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

@HasekuraIsuna

Жыл бұрын

... *nice*

@littlered6340

Жыл бұрын

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

@MrShiggitty

Жыл бұрын

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

@poppyseedsnuranium

Жыл бұрын

@@HasekuraIsuna ... Nice

@Son_Of_Atreides

Жыл бұрын

Would that make the stick man walking a plot twist?

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

@NeatNit

Жыл бұрын

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

@JordanBiserkov

Жыл бұрын

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

@vincentfreddoyle7555

Жыл бұрын

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

@JordanBiserkov

Жыл бұрын

@@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

Жыл бұрын

@@JordanBiserkov yes,polygons angles surface area

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

Жыл бұрын

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

Жыл бұрын

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

Жыл бұрын

@@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

Жыл бұрын

@@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

Жыл бұрын

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

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

Жыл бұрын

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

@marsovac

Жыл бұрын

@@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

Жыл бұрын

@@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

Жыл бұрын

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

Жыл бұрын

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

The triangular sequel! We want a triangular trilogy!

@dahawk8574

Жыл бұрын

*Triangle with a Vengeance*

@kaivalya931

Жыл бұрын

Triangle prequel: line revolution

@kaivalya931

Жыл бұрын

Triangle: the quadrilateral dynasty

@jplays8934

Жыл бұрын

yes!

@philipdmiller

Жыл бұрын

Triangle² or Triangle³

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

Жыл бұрын

His videos are amazing.

@elenplays

Жыл бұрын

An absolute God

@chromosundrift

Жыл бұрын

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

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

Жыл бұрын

😂

@oosmanbeekawoo

Жыл бұрын

I really like the art on your channel

@simonmultiverse6349

Жыл бұрын

I really like the chart on your anal.

@joshyoung1440

6 ай бұрын

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

Why not "Tri Harder"?

@Luna-wu4rf

Жыл бұрын

GENIUS

@sycration

Жыл бұрын

TriHard 7

@cs8712

Жыл бұрын

Do or do not - there is no triangle

@jimmygarza8896

Жыл бұрын

Triangle 3: the Threequel: Tri Harder.

@user-dx5wm1uz9d

Жыл бұрын

​@@sycration Cx

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

@johnsarthole

Жыл бұрын

I was going to say that

@NoNameAtAll2

Жыл бұрын

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

@lillilacac

Жыл бұрын

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

@aceman0000099

Жыл бұрын

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

@PureAsbestos

Жыл бұрын

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

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

@jpdemer5

Жыл бұрын

A Parker triangle, innit?

@ieatdirtwasntavailable

2 ай бұрын

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

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

@beskamir5977

Жыл бұрын

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

@himynameisdavenicetomeetyou

Жыл бұрын

@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.

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

Жыл бұрын

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

@__8120

Жыл бұрын

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

Жыл бұрын

_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

Жыл бұрын

​@@__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

Жыл бұрын

@@NitzanBueno oh absolutely

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

Жыл бұрын

Recognized it too!!

@leif1075

Жыл бұрын

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

@himynameisdavenicetomeetyou

Жыл бұрын

@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.

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

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.

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

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

Жыл бұрын

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

Жыл бұрын

@@anomaliecosmos The circle equation is the same.

@the1exnay

Жыл бұрын

The sign function being undefined at 0 feels very appropriate.

@ollllj

Жыл бұрын

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

@ollllj

Жыл бұрын

@@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)

I love these community contribution videos!

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

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

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 KZread. Thanks for making me want to learn more about maths ❤

@theoriginaltubeofyous

Жыл бұрын

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

@DingbatToast

Жыл бұрын

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

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

@ollllj

Жыл бұрын

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.

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!

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!

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.

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.

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

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!

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

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

Thanks for providing captions/subtitles!

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

@kazedcat

Жыл бұрын

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

@hughobyrne2588

Жыл бұрын

@@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.

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.

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.

The circumcircle solution is so nice. Love the creativity.

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

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.

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

Жыл бұрын

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°.

Conversations for Math aficionados. Truly fantastic.

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.

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.

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.

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

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

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

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

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.

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

5:30 love the last words and transition

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

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

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.

The funniest video from you last several years

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

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

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

Great maths content on KZread today for sure.

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

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

@Zolbat

Жыл бұрын

Inigo is an absolute legend

@__dane__

Жыл бұрын

@@Zolbat Inigo, Keijiro, Ben Golus and more

i really like that see-through Desmos!

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

Yoooo!! New equations for triangles just dropped🔥

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

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

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

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

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

Жыл бұрын

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

@kindlin

Жыл бұрын

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

@asheep7797

Жыл бұрын

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

@simonmultiverse6349

Жыл бұрын

It also says "K9" instead of "canine"

@Jivvi

Жыл бұрын

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

:) Fun video. Very enjoyed

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.

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

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

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

The triangle lore deepens

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

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

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?

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

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

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

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

In this video: Matt is excited because he learned a new video editing trick.

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).

"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.

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

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.

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.

Triangle videos are your best

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).

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.

Open source maths, I love it!

Looks like we improved on the Parker triangle.

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!

you should do a video just on that stick figure walking plot, thats so entertaining i bet it could be its own video on the math involved!

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

I was NOT read for the Matt Parker jumpscare in the new Captain Disillusion video. Very spooky

Shader code is like a different universe of code, it's so much fun.

Math aside your videos make me happy Matt. Thank you. - A math person.

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

I was all excited to see the triangle equations... I mean... You had me at "triangle" but as the saying goes: if you first don't succeed, tri-tri-tri again LOL...

15:52 would clever use of inequality signs and/or absolute values solve the better problem? Rather than forcing the function to be undefined outside the circle, you could use the inequality sign to specify only points inside (and on) the circle.

@Grable4PC

Жыл бұрын

That's how I started it, but I wanted to make a true triangle "equation," not a piecewise function, inequality, or anything with the absolute value or sign operators. Just 5 operators (+, -, *, /, ^) and an equals sign.