­ary over time, for the du­ra­tion of the fore­cast into late September.

You can clearly see the two main ar­eas of ris­ing and sink­ing mo­tion: Low pres­sure over the trop­i­cal Pacific and the sta­ble sink­ing air over the Indian Ocean, which re­veal the stand­ing wave for­ma­tion.

But just look­ing at two dif­fer­ent col­ors does­n’t re­veal the full pic­ture. For that rea­son, NOAA has cre­ated the Multivariate ENSO Index (MEI). This in­dex com­bines oceanic and at­mos­pheric data into a sin­gle mea­sure of the ENSO state. This re­veals how strongly the El Niño sig­nal is es­tab­lished across both the ocean and at­mos­phere.

You can see the MEI table be­low, which shows the bi-monthly value for 2026, com­pared to the last 3 Super El Niño events. The lat­est value shows that 2026 reached a record-high June-July (JJ) value of +2.4, the high­est for this pe­riod in the NOAA record since it be­gan in 1979.

Values above +2 are found only dur­ing the strongest El Niño events, but not this early. This con­firms just how strongly the oceanic and at­mos­pheric sig­nals have al­ready de­vel­oped in 2026. The in­dex un­der­went an ex­cep­tional +3.4 point jump in just four bi-monthly pe­ri­ods, climb­ing from a cool -1 in February-March to +2.4 in June-July.

This means the weather in your back­yard is di­rectly or in­di­rectly con­nected to what’s hap­pen­ing in the trop­i­cal Pacific, no mat­ter how far away you live.

With a his­toric 2026/2027 Super El Niño event un­fold­ing, we are en­ter­ing al­most un­charted ter­ri­tory in terms of at­mos­pheric im­pacts. The biggest im­pact in the Northern Hemisphere ar­rives dur­ing Fall and more in Winter, when the pres­sure sys­tems are at their strongest.

Fall 2026 Forecast: El Niño Winter Pattern Appears Early

The lat­est Fall pres­sure pat­tern fore­cast shows a much more evolved pat­tern than nor­mally ex­pected, look­ing more sim­i­lar to an El Niño Winter sig­na­ture.

You can see be­low that it shows a stronger-than-usual at­mos­pheric im­pact in Fall al­ready, due to the strength of this El Niño event. The key is a high-pres­sure anom­aly over Canada, with the El Niño Pacific low, and a wave of low-pres­sure sys­tems over the south­ern United States.

Further east, a low-pres­sure area sits over the North Atlantic, reach­ing into the UK and Ireland. This cre­ates a pro­nounced west­erly flow over the con­ti­nent and brings a warmer southerly air­mass ris­ing to­wards the north.

This is re­ally in­ter­est­ing to see, be­cause it breaks the usual Fall El Niño pat­tern, and in­stead looks more like an El Niño Winter pat­tern, seen be­low. It has the same low-pres­sure area in the North Pacific, a high-pres­sure zone over Canada, and a low-pres­sure storm track across the south­ern United States and into the Atlantic, just as the Fall fore­cast above.

This re­ally shows just how strong the Super El Niño global forc­ing is, al­ready ev­i­dent in the cur­rent Summer sea­son with the for­ma­tion of an at­mos­pheric stand­ing wave.

The tem­per­a­ture fore­cast be­low shows warmer tem­per­a­tures over the north­ern United States and Canada un­der the main high-pres­sure area. Temperatures are mostly around nor­mal in the south-cen­tral and east­ern United States, due to a more per­sis­tent low-pres­sure storm track start­ing in the south­ern half of the United States.

This is the October-December pe­riod in the fore­cast, which cov­ers the core Fall sea­son and the early tran­si­tion into Winter.

The pre­cip­i­ta­tion fore­cast for the same pe­riod also shows a very evolved El Niño sig­na­ture, with in­creased rain­fall over most of the United States, es­pe­cially in the south­east and east, due to the am­pli­fied Pacific jet stream. Drier con­di­tions are fore­cast over the north­west­ern U.S. and south­ern Canada.

Over Europe, we can see the im­pact of the low-pres­sure area in the North Atlantic, bring­ing warmer-than-nor­mal sur­face tem­per­a­tures over much of the con­ti­nent. Warmer anom­alies are fo­cused on the cen­tral, west­ern, and south­east­ern re­gions, dri­ven by the west­erly and south­west­erly flow.

