Behold all-new equations for triangles!

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Thanks to Jane Street for supporting this video. Check out their open roles, programs and events: www.janestreet.com/join-jane-...
Here is the original "Is there an equation for a triangle?" video. • Is there an equation f... (I'm taking suggestions for what Part III should be named. Comment below.)
And thanks to everyone who contributed to the triangleness. Here are all the triangles and non-triangles mentioned in this video:
Mohammed Jafari's close-but-not-triangle: www.desmos.com/calculator/hr1...
Generic Viewer's close-but-not-triangle: www.desmos.com/calculator/kzp...
Harrison's non-generalised triangle: www.desmos.com/calculator/cvp...
Idiotinium's non-generalised triangle: www.desmos.com/calculator/svf...
Jaden's non-generalised triangle: www.desmos.com/calculator/uas...
Tristan's non-generalised triangle: www.desmos.com/calculator/pdu...
And Tristan's Zombie Walker: www.desmos.com/calculator/zkc...
Reddit thread generalising triangle equation: / response_to_matt_parke...
Anson Mansfield's generalised triangle: www.desmos.com/calculator/pbz...
Graham Goble's generalised triangle: www.desmos.com/calculator/eax...
Steve's generalised triangle: www.desmos.com/calculator/qy4...
Inigo's triangle in ShaderToy: www.shadertoy.com/view/XsXSz4
Huge thanks to my Patreon supports. They encourage me to triangle my best. / standupmaths
CORRECTIONS
- None yet, let me know if you spot anything!
Filming by Alex Genn-Bash
Editing by Michelle Martin
Diagrams by Sam Hartburn
Written and performed by Matt Parker
Produced by Nicole Jacobus
Music by Howard Carter
Design by Simon Wright and Adam Robinson
MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
US book: www.penguinrandomhouse.com/bo...
UK book: mathsgear.co.uk/collections/b...

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Опубликовано:

25 апр 2023

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Комментарии : 838

@dougalstanton Год назад

I can't believe you didn't call the missing vertices "plot holes".

@HasekuraIsuna Год назад

... *nice*

@littlered6340 Год назад

MASSIVELY underrated comment. Not even 69 smh Edit: I see it now has over 69 likes but it is still not enough.

@MrShiggitty Год назад

Someone gonna steal this idea for a T-Shirt, its so good lol. Bruh hurry up and secure the movie rights xD

@poppyseedsnuranium Год назад

@@HasekuraIsuna ... Nice

@Son_Of_Atreides Год назад

Would that make the stick man walking a plot twist?

@ThisSteveGuy Год назад

Why not "Tri Harder"?

@Luna-wu4rf Год назад

GENIUS

@sycration Год назад

TriHard 7

@cs8712 Год назад

Do or do not - there is no triangle

@jimmygarza8896 Год назад

Triangle 3: the Threequel: Tri Harder.

@user-dx5wm1uz9d Год назад

@@sycration Cx

@rev6330 Год назад

10:00 To get rid of the "annoying dot" at the origin, just multiply the entire equation by (|x|+|y|)/(|x|+|y|). This factor is 1 everywhere except for the origin where the denominator is 0 and therefore the equation becomes undefined.

@kyay10 Год назад

You could also multiply it by 0^((0^(x^2)) * (0^(y^2))) which basically (ab)uses the convention that 0^0=1 as a logical not gate, and so it's only 1 (true) when not(not(x^2) and not(y^2)) which means not((x^2 = 0) and (y^2 = 0)) and so it's only true when not on the origin

@hughobyrne2588 Год назад

But some triangles have edges that legitimately go through the origin. *EDIT:* Never mind, I was ignoring some context of that particular part of the video, as others have pointed out below.

@kyay10 Год назад

@@hughobyrne2588 oh true. Hmmmm, maybe what you can do is use a complicated logical-or kind of equation by basically checking if any of the edges actually include the origin, and if so then calculate the distance from the centre of the triangle, borrowing from another equation from the video

@rev6330 Год назад

@@kyay10 Not exactly. Your solution is elegant for sure, but doesn't quite do the trick. The goal is to get rid of the 0 at the origin, so multiplying by another 0 doesn't really help. :D However, you could still use your approach by adding (1-0^((0^(x^2)) * (0^(y^2)))) to the equation. It adds 0 everywhere, excapt at the origin it adds 1.

