Finally, after ALL THESE years, I now know why the hell the "curved line" on MS Paint moves haphazardly sometimes (being MS) on clicking the 4th time. It ruined so many of my creations and had to undo-redo the curve a million times - Bezier Curve. Researcher level Maths is required for arts, or a teacher who knows practical application rather than theoretical maths
@@yomumma7803 lol you're right that was harsh. I respect artists as a backbone of every community, it just sucks that the art buying market is messed up and over saturated. Also plenty of artists can do math. I just think being able to combine practical engineering with design should automatically get you labelled as an artist of some sort.
If you wanna learn more about how they're made and what other applications they have, go watch the video "The Beauty of Bezier Curves". Should be the top result.
More please. There isnt enough video content on non traditional school-math. You should make introductory type videos for different areas of math, like "what is topology?" Or "what is chaos theory?". You already go into a lot of that type of content, but it would be nice to have a good starting point into all of the different topics.
When I learn math, I like to imagine everything to have some form of application, it's just that many of those applications haven't been discovered yet. After all, almost every math theorem were discovered purely for the sake of math and the applications found later.
It's the year 5132 and a physicist has discovered that a strange tachyon particle is perfectly described by the batman curve authored by ancient bored highschoolers.
@@anshumanagrawal346 What are Bezier-curves? When that name came up, I knew what curves he was going to talk about, as in various 3D-programs you use these curves to give the shape you're looking for. The beauty with Bezier-curves is how it transitions from one type of curve into another type of curve without break-points or corners.
As a maths teacher, I love Bézier curves. I use them all the time to create examples. You can easily use it to get beautiful smooth examples of functions with the values and derivatives you want. Very helpful for early calculus when you want your students to learn to read the slopes of tangents for instance.
When you said "you've probably never seen the (lemniscate) like this" there was immediately a chicken nuggets ad. Yeah I haven't seen it like that before.
Aren't the art (einstein, yoga, etc.) curves just fourier transforms? Those are super applicable to a lot of things like audio engineering and control theory! 3b1b has a super awesome video about them (it might even be a series I can't remember)
The first thing I thought when I saw the weierstrauss function was “hey! This must be useful in procedural generated worlds somehow”. But then probably not. It is intriguing enough that I want to hear more about it!
@@maxonmendel5757 Typically when the name "Lagrangian" is used alone, it refers to the Lagrangian function of Lagrangian mechanics, which is kinetic energy minus potential energy. So in this case, the formula ½mv² = mgh. For a brief overview, check the wikipedia article on Lagrangian mechanics; the part around the equation L=T-V directly addresses the formula mentioned here. If instead you meant "which Lagrangian function" (out of many possible), this one is the one describing a free falling test particle in a central force gravitational field; a very good approximation of orbital mechanics, or simply dropping stuff.
Not *so* much, because the route that takes minimum time or fuel is rarely a geodesic. Aircraft go out of their way to avoid headwinds and to get tailwinds, and there are elaborate tracking and monitoring systems to help them do that. Sure, the geodesic is still in there, but as the track they're deliberately deviating from, not the track they're flying.
I always stayed away from maths. some how I managed to crack all maths exam during my engineering and now after watching videos of your and other I thought we haven't learned anything or taught anything like this or this way. Keep it up.
@@roygalaasen I know right!! I hope school taught us Maths in this way, I am so sure students would go an extra step to learn , just tell them the applications. I had learnt calculus the Physics way, I mean calculus was introduced to me in Physics class and taught with applications.
Nice video. You can add that the structural benefit of a catenary arch is that it is either in pure tension (hanging down) or pure compression (standing up). So, no bending. This makes these shapes extremely efficient.
Well, not really :-( The name catenary comes from "cadena" = chain (spanish or latin). This means, that each element of the hanging string, or whatever must have the degree of freedom of each (infinitesimal) element of a chain, which the cable definitively does NOT have. At least at that scale. A hanging bridge cable(s) definitively are at a scale where the stiffness is "negligent".