The west­erly and south­west­erly flow from the Atlantic low-pres­sure area also brings in a lot of mois­ture, in­creas­ing pre­cip­i­ta­tion po­ten­tial over much of the con­ti­nent. This is vis­i­ble in the pre­cip­i­ta­tion fore­cast (right), in­di­cat­ing above-nor­mal pre­cip­i­ta­tion over most of Europe.

All these fore­casts show that we can ex­pect a strong pat­tern evo­lu­tion from the his­toric El Niño event. But the full strength forc­ing usu­ally oc­curs in Winter, with the lat­est round of fore­casts in­di­cat­ing a highly am­pli­fied Winter pat­tern.

Winter 2026/2027 Forecast: An Amplified Pattern Builds Across North America

The win­ter sea­son is the most high-im­pact part of the year, and usu­ally of most in­ter­est to most peo­ple. It also packs the most en­ergy in the weather sys­tems, mak­ing it the most im­pact­ful part of the year in a Super El Niño event.

Below is the very lat­est ECMWF pres­sure anom­aly fore­cast, re­leased in the past few days. This is the win­ter fore­cast for the December-February pe­riod, and it shows a strong El Niño forc­ing at play. The key parts are the deep low-pres­sure zone in the North Pacific and the block­ing high-pres­sure area over Canada.

The fore­cast also shows a high-am­pli­tude pres­sure pat­tern across the south­ern and east­ern United States. This is aligned with a strong Pacific jet stream, the key com­po­nent of a Super El Niño Winter. It’s worth adding that this pe­riod is 4 – 6 months in the fu­ture, but al­ready shows such strong anom­alies.

Such a pres­sure pat­tern trans­lates into a sharp tem­per­a­ture dif­fer­ence be­tween the United States and Canada. You can see above-nor­mal tem­per­a­tures un­der the high-pres­sure zone in Canada and the north­ern United States, also in­clud­ing the U.S. West Coast, and the Northeast.

The deep south­ern low-pres­sure sys­tems al­low cooler-than-nor­mal tem­per­a­tures across Texas, the Gulf Coast, and the Southeast. This is pri­mar­ily dri­ven by an ac­tive south­ern jet stream zone, which brings per­sis­tent cloud cover and rain.

As we move into mid-late win­ter, there are some in­di­ca­tions of the Super El Niño pat­tern go­ing into over­drive. The February fore­cast be­low shows a much deeper pres­sure wave over the United States for mid-late Winter, with the whole at­mos­pheric wave shifted east.

This north­ern block­ing is forc­ing a deep, highly ac­tive low-pres­sure zone from the Pacific straight through the cen­tral, south­ern, and east­ern United States. In such a con­fig­u­ra­tion, this sig­nals a po­ten­tially colder-than-nor­mal mid and late win­ter sea­son over parts of the United States.

The cor­re­spond­ing tem­per­a­ture fore­cast for February shows a sur­pris­ing cold anom­aly over a large part of the United States, reach­ing up into south­west­ern Canada. This would make for a very in­ter­est­ing Winter sea­son, as it could lead to po­ten­tially good snow­storm sce­nar­ios across the cen­tral, east­ern, and north­east­ern United States.

I do have to add that this is just a re­cent de­vel­op­ing trend, so it’s not a fixed fore­cast by any means. But it is a cal­cu­la­tion based on real oceanic and at­mos­pheric con­di­tions. I will fur­ther mon­i­tor this de­vel­op­ment, es­pe­cially since it is gain­ing sup­port from other long-range pre­dic­tions.

A Super El Niño pat­tern also strongly im­pacts the snow­fall po­ten­tial. When an ac­tive south­ern jet stream over­laps pe­ri­odic cold air drops from Canada, it can shift the pri­mary win­ter storm cor­ri­dor far­ther south than usual.

Below is the lat­est ECMWF snow­fall fore­cast, and it shows ex­actly this de­vel­op­ment. We can see re­duced snow to­tals across the Northern United States and south­ern Canada, but above-nor­mal snow­fall po­ten­tial across the Central and Eastern United States, and south­east­ern Canada.