@skyjoe55 Год назад

@@hughobyrne2588 That is true in general but not in the specific graph in question

@NoahStolee Год назад

“‘Every triangle’s a love triangle when you love triangles’ - Pythagoras” -James Acaster

@NeatNit Год назад

Oh wow, that's so James Acaster. Can you say where this joke is from?

@JordanBiserkov Год назад

"Every love triangle has at least one obtuse angle" (obtuse ~ stupid)

@vincentfreddoyle7555 Год назад

@@JordanBiserkovmeaning none of them are right (As in morally)

@JordanBiserkov Год назад

@@vincentfreddoyle7555 It doesn't follow from the axioms. Morality is complicated. You can't control who you fall in love with. You can (and should) control your actions. But you also can (and should) fight for your love ones. And in love (as in war) everything is allowed. Nobody will be judging the winners. Rant over, back to geometry.

@vincentfreddoyle7555 Год назад

@@JordanBiserkov yes,polygons angles surface area

@JoshZanders Год назад

That walking-stickman plot was BONKERS. 🤣 I never would've thought to create an equation that animates a figure with a single dynamic value.

@aceman0000099 Год назад

Wait until you hear about Pixar. And a little variable called T for time 😂

@marsovac Год назад

@@aceman0000099 rest assured pixar is not using equations with one variable to render their movies. A game or rendering engine has the game time parameter but that is also not used to run equations, but rather to define the velocity of transformations applied to 3D meshes to render each frame. Each frame per se has no T as input, it just has static vertex positions which get in memory modified every T time.

@aceman0000099 Год назад

@@marsovac they get modified in memory according to the curved paths of interpolation, which is an equation that takes position A and Position B and gives you a position in between based on T. Whether it's flash animation or 3D motion capture, there is undoubtedly interpolation and splines involved. I've made animations before.

@simonmultiverse6349 Год назад

Personally, I would start with something smooth... some kind of series... I know EXACTLY what I mean, but you'll appreciate it better if you work it out for yourself. *FURTHERMORE* ... there's a finesse which makes it even better.

@simonmultiverse6349 Год назад

*MATT* ... it's not "arbitary" ...the word is arbit *R* ary !!!

@tuna5618 Год назад

What a relief, I was lying awake at night wondering when new equations for triangles would come and this is just what I need. Finally, something useful on the internet for once.

@DhirC35 Год назад

😂

@oosmanbeekawoo Год назад

I really like the art on your channel

@simonmultiverse6349 Год назад

I really like the chart on your anal.

@joshyoung1440 6 месяцев назад

I mean your joke kind of defeats itself a tad at the end when you point out that the internet is known to be full of much, MUCH more information much more useless than this, so it begs the question why the joke would apply to this video in particular

@asheep7797 Год назад

The triangular sequel! We want a triangular trilogy!

@dahawk8574 Год назад

*Triangle with a Vengeance*

@kaivalya931 Год назад

Triangle prequel: line revolution

@kaivalya931 Год назад

Triangle: the quadrilateral dynasty

@jplays8934 Год назад

yes!

@philipdmiller Год назад

Triangle² or Triangle³

@oleg-avdeev Год назад

Of course it’s Inigo Quilez! You can expect the guy to pop up every time something resembling signed distance fields happens. An absolute legend.

@jonathansharret4900 Год назад

His videos are amazing.

@elenplays Год назад

An absolute God

@chromosundrift Год назад

Yeah and he not only has equations for a triangle but also for spectacular animated 3d characters with lighting and shadows and scenery!

@Janokins Год назад

It's good to see Inigo getting recognition. He's the shader programming GOAT

@johnsarthole Год назад

I was going to say that

@NoNameAtAll2 Год назад

his name is Inigo Mantoya you killed his father prepare to die

@lillilacac Год назад

Inigo has worked on many Pixar films as well (Brave, The Good Dinosaur, Lava)

@aceman0000099 Год назад

In 2050 they will teach about Inigo Q in the "history of computing" lessons

@PureAsbestos Год назад

I believe I had commented about Inigo's SDF on the original video. He is indeed a shader god and deserves all the recognition.

@wallyhall Год назад

"It was like a triangle that was in a hurry" - absolutely killed me. Coffee spat out everywhere. Thanks Matt.

@jpdemer5 Год назад

A Parker triangle, innit?

@ieatdirtwasntavailable 2 месяца назад

They dont call him "stand up maths" for no reason.