Well, not really :-( The name catenary comes from "cadena" = chain (spanish or latin). This means, that each element of the hanging string, or whatever must have the degree of freedom of each (infinitesimal) element of a chain, which the cable definitively does NOT have. At least at that scale. A hanging bridge cable(s) definitively are at a scale where the stiffness is "negligent".
12:30 *Everything* travelers along geodesics in GR as long as there is no force acting on it. (In GR gravity is more like a fictitious force that is a result of the spacetime geometry)
Excellent video. There is a great story about a challenge question published by Bernoulli. The challenge was delivered to Newton upon arriving home after a long day as Master of the Mint. He worked on the problem that night and sent the solution to Bernoulli without signing the paper. Bernoulli knew it was the work of Newton stating that one can discern the work of the lion by its paw print. The curve, as I recall, was a catenary.
I keep checking your channel hoping something has changed. RUclips needs to get on this. Why can't you send them a video saying 'hey it's me, the guy in every fucking video on my channel, I'm legit and I was hacked'. I feel like that would be proof enough that it's your account lol.
Hey Zach, love your videos! As a mechanical engineer who works on aerodynamic optimization I wanted to add another use of the Bezier curves (and parametric curves in general). In shape optimization (say we want to optimize the shape of an airfoil to get minimum drag) we often use Bezier curves to parameterize the shape and we move the control points to optimize it. We do that because if we optimized the shape by moving each node of the geometry, the resulting shape would not be smooth and pretty much impossible to manufacture. So there you go, have a great day!
Weiertrass isn’t application but demonstrates that we can’t rely on assumptions or intuition because it turns out most curves are miserable But most common and natural curves (polynomials, trig and exponential) are nice and we really got lucky that they are in fact analytic.
Do you have other recommended readings to gain this kinda insight on these kinds of topics and the things you talk about on your channel? Also some information on what prerequisites are necessary to understand the stuff would be nice
Fyi the brachistochrone is not drawn correctly as the optimal rolling path, it needs to be exactly half a period of the cycloid, such that it never has to roll back up.
Look up the Limacon curve. With the right parameters it nicely models the cross section of the human belly :-) . The cardioid is a special case of the Limacon.
Photoshop doesnt use vector graphics, its a raster progrqm meaning it uses pixels to represent images, illustrator uses vector graphics meaning it uses curves
In Barcelona, they are building a cathedral currently (so far, it took just a century), authored by Antoni Gaudi. Gaudi had some amazing design ideas, one of which was hanging strings upside down, and putting weights in specific places. Like this, he built an upside down string model with the optimal arches for weight distribution - catenary curves. This string model can be seen there and I was totally taken away by that. I highly recommend visiting and seeing Sagrada Familia in person.
What a great video. I always thought someone should talk about interesting curves in general as a youtube video, and the animations make the video a perfect execution of that idea:)
I do motion graphics for a living and Bézier curves are a big part of my life haha. To the point where after many hours sitting in front of the computer, I close my eyes and my brain is just making them
"I'm sure many of you know where this is going..." Ah yes, something relatable "...but this is relates to.." Bring me those sinoids, baby! "...Brachistochrone curve." *sigh* hope is an illusion
I liked the engineering approach to these curves. I've always thought of the bezier curve as being the curve that solves the parametric cubic for 2 tangents (point and slope), but that's a much more mathsy way of thinking of it. I remember coding different ways to interpolate sets of points in pov-ray before it got anything clever to do it. Fun to try your own ways (I liked connecting sets of points with just quadratics: get some crazy results!)
Internally, adobe only uses quadratic curves - it’s just that for the curve that looks cubic, one of the control points it at the same location as one of the end points. Mathematically though, that reduces to the same thing as a cubic curve. This is also true for straight lines - both control points are at the same locations as the end points.