Good snow­fall po­ten­tial is in­di­cated across the Southwestern and Southeastern U.S., the Central Plains, parts of the Midwest, East, and into the Mid-Atlantic.

Areas across Canada, the Pacific Northwest, and the north­ern Great Lakes show be­low-av­er­age snow­fall anom­alies, dri­ven by warmer tem­per­a­tures and a north­ern ridge that pushes the po­lar air away.

The most im­por­tant thing for snow­fall is where the mois­ture flow in­ter­sects cold air. This fore­cast setup el­e­vates the po­ten­tial for ma­jor win­ter storms, ice events, and heavy snow­fall from the Southern Plains and Mid-Atlantic into the in­te­rior Northeast. But it does rely on hav­ing a cold enough air mass to work with.

Europe Winter 2026/2027: Westerly Pattern Favors a Milder Season

Below is the very lat­est win­ter pat­tern fore­cast by ECMWF. It shows the highly am­pli­fied pres­sure over North America, also im­pact­ing the weather down­stream. The main fea­ture for Europe comes from the low-pres­sure area ex­ten­sion into its north­west­ern and north­ern parts.

This cre­ates a strong pres­sure dif­fer­ence from north to south, boost­ing the west­erly flow into Europe from the Atlantic, while also al­low­ing some northerly flow over the north and north­west.

This is re­flected in the lat­est December-February tem­per­a­ture fore­cast be­low, where you can see mostly above-nor­mal tem­per­a­tures dur­ing the win­ter sea­son. This is the re­sult of a dom­i­nant mild west­erly flow. It still al­lows some northerly flow into the UK and Ireland, as low-pres­sure ar­eas move from the Atlantic into north­ern Europe.

The main snow­fall po­ten­tial in such a pat­tern comes with in­di­vid­ual low-pres­sure sys­tems mov­ing fur­ther in­land and to the south, bring­ing along a more northerly flow. The over­all sea­sonal pat­tern is not that fa­vor­able for broad snow­fall over Europe. The ex­cep­tions are the north and north­east, and the cen­tral higher el­e­va­tions.

I will write full in-depth fore­cast ar­ti­cles for Fall and Winter 2026/2027 over the United States, Canada, and Europe, once all the nec­es­sary data is avail­able.

Scientific Research Used in this Article

Super El Niño Development: Formation Mechanism for 2015/16 Super El Niño — Chen et al. (2017).

Extreme El Niño Teleconnections: A Distinct and Reproducible Teleconnection Pattern over North America dur­ing Extreme El Niño Events — Beniche et al. (2024).

Westerly Wind Bursts: The Essential Role of Westerly Wind Bursts in ENSO Dynamics and Extreme Events Quantified in Model “Wind Stress Shaving” Experiments — Yu and Fedorov (2022).

Forecast and analy­sis im­ages in this ar­ti­cle are from ECMWF, CyclonicWX, weath­er­mod­els.com, and WeatherBell (using a com­mer­cial li­cense).

El Niño dri­ving the August weather shift: Super El Niño Drives an August Weather Shift as Fall and Winter Signals Strengthen

New ocean anom­alies emerg­ing: A New Ocean Anomaly Joins Super El Niño, Reshaping the Winter 2026/2027 Forecast

We will keep you up­dated on the global weather pat­tern de­vel­op­ment, so book­mark our page. Also, if you have seen this ar­ti­cle in the Google App (Discover) feed, click the like (♥) or the star but­ton there to see more of our fore­casts and our lat­est ar­ti­cles on weather and na­ture in gen­eral.

Raph Levien, August 8, 2026

I have for many decades been fas­ci­nated by the prospect of a curve fam­ily bet­ter suited for in­ter­ac­tive de­sign than cu­bic Béziers. In that search, I have come to a new-found re­spect for those Béziers. In par­tic­u­lar, though other curves like Euler spi­rals are bet­ter at rep­re­sent­ing smooth curves, they fall short at rep­re­sent­ing re­gions large cur­va­ture vari­a­tion, where cu­bic Béziers ex­cel. The great strength of Béziers is their ver­sa­til­ity.