@himynameisdavenicetomeetyou Год назад

Inigo, if you're reading this, I miss your paint by maths videos they were spectacular. Thanks for the content and for shadertoy :)

@beskamir5977 Год назад

Agreed. Inigo is an absolute genius especially with how he breaks everything down in his explanation videos.

@himynameisdavenicetomeetyou Год назад

@Beskamir it's the fact that everything is essentially explained from first principles that does it for me. That and the results are visually spectacular.

@arirahikkala Год назад

Shadertoy is so much fun. You go through your life barely using any trigonometry, then you draw even the tiniest shader effect, and oh god so much trig, suddenly everything is trig. It's especially mind-bending as a programmer, because you're so used to the idea of having explicit objects that you render, and it's a complete shift in perspective to fake the existence of objects with just a function from time and screen position to pixel colour.

@twinkyoctopus Год назад

trig is like the algebra of higher maths, it apears everywhere

@__8120 Год назад

Yeah but unfortunately the idea of rendering objects implicitly to the screen pixel by pixel doesn't seem to really be a thing outside of shadertoy. Fragment shaders like the one you get there are generally used to apply final effects to an individual face pixel by pixel, rather than the entire screen. It's a great playground for sure, but you do typically have real explicit objects to render

@MichaelPohoreski Год назад

_Rendering Worlds with Two Triangles_ is a good introduction for non-graphic programmers, along with my _HOWTO: Ray Marching_ which has tons of examples to play with.

@NitzanBueno Год назад

@@__8120 Rendering with fragment shaders (and raymarching) really isn't very popular, but it allows rendering complex shapes using only equations that describe them, which can be very short. That's why it's extensively used in the demoscene, specifically for 4KB/1KB intros. I'd say that is indeed its most prevalent place, however - it's mainly a thing for artsy programmers and not really used in the graphics world, but I still love it 🥰

@__8120 Год назад

@@NitzanBueno oh absolutely

@amyshaw893 Год назад

inigo has their own youtube channel hwere they explain a lot of their code. it really helped me when i was in uni and i was doing some work on 4d shape rendering using raymarching, and he works from first principles which is amazing

@killhour Год назад

I've been working with HLSL to make arbitrary real time SDF combinations for the last month or so and Inigo's writings on the topic have been invaluable. I recognized the thumbnail immediately. Was quite surprised to see it here. SDFs are definitely worth an entire video.

@snurffff Год назад

Recognized it too!!

@leif1075 Год назад

Am I the only one who doesn't know what anyone those abbreviations mean? hlsl ? Sdf? How does anyone know those?

@himynameisdavenicetomeetyou Год назад

@Leif for folks who do programmatic graphics or video game work, they're somewhat common-speak. For everybody else, they're nonsense. HLSL = High-Level Shader Language SDF = Signed Distance Field To learn about the former, you can read online in tutorials anywhere. For Unreal Engine or Unity, you'd be writing shaders in HLSL, so there are plenty of tutorials on the topic. For learning about SDFs, I'd go straight to Inigo's channel and watch some of his wonderful videos.

@Hermaniac8 Год назад

Inigo Quilez's SDF videos are so good. Masterpieces in educational content

@christianzander6547 Год назад

Good luck with the cinematic universe. But I am holding out for 'Triangle Hard with a Vengeance' caus I love a New York settings for my math thrillers. Can't wait for the scene where you walk through the Financial District with a sandwich board describing a controversial conjecture about triangles infuriating the local population.

@lillithplays6514 Год назад

The sign function shouldn't be a problem. You can represent it as x/abs(x) for any nonzero x. The absolute value can also be the positive solution to sqrt(x^2)

@anomaliecosmos Год назад

Does the sqrt(x^2) version technically leave some extra values in the imaginary/complex plane? Best case it ends up a second triangle, but that might count as similar to the one with the extra origin dot or the rays out the corners. (Didn't think of x/abs(x) though, clever!)

@friedrichhayek4862 Год назад

@@anomaliecosmos The circle equation is the same.

@the1exnay Год назад

The sign function being undefined at 0 feels very appropriate.

@ollllj Год назад

@@anomaliecosmos yes, this fails for nonzero imaginary values, but these do not occur here. shortcuts are important.