Here are some random curves: Curve 1: x=cos(y^y)+cos((y^4y)-3)+cos(y^-4) Curve 2: x=sin(y^8)+cos(xy^9) Curve 3: x=sin(xy) Curve 4: x=cos(y)+cos(y^7)+sin(x^4)+sin(x^-5)
0:18 “It also has some interesting applications.” “No, I’m kidding.” “But what if I’m not?” “...” “...yeah, I’m just kidding.” Why do I hear boss musi- _”But what if I’m not?”_ *”Hey, Vsauce! Michael here.”*
7:57 moment is misleading. Ball on brachistochrone curve gets to zero level (where point B is) faster then free-fall ball. That is impossible. Guess we can call it artistic exaggeration.
here a parametric curve I discovered I call it a triangular curve heres the mathematica code demonstrating it. Manipulate[ ListPlot[Table[ If[t > d, {(e + 1) Cos[t], (e + 1) Sin[ t]}, {(e + 1 - c (t/(d))) Cos[t], (e + 1 - c (t/(d))) Sin[ t]}], {t, 0, Pi, 0.011}]], {c, 0, e + 1}, {d, 0.011, Pi}, {e, 0, 4}]
LOL Sin of sin of sin..... batman function. Those graphs aren't functions according to the vertical line test. LOL. The variables and dimension of the information matters. Weierstrass function. Never differentiable any where, infinitely jagged and continuous everywhere. Fractal-ish. Huh. Bezier Curves. Lemniscate - watt's linkage, robotics. Cycloid - Brachistochrone curve (sine wave) Catenary curve (hyperbolic sine)
You can really see that this guy is an engineer since he keeps talking about the practical uses of math.
I don't care what his reasons are. It's helpful for me to learn something if I can at least picture myself using that knowledge.
@@CrasterFamily He could be one, but he majored in Electrical Engineering
Chef 11:49
Practical Vs Theoretical strengths of math pls??
Lol tru
*I mean who doesn't like them curves?*
That's why we got this natural instinct to study all of them!
@@datchnac4577 I like how Evolution and Mathematics are connected
When I was a teenager, I definitely discovered all sorts of curves via certain videos
😳
actually, this is about math. not certain videos.
@@juliaanimates9765 no, it's about certain videos. Don't be fooled
@@juliaanimates9765 don’t be fooled Julia
@@juliaanimates9765 yeah Julia, this is secretly a sex education video, always gotta stay vigilant for things like this
Finally, after ALL THESE years, I now know why the hell the "curved line" on MS Paint moves haphazardly sometimes (being MS) on clicking the 4th time. It ruined so many of my creations and had to undo-redo the curve a million times - Bezier Curve. Researcher level Maths is required for arts, or a teacher who knows practical application rather than theoretical maths
Engineers are just artists that know math and have a job.
@@chaotickreg7024 wow, harsh bro lol
@@yomumma7803 lol you're right that was harsh. I respect artists as a backbone of every community, it just sucks that the art buying market is messed up and over saturated. Also plenty of artists can do math. I just think being able to combine practical engineering with design should automatically get you labelled as an artist of some sort.
@@chaotickreg7024 At least engineers make useful stuff.
If you wanna learn more about how they're made and what other applications they have, go watch the video "The Beauty of Bezier Curves". Should be the top result.
“Not applicable but definitely artistic”
All my friends call me artistic! They say I'm so colorful I belong on the spectrum.
His non applicable curves are hunted for in stocks charts.
@@ObjectsInMotion sameeee
0:22
Hey Vsause, Zach here
More please. There isnt enough video content on non traditional school-math. You should make introductory type videos for different areas of math, like "what is topology?" Or "what is chaos theory?". You already go into a lot of that type of content, but it would be nice to have a good starting point into all of the different topics.
When I learn math, I like to imagine everything to have some form of application, it's just that many of those applications haven't been discovered yet. After all, almost every math theorem were discovered purely for the sake of math and the applications found later.