So, to find a strictly bet­ter curve fam­ily, one re­quire­ment is clear: the fam­ily should con­tain both smooth cur­va­ture vari­a­tion and higher-ten­sion re­gions where cur­va­ture peaks. Polynomial spi­rals, or Spiro curves, the sub­ject of my PhD the­sis, fail to achieve this goal.

After con­sid­er­able search and re­ject­ing a num­ber of can­di­dates, I now bring a pro­posal for a curve fam­ily which I think is a very strong can­di­date for sup­plant­ing cu­bic Béziers in 2D vec­tor graphic de­sign.

Without fur­ther ado, the curve fam­ily is rep­re­sented by the Cesàro equa­tion, spec­i­fy­ing cur­va­ture as a func­tion of arc length:

$$\kappa(s) = \frac{as+b}{(cs^2+ds+1)^{1.5}}$$

This curve be­haves sur­pris­ingly sim­i­larly to a cu­bic Bézier, es­pe­cially at smaller an­gles, but when pushed has very dif­fer­ent be­hav­ior. Overall it has smoother cur­va­ture vari­a­tion and is more likely to have mo­not­o­nic cur­va­ture. It con­tains within it a few valu­able an­a­lytic curves, and is also good at ap­prox­i­mat­ing a wide range of oth­ers. The re­main­der of this blog post is de­voted to show­ing its be­hav­ior in a wide range of con­texts.

Approximation of cu­bic Béziers

The hy­per­bezier closely ap­prox­i­mates cu­bic Béziers at low de­flec­tion an­gles at the end­points. I will show rather than try to pre­sent math­e­matic rea­son­ing. Below is an in­ter­ac­tive tester that maps cu­bic Bézier con­trol points to a cor­re­spond­ing hy­per­bezier. The Bézier is shown in gray for com­par­i­son.

op­tions

cu­bic Bézier cur­va­ture comb

At larger an­gles, and also when the “arm lengths” of the con­trol han­dles get larger, the fit with the cu­bic Bézier is not par­tic­u­larly close, but the con­trol scheme is still a use­ful way of set­ting the pa­ra­me­ters for the hy­per­bezier curve (setting the poly­no­mial co­ef­fi­cients di­rectly is not at all in­tu­itive).

Note that this pa­ra­me­ter map­ping is a first us­able draft, and may not be the fi­nal ver­sion. That said, in an in­ter­ac­tive edit­ing con­text, a per­fect map­ping is not re­quired, as it’s al­ways pos­si­ble to tweak the con­trol points to reach any de­sired curve shape.

The de­tails of the map­ping can be found in the JavaScript source for this page, but the ba­sic prin­ci­ple is that the arm lengths set the de­nom­i­na­tor, then the nu­mer­a­tor is solved to make the end­point tan­gents match.

Exact an­a­lyt­i­cal curves

The most ob­vi­ous an­a­lyt­i­cal curve con­tained in the fam­ily is the Euler spi­ral, which is clearly at­tained when $c == d == 0$. The Euler spi­ral has smooth cur­va­ture vari­a­tion (it is a so­lu­tion to the Minimum Variation Curve prob­lem) and mo­not­o­nic cur­va­ture. An ex­act cir­cu­lar arc is also within the pa­ra­me­ter space, sim­ply when $a == 0$ as well. Cubic Béziers, by com­par­i­son, only ap­prox­i­mate cir­cu­lar arcs.

There a few other log-aes­thetic curves lurk­ing in the pa­ra­me­ter space, mean­ing cur­va­ture is sim­ply the ar­clength raised to a par­tic­u­lar power. Reachable ex­po­nents in­clude -3, -2, -1.5, and -0.5 (alternatively, us­ing the con­ven­tion from the log-aes­thetic pa­pers, α can be 1/3, 1/2, 2/3, and 2). The last of these is the cir­cle in­vo­lute, which is in­ter­est­ing be­cause it is its own par­al­lel curve, among other things. And the first of these is the evo­lute of the Euler spi­ral.

The pa­ra­me­ter map­ping de­scribed above is de­signed so that when the con­trol points lie on the “double parabola” in my Euler spi­ral par­al­lel curve blog post, then the re­sult is an ex­act Euler spi­ral.