@ollllj Год назад

@@the1exnay negative zero is fun, but not useful, so positive nulls are asserted. MUCH worse is that the first derivative of abs() is discontinuous, any 3d shapes with "abs" will always have a shiny-kin/corner at their 1st derivative discontinuity, and that always looks "to oartificial" , unless you enforce "smooth abs" (seach "sabs" on shadertoy)

@griffinblades8475 Год назад

I love these community contribution videos!

@heighRick Год назад

Thanks Matt, I was not expecting such an excellent and informative video about triangles on a Wednesday, helps a lot!

@soffeebeans Год назад

I just wanted to compliment the way you displayed and referred to the desmos equations. It felt really very natural and was a lot more engaging than a box off to the side!

@DingbatToast Год назад

I don't know how you continue to take one of my most hated subjects at school and make some of the most enjoyable and entertaining content on RU-vid. Thanks for making me want to learn more about maths ❤

@theoriginaltubeofyous Год назад

you probably didn't hate maths in school because of maths, you probably hated it because of school.

@DingbatToast Год назад

@@theoriginaltubeofyous true, i hated it because of my maths teacher

@Pogonateur Год назад

Your video editing is always so amazing, i really need to say it, it's such a pleasure to not only have interesting videos about maths but also so well edited like yours. Thank you so much for your work!

@macronencer Год назад

This was a LOT of fun! My favourite was the affine transformation with the homogeneous co-ordinate matrices. That approach occurred to me immediately as soon as you showed the base triangle for it, because when I was young I spent a lot of time studying computer graphics techniques. Homogeneous co-ordinates are very useful.

@MateusAntonioBittencourt Год назад

Wow... just re-watched yesterday your original video. Now today there's a part two.

@kylerichardson514 Год назад

This episode was fascinating to me. I'm in the FEA world working on a smooth potential to describe contact. A very closely related problem. The code at the very end looked like it was possibly based on the old algorithms in FEA with all sorts of problems and issues. Mainly they are only C^0 continuous. Using Non-Newtonian calculus it is possible to build smooth potentials that can describe any arbitrary shape.

@djcsdy2 7 месяцев назад

I was super excited when Inigo Quilez popped up! Glad you took the time to talk a bit about shadertoy.

@vick229 Год назад

All Parker tri angles in one place.. love it 🤣

@vittorio1159 Год назад

15:30 You can just multiply by (1+sqrt(r^2-(x-h)^2-(y-k)^2)) removing that annoying denominator and this way including the vertices of the triangle.

@aleksitjvladica. 9 месяцев назад

By far the best video I watched recently, I had a blast, I had a lot of fun! Although I knew most of this. Thank you my mate! I love you!

@Jetpans Год назад

The 11:15 solution is just so beautiful. Using such simple concepts and yet getting such a magnificent result.

@unvergebeneid Год назад

Seeing those shapes that are not made up of triangles on Shadertoy was actually a thing of beauty. Maybe one day games will have actual curves being rendered on our GPUs.

@DerToasti Год назад

well, the one at 16:50 is the perfect one. very nice, straightforward and i assume it generalizes to any polygon. at 15:00 it shouldn't be too hard though to find a function that works at the ends too, like 1+sqrt(1-abs(x-1)-abs(x)) for example.

@mananagrawal6855 Год назад

The circumcircle solution is so nice. Love the creativity.

@cannot-handle-handles Год назад

Thanks for providing captions/subtitles!

@alanmiller7875 Год назад

First off... it's "Triangle 2: Electric Boogaloo." Secondly, I love that you started with triangles that were Parker Squares.

@ecospider5 Год назад

Conversations for Math aficionados. Truly fantastic.

@literallyjustayoutubecomme1591 Год назад

I remember stumbling upon inigo’s channel some months ago, some really cool stuff

@thebrickcraft Год назад

It's not defined for zero but x/abs(x) will give you a one with the sign of x. And since abs can be done by using sqrt(x^2) using sign isn't a bad way.