It's the year 5132 and a physicist has discovered that a strange tachyon particle is perfectly described by the batman curve authored by ancient bored highschoolers.
NortheastGamer lmao
It just code make it develop and design that it they done this long time ago it was hard
I was born in middle Get in calculus this is easy theory must theory get hardest in math ever cuvre or right scission
It still think 3D we are thrid Dimension where not that far
2:39 Holy crap. In less than 30 seconds you answered a question I had for over a decade. Thanks!
@liebe please what was the question please
What question?
@@anshumanagrawal346 What are Bezier-curves? When that name came up, I knew what curves he was going to talk about, as in various 3D-programs you use these curves to give the shape you're looking for. The beauty with Bezier-curves is how it transitions from one type of curve into another type of curve without break-points or corners.
In the first curve ( 0:12 ) add some coefficients before x ,y ,xy ... (with sliders) the result is truely fascinating
Hmm yes, these curves do seem pretty cool
I didnt know bigfoots were into math
why are you everywhere?
lmao where u to next? Ryan's Toy Review??
By the videos on which I see you you definitely seem like an engineer
Agreed!
As a maths teacher, I love Bézier curves. I use them all the time to create examples. You can easily use it to get beautiful smooth examples of functions with the values and derivatives you want. Very helpful for early calculus when you want your students to learn to read the slopes of tangents for instance.
Curves are an interesting concept. You definitely have to utilize more techniques to accomplish them, but they are wonderful! Cheers man
When you said "you've probably never seen the (lemniscate) like this" there was immediately a chicken nuggets ad. Yeah I haven't seen it like that before.
0:58 "It just looks pretty (shows a fractal), ... or cool (shows batman) ... or, really weird (shows himself)"
Aren't the art (einstein, yoga, etc.) curves just fourier transforms? Those are super applicable to a lot of things like audio engineering and control theory! 3b1b has a super awesome video about them (it might even be a series I can't remember)
With the same amount of money I pay for Netflix I can pay Curiosity Stream, Brilliant. and still, have a leftover. Goodbye Netflix
9:18 you scared me into thinking my computer screen froze D:
Bruh;hhhgddh
The first thing I thought when I saw the weierstrauss function was “hey! This must be useful in procedural generated worlds somehow”. But then probably not. It is intriguing enough that I want to hear more about it!
10:35 Isn't that the Lagrangian?
Yes it is.
Yes, he probably didn’t say as much because it would make it seem complicated and obscure
Which one?
@@maxonmendel5757 Typically when the name "Lagrangian" is used alone, it refers to the Lagrangian function of Lagrangian mechanics, which is kinetic energy minus potential energy. So in this case, the formula ½mv² = mgh. For a brief overview, check the wikipedia article on Lagrangian mechanics; the part around the equation L=T-V directly addresses the formula mentioned here.
If instead you meant "which Lagrangian function" (out of many possible), this one is the one describing a free falling test particle in a central force gravitational field; a very good approximation of orbital mechanics, or simply dropping stuff.
And then he continues to show the "action" or rather the principle of stationary action.
Geodesics are also widely used in flight trajectories as he shortest distance between two points on an oblate earth
Not *so* much, because the route that takes minimum time or fuel is rarely a geodesic. Aircraft go out of their way to avoid headwinds and to get tailwinds, and there are elaborate tracking and monitoring systems to help them do that. Sure, the geodesic is still in there, but as the track they're deliberately deviating from, not the track they're flying.
Jokes on you, everyone knows the earth is an hyperbola 🙄
"even less useful tho(...)"
Now you have my attention
I always stayed away from maths. some how I managed to crack all maths exam during my engineering and now after watching videos of your and other I thought we haven't learned anything or taught anything like this or this way.
Keep it up.
I mean honestly all I learnt is that maths I thought wasn't useful usually explains maths that is useful
As a Graphic designer I've used Bezier Curves for years and years, but seeing the process behind them is marvellous, it's so cool.