Superellipses and squir­cles

A cu­bic Bézier can do super­el­lipses up to a cer­tain point, but does not ap­proach a sharp cor­ner; some­where be­fore then it starts de­vel­op­ing ad­di­tional in­flec­tion points, while a true super­el­lipse or squir­cle is of course con­vex.

The hy­per­bezier can go all the way to a sharp cor­ner, and vi­su­ally looks pretty close to the super­el­lipse. It’s not in­cred­i­bly ac­cu­rate (for mod­er­ate ex­po­nents like 5, the best cu­bic Bézier fit is slightly bet­ter, in fact), but vi­su­ally does the right thing. It’s prob­a­bly best to say that the hy­per­bezier is a sub­tly dif­fer­ent squir­cle than the super­el­lipse, nei­ther bet­ter nor worse.

Hyperbola

Another fun­da­men­tal curve is the hy­per­bola. It is round at the turn, but cur­va­ture tails off and the curve reaches a lin­ear as­ymp­tote on both sides. Cubic Béziers do not fit this curve well, and do not ex­hibit that as­ymp­totic be­hav­ior. But the hy­per­bezier fits it nat­u­rally and with very high ac­cu­racy.

We can also ap­pre­ci­ate the re­la­tion­ship math­e­mat­i­cally. In both, the cur­va­ture tails off as $\kappa \approx s^{-3}$ as­ymp­tot­i­cally. In ad­di­tion, us­ing the small an­gle ap­prox­i­ma­tion $\sin \theta \approx \theta, \cos \theta \approx 1$, in­te­grat­ing the Whewell equa­tion (see be­low) of the even-sym­met­ric hy­per­bezier yields the hy­per­bola ex­actly. This close re­la­tion­ship is one of the in­spi­ra­tions of the name; it is some­thing of a fu­sion be­tween a hy­per­bola and a Bézier. Another in­spi­ra­tion, in­ci­den­tally, is that the log-aes­thetic so­lu­tions, in­clud­ing the Euler spi­ral, are rep­re­sented by hy­per­ge­o­met­ric func­tions.

Elastica

An im­por­tant curve fam­ily to con­sider is the elas­tica, an ide­al­ized thin flex­i­ble strip. The math be­hind the elas­tica has hun­dreds of years of his­tory, but more to the point it is both a smooth curve (it can be char­ac­ter­ized as the ex­act so­lu­tion to the Minimum Energy Curve prob­lem) and can ex­hibit large cur­va­ture vari­a­tion when placed un­der ten­sion. To my eyes, it’s a more pleas­ing and nat­ural curve, as it’s based in phys­i­cal re­al­ity in a way that cu­bic Béziers aren’t. Ideally we’d be able to ap­prox­i­mate it well, and the fact that Spiro curves can’t is a strong case against them. Fortunately, our hy­per­bezier does rea­son­ably well. As in the case of the super­el­lipse, it’s a fairly de­cent vi­sual match, though again not a pre­cise ap­prox­i­ma­tion.

Some math­e­mat­i­cal prop­er­ties

This sec­tion is a grab-bag of some math­e­mat­i­cal prop­er­ties of the hy­per­bezier curve.

An ap­peal­ing prop­erty is that the Cesàro equa­tion is eas­ily in­te­grated, yield­ing a Whewell equa­tion:

$$\theta(s) = \frac{a’s+b’}{\sqrt{(cs^2+ds+1)}}$$

Here, the pa­ra­me­ters in the nu­mer­a­tor aren’t the same as the Cesàro equa­tion (though they are read­ily de­rived), but the qua­dratic poly­no­mial in the de­nom­i­na­tor is the same. Another round of in­te­gra­tion (which I be­lieve will be es­pe­cially use­ful for curve fit­ting) also yields sim­ple an­a­lyt­i­cal equa­tions, re­veal­ing that a close re­la­tion to trig func­tions.

The de­riva­tion of pa­ra­me­ters for the Whewell equa­tion fol­lows fairly straight­for­wardly from the fol­low­ing in­te­gral, which has a pleas­ing sym­me­try:

$$\int \frac{as+b}{(cs^2 + 1)^{1.5}} ds = \frac{bs-a/c}{\sqrt{cs^2 + 1}} + C$$

Going in the op­po­site di­rec­tion, the de­riv­a­tive is also straight­for­ward, yield­ing a qua­dratic over the same poly­no­mial in the de­nom­i­na­tor, this time with an ex­po­nent of 2.5. To com­pute roots, iden­ti­fy­ing the cur­va­ture ex­trema, in­volves merely solv­ing the qua­dratic.