@johnchessant3012 Год назад

The circumcircle method was clever! The fact that it misses just the three vertices is extremely funny to me haha

@kazedcat Год назад

There must be a way to add back those 3 missing vertices

@hughobyrne2588 Год назад

@@kazedcat If we take the 'sqrt' function as one that strictly works within the real numbers, i.e. it gives the result 'undefined' for less than zero, zero for zero, and a nonzero positive value for a nonzero positive value, then this trick can be used quite easily - instead of excluding the region outside a circle, excluding a region outside the triangle, as defined by the half-planes indicated by the sides of the triangle. Get nine values a1 b1 c1 a2 b2 c2 a3 b3 c3 such that aix+biy+ci=0 is an equation for line number 'i' of the triangle, and aix+biy+ci>0 is the half-plane that includes the triangle. Then, the equation sqrt(a1x+b1y+c1)*sqrt(a2x+b2y+c2)*sqrt(a3x+b3y+c3)=0, where the expression is evaluated at all stages in the reals, will give exactly the triangle. The square-roots ensure the expression does not give a value for any point outside the triangle (as long as you're not using complex-number-clever square roots), so if the point is on any line, and the product is defined, the product is zero, and if it's off every line, the product is nonzero.

@gasparliboreiro4572 Год назад

5:30 love the last words and transition

@cristhiangrundmann1226 Год назад

You can make an equation for a line segment easily: dist(a,p) + dist(p, b) - dist(a,b) = 0. You can then multiply a bunch of segment equations together to make the union of them.

@cmilkau Год назад

The easiest way to make a triangle I think is to use max or min (works for all convex polytopes). Basically just take the max or the min of a bunch of affine-linear functions on the plane and you can get a polygonal-base cone (a generalized pyramid) as a graph of that max/min. Now pick another affine-linear function, could be as simple as just 0, and equate the two. The intersection of the two graphs typically is the border of a convex polygon (and it still is when projected back down onto the domain). Example: min { |m₁x - y + b₁|, |m₂x - y + b₂|, |m₃x - y + b₃| } = 0 in the notation of the video. You can choose an appropriate sign for each of the absolute values depending on the m's and b's. Note: max {x, y} = (x + |x - y| + y)/2 and min {x, y} = (x - |x - y| + y)/2, for the guys who don't want special functions. Note: You can also get weird unbounded "polytopes" in general. This is analogous to how cone intersections may not just give you ellipses but also hyperbolas or even parabolas.

@ethanolcott3960 Год назад

There's a nice clean way to represent triangles in Desmos, just define a function l(A,B,t)=A+(B-A)t and then call [l(P,Q,t),l(P,R,t),l(Q,R,t)] where P, Q, and R are all called as points.

@aikumaDK Год назад

1:26 subtitles says 'James Street' instead of Jane Street 7:35 Subtitles says 'octan' instead of arctan 12:10 Subtitles says 'F9' instead of (presumable) affine 13:38 'Goebel' instead of Goble 19:56 'gene street'

@abigailcooling6604 Год назад

0:38 subtitles says 'sine function' instead of sign function.

@kindlin Год назад

I'm pretty sure they're just autogenerated, so if that's the worst they did a good job.

@asheep7797 Год назад

also 1:26 subtitles say "James tree" instead of Jane Street

@simonmultiverse6349 Год назад

It also says "K9" instead of "canine"

@Jivvi 11 месяцев назад

10:53 "okay good" instead of "so good".

@FlamingBehar Год назад

I literally just re-watched that triangle video today, how serendipitous 😁

@Zero-4793 Год назад

that n sided shape equation is insane. feel like we need a video just on it

@sighmon5640 Год назад

That first equation reminds me of the "trick" where rearranges some pieces in a triangle and seems to create an extra square of area in the process, but the resultant is just a quadrilateral that looks very close to a triangle

@Jivvi 11 месяцев назад

The original "triangle" is also a quadrilateral. The seemingly straight edge has an angle that is less than 180° by exactly as much as the angle in the final one is more than 180°.

@__dane__ Год назад

I would recognize that SDF shadertoy from anywhere. It’s not a video mentioning SDFs without Inigo Quilez

@Zolbat Год назад

Inigo is an absolute legend

@__dane__ Год назад

@@Zolbat Inigo, Keijiro, Ben Golus and more

@morejpeg Год назад

Inigo is a legend. I used his triangle SDF to do 3D collision detection!

@AmPlant Год назад

Triangle is my favorite shape! I'm so glad the sequel finally dropped :D

@RandyKing314 Год назад

i really like that see-through Desmos!

@teamruddy611 11 месяцев назад

8:00 I've seen prime computing equations (totally exist, but very long) smaller than that. Also I have a triangle myself but I can't post it here - it starts as a Isosceles Triangle and you can drag around the corners - it is made by drawing a line from each point to every other point.