I absolutely love this channel coz of it's content, I literally clapped at the end 😭 Great Job!!!
I wonder how he manages to be so productive yet still so fresh
@@roygalaasen I know right!! I hope school taught us Maths in this way, I am so sure students would go an extra step to learn , just tell them the applications. I had learnt calculus the Physics way, I mean calculus was introduced to me in Physics class and taught with applications.
Nice video. You can add that the structural benefit of a catenary arch is that it is either in pure tension (hanging down) or pure compression (standing up). So, no bending. This makes these shapes extremely efficient.
This channel is increasing in quality
But that cable at 8:45+ is stiff. That messes with the mechanics. So is it really a catenary?
Well, not really :-(
The name catenary comes from "cadena" = chain (spanish or latin).
This means, that each element of the hanging string, or whatever must have the degree of freedom of each (infinitesimal) element of a chain, which the cable definitively does NOT have.
At least at that scale.
A hanging bridge cable(s) definitively are at a scale where the stiffness is "negligent".
Well, not really :-(
The name catenary comes from "cadena" = chain (spanish or latin).
This means, that each element of the hanging string, or whatever must have the degree of freedom of each (infinitesimal) element of a chain, which the cable definitively does NOT have.
At least at that scale.
A hanging bridge cable(s) definitively are at a scale where the stiffness is "negligent".
12:30
*Everything* travelers along geodesics in GR as long as there is no force acting on it.
(In GR gravity is more like a fictitious force that is a result of the spacetime geometry)
Loxodrome: en.wikipedia.org/wiki/Rhumb_line
Excellent video. There is a great story about a challenge question published by Bernoulli. The challenge was delivered to Newton upon arriving home after a long day as Master of the Mint. He worked on the problem that night and sent the solution to Bernoulli without signing the paper. Bernoulli knew it was the work of Newton stating that one can discern the work of the lion by its paw print. The curve, as I recall, was a catenary.
Actually it was the brachistochrone
"But as far as I know, I am just kidding" - That was the first time someone earned an instant sub from me before watching the content.
I wish I could post videos again ;_;
I keep checking your channel hoping something has changed. RUclips needs to get on this. Why can't you send them a video saying 'hey it's me, the guy in every fucking video on my channel, I'm legit and I was hacked'. I feel like that would be proof enough that it's your account lol.
I’m glad you got your channel back!
"I'm sure many of you know where this is going..." oh yeah for sure.
Hm, always used bezier curves in video editing but never knew the math and reasoning behind it. Cool!
Hey Zach, love your videos!
As a mechanical engineer who works on aerodynamic optimization I wanted to add another use of the Bezier curves (and parametric curves in general).
In shape optimization (say we want to optimize the shape of an airfoil to get minimum drag) we often use Bezier curves to parameterize the shape and we move the control points to optimize it. We do that because if we optimized the shape by moving each node of the geometry, the resulting shape would not be smooth and pretty much impossible to manufacture.
So there you go, have a great day!
Weiertrass isn’t application but demonstrates that we can’t rely on assumptions or intuition because it turns out most curves are miserable
But most common and natural curves (polynomials, trig and exponential) are nice and we really got lucky that they are in fact analytic.
Wait, how can those create curves when sin(x) = x and cos(x) = 1?
Do you have other recommended readings to gain this kinda insight on these kinds of topics and the things you talk about on your channel? Also some information on what prerequisites are necessary to understand the stuff would be nice
physics phd student here. It's nice to see such a practical video.
Dang I wasn't ready for the video to end so abruptly, I could've watched another half hour of weird curves
3:42 did you use Desmos to make these curves with the 3 points?
If you mean the 4 points at 3:45 then yes. The page with the white background was a website I found (and linked in the description).
@@zachstar yes sorry I put the wrong time code, do you happen to have the link for that Desmos graph? That looks like a lot of fun to mess with
@9:00 If you turn the catenary curve upside down like the St Louis Arch does it then Maximize potential energy??