Because the de­nom­i­na­tor is pos­i­tive, there is at most one in­flec­tion point, at $s = -b/a$, when it is in range [0, 1]. That clearly sig­nals that it can’t ac­cu­rately ap­prox­i­mate all cu­bic Béziers, as those can have two in­flec­tion points. In my ex­pe­ri­ence, in­stances of cu­bic Béziers with two in­flec­tion points are rarely used in de­signs (it might be in­ter­est­ing to look at a cor­pus of vec­tor graph­ics to quan­tify this).

I have ex­per­i­mented with other val­ues for the ex­po­nent than 1.5. A value of 1 would be ap­peal­ingly sim­ple, as it would re­sem­ble Padé ap­prox­i­ma­tion, but I found it can­not rep­re­sent high-ten­sion curves in a nu­mer­i­cally sta­ble way. I also con­sid­ered an ex­po­nent of 2, but the in­te­grals don’t come out as cleanly and over­all it did­n’t match cu­bic Béziers as closely. Perhaps this is not sur­pris­ing, as the ex­po­nent of 1.5 re­sem­bles the equa­tion for cur­va­ture of a para­met­ric curve $\mathbf{x}(t)$:

$$\kappa = \frac{\mathbf{x}′ \times \mathbf{x}‘’}{(\mathbf{x}′ \cdot \mathbf{x}’)^{1.5}}$$

In gen­eral, arc length pa­ram­e­trized curves are in many ways more pleas­ant to work with than gen­eral para­met­ric curves; there’s a non­triv­ial cal­cu­la­tion to solve the in­verse arc prob­lem for Béziers.

Future work

In this post, I pre­sent the curve fam­ily to the world. I wel­come and en­cour­age ex­per­i­men­ta­tion. Of course, there is lots to be done to make the math­e­mat­i­cal idea into a prac­ti­cal tool. For one, it should be wired up into a spline. I ex­pect to fol­low the ideas in The hy­per­bezier pen tool by Colin Rofls, which was based on an ear­lier draft of the hy­per­bezier math. This ear­lier draft had some of the de­sired be­hav­ior of the cur­rent pro­posal, but also some flaws, in­clud­ing not be­ing able to ap­prox­i­mate sharp super­el­lipses, and also not be­ing closed un­der sub­di­vi­sion. On the other hand, it had a straight­for­ward so­lu­tion to the prob­lem of map­ping Bézier-like con­trol points to the pa­ra­me­ters.

One idea from the above spline that I think is worth pre­serv­ing is mak­ing splines G2 con­tin­u­ous by con­struc­tion. Such ap­proaches don’t work well with cu­bic Béziers be­cause there are mul­ti­ple lo­cal min­ima, cre­at­ing sad­dle points in the op­ti­miza­tion ter­rain.

Another promis­ing di­rec­tion is work­ing out curve fit­ting. While I haven’t got­ten deep into it yet, I ex­pect curve fit­ting to be fairly straight­for­ward, es­pe­cially com­pared with cu­bic Béziers. I’m es­pe­cially in­ter­ested in fit­ting touch or pen data. As usual for this prob­lem, a cen­tral part of the chal­lenge is find­ing good sub­di­vi­sion points.

I hope to land bet­ter im­ple­men­ta­tions in the spline repos­i­tory, but make no guar­an­tees how fast that will go.

Acknowledgements

I’ve been work­ing on this for years, and have ben­e­fited from dis­cus­sions with many peo­ple, in­clud­ing Colin Rofls, Alex Ionuț, Jacob Rus, and Trevor David Black (apologies if I’ve missed some­one; just let me know). Special credit goes to Muhammad Ragib Hasin, who im­proved many of the nu­mer­i­cal and solv­ing tech­niques for pa­ra­me­ter map­ping. AI as­sis­tance was used to re­fine the cur­rent ver­sion of pa­ra­me­ter map­ping and to pre­pare the vi­su­als for this post.