@PretzelBS Год назад

Today I bought both your books cause I’m in the uk for a family wedding! You’re my favourite educational RU-vid channel!

@micksam7 Год назад

IQ is a legend in the shader coding and demoscene communities. Props for showing off his work!

@ktbbb5 Год назад

Thumbnail looks like it is from Indigo Quilez's video. I'm already excited!

@EMAngel2718 Год назад

I think that if you're going to be fine with the absolute value function then you should be fine with the sign function because it's very easy to derive the sign function using the absolute value function

@cthutu Год назад

Signed distance fields are used for ray intersections and font rendering. Great area of maths.

@jmi967 Год назад

Concerning Graham Goble's entry, would either of the following put the vertices back? (1) Use his triangle equation as bounds on the same circle and then combine the two results. Essentially using his almost triangle to do what he did from the other side (2) Do what he did but use calculus to infinitesimally increase the radius of the circle to include the vertices

@jeremybuchanan4759 Год назад

Gen Eric is so much more easy-going than any of the other Gen's ... well done Gen Eric

@friiq0 Год назад

This is the earliest I’ve ever caught a video 😄

@EthanBradley1231 Год назад

I noticed that with the almost-triangle equation featured around 5:00, there are certain non-integer values of *a* for which it is *extremely* not a triangle. At *a* = 12.5, for example, the almost-vertical almost-line at x = -1 turns into two (almost) lines shooting out from (-1, 2) and (-1, -2).

@TheSmegPod Год назад

Curious to know what those n-gon equations give you for non-integer values What does a shape with pi or sqrt(2) or i sides look like

@mgpalpha Год назад

As soon as I saw the thumbnail I knew the video would end in Inigo Quiles. Absolutely insane shader artist

@taranmellacheruvu2504 Год назад

7:59 This here could definitely use some simplification. Unfortunately, you’d probably have to be a bit clever about it. Its various components look very similar to the steps taken when deriving the general area formula for regular n-gons. For that one, you need to cleverly mess around with some sin^2 s and turn them into a cot. For this one, I haven’t started yet.

@curiousfirely Год назад

I'm just over here, being amazed at the overlay of the graphs on top of your face.

@monstronamaguederaz Год назад

5:27 That's what I'd call a mathematically love triangle

@stevenchristy6156 Год назад

The sign function is legit. I use it sometimes in my own programming as its very cheap to obtain the high bit of a float or double.

@sp0_od597 11 месяцев назад

At 2:57, I believe that skewing and stretching the triangle can get you all triangles albeit, rotated and translated in the xy plane, so to fix this you could transform x and y into a new coordinate space x' y' and get all triangles that way. Basically I am stating that by stretching and skewing the triangle, you can generate any combination of internal angles of the triangle. Then by change of coordinate space, you can scale, translate, and rotate the triangle to wherever you want in the new coordinate space.

@philipmurphy2 Год назад

Great maths content on RU-vid today for sure.

@williampatrick8814 11 месяцев назад

The circle capture function is the most elegant attempt so far.

@Sebal007 Год назад

18:03 Those, and what follows on the shadertoy example, are called signed distance fields. Pretty usefull for some stuff :)

@edwardhoppe4294 Год назад

Parametric equation of a triangle: p = p0 + r(p1 - p0) + s(p2 - p0), where pn are the 3 vertices, and r, s are scalars. The point p lies within the triangle for 0

@Beokabatuka Год назад

There's something so pure about the joy of math(s) nerds trying to solve somewhat meaningless yet captivating problems. I love it.

@DendrummerMC Год назад

Regarding the second solution aka the anonymous one. Wouldn't a quick workaround to make that "disc"-triangle into a "circle"-triangle be to put it into a modulo?

@quillaja 9 месяцев назад

From the very start of this video, I just knew Inigo Quilez would have the best, simplest, most precise answer (even if he hadn't submitted one). The one with the circle "cropping" the triangle was also good.

@randomname9291 Год назад

1:50 you can actually make a square parallel/perpendicular to the axes like so: abs(x+y) + abs(x-y) = n where n is the square’s side length

@haph2087 Год назад

Can one just take any equation for an arbitrary (non-colinear) triangle, and turn that into an equation for any other triangle, by changing your basis? (Any function that applies a transformation to the X-Y plane also works as a transformation to all functions on the X-Y plane).