Yep! I believe it would.
Woah, that's cool
Hahahaha beginning had me dead 💀
Could you make video Hilbert curve ?
That intro though
perfect 😂😂👌
7:21 *Vsauce has entered the chat*
Fyi the brachistochrone is not drawn correctly as the optimal rolling path, it needs to be exactly half a period of the cycloid, such that it never has to roll back up.
Look up the Limacon curve. With the right parameters it nicely models the cross section of the human belly :-) . The cardioid is a special case of the Limacon.
I could put sin(sinx + cosy) = cos(sinxy + cosx) on my wall and nobody would realize it was a function and not an abstract work of art.
Thank you for explaining Bezier curves. I only vaguely remembered how they work. This clears it up.
Photoshop doesnt use vector graphics, its a raster progrqm meaning it uses pixels to represent images, illustrator uses vector graphics meaning it uses curves
I had more of an interest in curves in college. Most of the girls in HS were meh.
yoooooooooooooooooooo
That first formula is so cool. Typed it into a graphing app and you can explore it like a fractal.
I'll bet the Einstein curve works by using the Fourier Transform.
@12:05 - That is not a spiral, it is a helix :p
Nice guided tour of some interesting curves though, well done!
the infinite complexity of curvature is astounding - good video
The lemniscate and watts curves remind me of planetary orbits.
Incredible content man, loving all these videos you have been putting up lately. I really appreciate the work you do here.
0:20 "No, I'm just kidding. Or maybe I'm not...
[Mystery music begins]
Hey Vsauce, Zach here"
In Barcelona, they are building a cathedral currently (so far, it took just a century), authored by Antoni Gaudi. Gaudi had some amazing design ideas, one of which was hanging strings upside down, and putting weights in specific places. Like this, he built an upside down string model with the optimal arches for weight distribution - catenary curves. This string model can be seen there and I was totally taken away by that. I highly recommend visiting and seeing Sagrada Familia in person.
You should've at least mentioned the principal of least action when you were talking about the projectile motion. :(
Oh yeah it pressure pump fusion with engine gas with time
And tech
It what creative is the important what engine well look like
Yea they never taught us how to complement them without being a creep.
So this is how the pen tool works
Damn it , Bezier Curves
Does that mean that a chain-fed machine gun is an automatic kalashnicosh?
The only curve to know is the grading curve.
What a great video. I always thought someone should talk about interesting curves in general as a youtube video, and the animations make the video a perfect execution of that idea:)
Missed and opportunity to pull Fourier transforms into the discussion of the Einstein Curve as that is how you generate such a curve.
Flammy got hacked:(
Dear sir can you suggest a best book for calculus.
The Bezier curves remind me of "splines" in AutoCAD
I thought of the same thing! May be they are the same thing
The only curves I learned in school was me getting curved by girls
I do motion graphics for a living and Bézier curves are a big part of my life haha. To the point where after many hours sitting in front of the computer, I close my eyes and my brain is just making them
This is hilarious! But I also feel it opened my mind as to the array of curves that exist! Thank you!
1:08 you know you pronounce it correctly if auto-caption got it
bruh ive always wondered how that pen tool worked
Yoo WTF. Never thought I would see you in a RUclips comment. Cool to see you watch Zach Star too.
"I'm sure many of you know where this is going..."
Ah yes, something relatable
"...but this is relates to.."
Bring me those sinoids, baby!
"...Brachistochrone curve."
*sigh* hope is an illusion
Wait wait wait hold up you can minimize potential energy with a string time to test that out right now with. Rope jk lol omg .
That was the best explanation for the photoshop pen tool I have ever heard, thank you very much for that
Also, my favorite curve is Perlin Noise :3
Can you make a video on pursuit curves? Looks like they're the secret to winning a game of manhunt.
I actually have been wanting to make a video on those!