@besusbb Год назад

Nice video! I really should've expected iq and shadertoy to be in a video when there's the triangle sdf in the thumbnail haha

@psychoDon525 Год назад

"You're missing something infinitely small, which some would say, 'Does that really count?' I think it does." Seems like a bit of a change of heart on the significance of infinitesimals from Mr. Matt Parker.

@johncage5368 Год назад

I liked that safety triangle with the rounded corners. ;)

@LyuboRyuk Год назад

The funniest video from you last several years

@wikingagresor Год назад

I'm not a specialist or anything, but for me the simplest way to define a triangle is to have 2 vectors and a 3rd vector that is a product of their vector multiplication.

@frankharr9466 Год назад

Hey, that is exactly how I look and how I walk. It's amazing!

@cragnog Год назад

"As a approaches infinity this will approach a triangle" was a way funnier line than one would expect

@ollllj Год назад

this is a VERY overused phrase in 8th to 13rth grade maths, usually not with triangles. But i guess they stopped teaching that maths after the 90s.

@harrywoodell7008 10 месяцев назад

I use nonlinear in my music all the time. Signum can make beautiful music. Or nerve impulses that let me play music. Absolute value is ... absolute value!

@Ethan-sc7gq Год назад

Instead of the sign function, it could use |cos(pi t) - 1| /2 . That outputs 1 for odd values and ) for even, just switch the - to a + to reverse it. Multiply the individual multipliers by this to use or negate them. I made it up while making a Collatz Conjecture Desmos graph :) .

@karibui494 Год назад

I got it without trig functions with: sign(x) = -2mod(x, |x|+1)+|x|+1+x, this gives 1 for x>=0 and -1 for x

@satindra.r Год назад

@@karibui494 ideally you would want to get 0 for x=0 but i think its possible to convert this into something we can define floor as x-((cot^-1(cot(πx)))/π) and then use floor(tanh(x))-floor(tanh(-x)) to get sign(x)

@angeldude101 Год назад

sign(x) is actually exactly the same as normalization for vectors, which is usually written as v/|v|. This works for real numbers as well though doesn't include 0. Absolute value can be found the same way as other things as well with sqrt(x²). (Technically sqrt(x*conj(x)), but the conjugate of a real number is itself.)

@karibui494 Год назад

@@satindra.r setting sign(0) = 0, wouldn't remove the edges in this case?

@karibui494 Год назад

@@satindra.r I played around and found: mod(-x, |x|+1)+(x-|x|)/2 this is 1 for x > 0 and 0 for x

@davidanoble Год назад

I found an easier solution. Use a variation on polar coordinates where θ is the angle but r is the distance from the desired triangle. Then every triangle is just 'r = 0' but with a different coordinate system 🙃

@Yupppi Год назад

Really cool and fun video. I think it deservers a Triangulogy.

@switchisnt Год назад

This is fascinating. I have no idea what any of those equations actually mean. Or most of what was said.

@marlou169 Год назад

I liked the Graham solution, three lines combined with the circle✨

@keeperofthegood Год назад

:) Fun video. Very enjoyed

@matthewyoung2554 Год назад

It should be possible to write a triangle function (not equation) if you use polar coordinates (though you'd be constrained to triangles that contain the origin).

@oscidleus Год назад

I've seen his book translated to polish few days ago and got proud that some publisher decided that theres enough people intrested in math in Poland

@freshIight Год назад

Yoooo!! New equations for triangles just dropped🔥

@ollllj Год назад

sign02(a)=floor(floor(sqrt(a*a))-floor(a)) , where the floor() function just ensures good precision and floor() or fract() are ESSENTIAL for modulo-arithmetic on type float. sign02(a) //returns 0 or 2 (0 for negative) sign01(floor(sign02(a)*.5)) //returns 0 or 1 sign(floor(sign02(a)-1)) //returns -1 or 1 (-1 for negative)

@spacebarium Год назад

The triangle lore deepens

@ParadoxProblems Год назад

For the "filled in triangle" can you not take the derivative of the function. It would then be a triangle with the graph plotting infinity along its boundaries and 0 everywhere else.

@bentationfunkiloglio Год назад

Triangle videos are your best

@MrSilbarita Год назад

You can only be mad at the sign function if you are also mad at the absolute value function, since you can pretty much construct sign(x) as |x|/x (except for the case x=0). And you con only be mad at the absolute value function if you are mad a sqrt(x) and x^2. So really, there is no basis for being mad at sign.