I liked the engineering approach to these curves. I've always thought of the bezier curve as being the curve that solves the parametric cubic for 2 tangents (point and slope), but that's a much more mathsy way of thinking of it. I remember coding different ways to interpolate sets of points in pov-ray before it got anything clever to do it. Fun to try your own ways (I liked connecting sets of points with just quadratics: get some crazy results!)
That's the Principle of Least Action at about 10:30
You can derive F=ma with it
I think this curve is pretty cool too: curve25519
Yeah or elliptic curves in general.
did u change name?
Internally, adobe only uses quadratic curves - it’s just that for the curve that looks cubic, one of the control points it at the same location as one of the end points. Mathematically though, that reduces to the same thing as a cubic curve. This is also true for straight lines - both control points are at the same locations as the end points.
Hmm if you put the right side of the equation at 0:13 to the power of infinity, you get a perfect diagonal lattice grid...
Here are some random curves:
Curve 1: x=cos(y^y)+cos((y^4y)-3)+cos(y^-4)
Curve 2: x=sin(y^8)+cos(xy^9)
Curve 3: x=sin(xy)
Curve 4: x=cos(y)+cos(y^7)+sin(x^4)+sin(x^-5)
I was just thinking how it resembles steam engines and there, he said it! I might have gotten too excited for a second there 😆
Excellent and worthwhile video on curves of various types. A must see for everyone to view, especially mathematics and science students.
Great content! I particularly love the Bezier curves! Fascinating and useful
0:18
“It also has some interesting applications.”
“No, I’m kidding.”
“But what if I’m not?”
“...”
“...yeah, I’m just kidding.”
Why do I hear boss musi-
_”But what if I’m not?”_
*”Hey, Vsauce! Michael here.”*
My curves.
For electronic drawing
Mathematica code 1.
Plot[Piecewise[{{((0.5 - x)/
0.5) (x/2) + (x/0.5) (0.25 + (x - 0.5) 1.1),
x 0.5 && x 0.75}}] , {x,
0, 1}]
Hopefully for in 3d one surface of a tetrahedron but curved.
Mathematica code 2.
Manipulate[
ListPlot[Table[
If[t > d, {0,
0}, {(e + 1 - c (t/d)) Cos[t], (e + 1 - c (t/d)) Sin[t]}], {t, 0,
2 Pi, 0.0011}],
AspectRatio -> Automatic,
PlotRange -> {{-(e + 1), (e + 1)}, {0, (e + 1)}} ,
PlotTheme -> "Minimal"], {c, 0, e + 1}, {d, 0.0011, Pi}, {e, 0, 4}]
7:57 moment is misleading. Ball on brachistochrone curve gets to zero level (where point B is) faster then free-fall ball. That is impossible. Guess we can call it artistic exaggeration.
Or perhaps a fault caused by animating each ball with a constant velocity?
very cool video man, I love these
Now a days exponential graph is famous because of covid 19
here a parametric curve I discovered I call it a triangular curve heres the mathematica code demonstrating it.
Manipulate[
ListPlot[Table[
If[t > d, {(e + 1) Cos[t], (e + 1) Sin[
t]}, {(e + 1 - c (t/(d))) Cos[t], (e + 1 - c (t/(d))) Sin[
t]}], {t, 0, Pi, 0.011}]], {c, 0, e + 1}, {d, 0.011, Pi}, {e,
0, 4}]
Kinda reminds me of my sister 😏
"I'm sure many of you know where this is going" nope
Yeah, I know
Thought girls were hot in high school
20-25 yo are much hotter
LOL Sin of sin of sin..... batman function. Those graphs aren't functions according to the vertical line test. LOL. The variables and dimension of the information matters.
Weierstrass function. Never differentiable any where, infinitely jagged and continuous everywhere. Fractal-ish. Huh.
Bezier Curves.
Lemniscate - watt's linkage, robotics.
Cycloid - Brachistochrone curve (sine wave)
Catenary curve (hyperbolic sine)