For those looking for clarification, the parabola is _not_ chaotic. The divergence over time in the parabola is constant (a linear development), whereas it is non-constant and non-periodic in the circle.
When you say the parabola's rate of divergence is linear, is it really fully linear? or just much closer to linear than usual for a curved structure? I just mean to ask if: the rate of divergence = a*t + b , where t is time and a&b are some constants? Not even the tiniest exponential variable like a 0.0001t^2? Not trying to be a stickler here for the sake of being a stickler lol. But, if its fully linear, then it would be even cooler to me and worth investigating further the properties of a parabola that make it so.
@@chasmurphy1227 That is a great point. Suppose the parabola is modeled by f(x)=x^2 and the circle is modeled by g(x)=-(1-x^2)^0.5. Then at any point x=a where there is a collision of a “ball” bouncing inside the parabola, the normal vector points in the direction of a line with a slope of -1/(2a), since f’(x)=2x. In the circle, the normal vector “slope” is g(a)/a. The important distinction here is that the way balls bounce inside the circle is governed by a recursive function. Chaos arises very easily from recursively defined dynamical systems. I would venture to say that any polynomial curve would not produce chaotic bouncing behavior whereas almost any other type of function would.
Quelle 200 palle mi sembrano palle virtuali con elasticità infinita, senza attriti, giusto? ma io vorrei capire come le ha lasciate: aggregate in un grumo compatto? o una morula vuota dentro? o allineate in un segmento?
@dejuren agreed. It spelled "hyperbola" not "hiperbola". You can rest easy now that I've defended the correct spelling of these shapes that I'm sure you're very familiar with
I think it makes slightly less sense than you let on. You speak in terms of optics, but the simulation here is kinematic. It's not obvious to me that a parabolic shape would play nice in this context.
@@joeg579 you can approximate how light behaves using individual balls bouncing off the object. In this case, the ball is affected by gravity, but since energy is conserved, the same concepts should apply, at least in terms of angles.
Ooo that's really interesting. Parabolas are certainly an interesting shape! I've been doing a few of my own over the past few days, thank you for inspiring me!
Oh, interesting. I wonder if the parabola will eventually get to chaos comparable to the circle or if it just ends up back where it started without much variation.
@Peter J. So in other words the ball will go through chaos, then order, then chaos, then order, etc. The reason we only see the chaos part of the balls is because the video is not long enough to contain the full cycle.
The parabola has a ‘focal point’ meaning that all lines bouncing off the side of a parabola (obeying that entrance angle equals exit angle) will go through the same point. This is also why they are used to pick i.e. television signals - the parabolic antennas you see on the sides of apartment complexes. They can pick up signals from all sides because of this focal point mathematical quality. I assume that is the reason for this.
@@distrologic2925 a circle has a focal point at the centre, but since the balls are dropped below the centre, they can never pass through it. that’s my guess but idk for sure
I think the reason is that gravity tends to create parabolic trajectories, so the parabola keeps everything in "sync" by maintaining the stability of motion. This reminds me of the interference between sines, with same frequency, but different phase
You'll notice that the balls all regularly meet up in the same place again. Their behaviour in the parabola continually loops back over again, following the same pattern.
It isn’t preventing it…. You literally see it just takes a bit longer. I am honestly confused by people. You see that chaos is happening and then boldly assert that it isn’t.
@@pyropulseIXXI Chaos means that slight changes in starting conditions eventually lead to wildly different behaviors in the long term (and so, if you have a limit to the precision in your measurements of the initial conditions, it is unpredictable). For the parabola, there is a periodic expansion and contraction of the distances between the points, but this is not chaotic behavior.
@@tracyh5751 stating something is chaotic alone is not meaningful. these kind of kinematic systems are deterministic so they cannot be truly chaotic. bottom one is more chatoic compared to the upper one. capability of predicting the outcome determines how chaotic we assume it to be. but ideally, with all the capabilites, no kinematic system can be chaotic since we have the capability to predict.
The parabola is diverging slightly, but the reason it is not chaotic is that it is diverging *along the color gradient* and still is essentially a line. So you could project the ending location of any arbitrary precision because it would be along the same designated spot on that line. However in chaotic systems, an arbitrarily small change in the starting position will result in completely unpredictable ending position.
Kinda neat to think about this in the context of optics. The spread could be similar to chromatic (or spherical?) aberration, best in the center near the focal point, worse at the edges. Of course the mechanism is completely different. But consider the shape of reflector telescope lenses, radar dishes, etc. Pretty neat.
Here's an idea: Create a shape that is on one axis a circle, and at each point on the circle affix the apex of a parabola, and then run this simulation in 3d. For reference, graphing this as a function might look like z=(y^2 /4)-sqrt(1- x^2)+1{x^2≤1} [Clarification]: This function is constructed as follows; z=p(y,s) - c(x,r){in bounds for non-imaginary numbers}+r Where r represents the circle's radius, and s represents the parabola's scale-- note that both functions place the shape on the z=0 plane at f(0) c(g,r) is the function for the circle = sqrt(r²-g²) p(g,s) is the function for the parabola = sg² Reduced form: z=r+(s*y²)-sqrt(r²-x²) {x²≤r²}
@@prsimoibn2710 well there's multiple possibilities. One could simply be experimenting to see how it would work for the sake of doing so. Depending on how it functioned, could be a pretty interesting thing to know and be applicable in... Somewhere I guess haha
That's really interesting, it would be good to see if the same non-chaotic behaviour holds for a 3d revolution parabolloid; my hypothesis is that it will do exhibit chaotic behaviour. My intuition comes as of my knowledge of when you run a 2d simulation of a brownian motion, you almost surely reach the origin point, but the same doesn't hold true for higher dimensions.
I wonder if this is why spherical lense in telescopes cause defects. The parabola seemed to keep a preferred direction on landing, but the circle seems to be unbiased in terms of what particular angle it lands at. Very interesting
from this video we learn that a stronger curvature reflects more dispersion, and as a consequence, more chaos during a time period . in the parabolic shape the strongest curvature is located in the lowest point and decreases as we move away from it. it is that zone around that point responsible for introducin that much chaos or variety. in the case of the circular shape the curvature is constant and evenly distributed along all the line which means it will introduce more chaos faster (with every bump or reflection)
That is why some antenna are parabolic and not circular. Parabola has properties that make points to converge to the center, no matter the direction, as well as the electromagnetic wave that carries the signals.
This is actually why the most expensive telescopes and dishes use a parabolic shape as a paraboloid is a curve that no matter where you intersect the reflection will always pass through the focus. The focus is thus where we put the measuring device that receives the signal
Does the parabola work differently for point masses vs balls? If the balls bounced against their centers instead of their surfaces, world they diverge more slowly, or perhaps not at all?
The parabola shape is focusing. If a ball goes a little bit more left or right than an other, it will be directed back a little bit to the middle, counteracting little differences. Wile the circle doesn't, so any little differences can accumulate and let it run more and more different paths.
Can you explain why the 200 balls that are identical except for color dropped at the same exact time and velocity and direction seem to differentiate in their paths at all. Every ball would follow the same physics and therefore the same path, therefore the balls would never differentiate in their paths. The only way to get the bottom pattern is if each ball had a slight difference in elasticity or mass. Then it wouldn't matter what the shape was you dropped them into they would take different paths eventually. So I'm just not sure what the shape of the half-pipe matters.
They all have a very little difference in initial conditions (i.e. starting position). And the property(by definition actually) of chaotic systems is that a tiny difference in initial conditions will lead to huge difference in the behavior.
It has to be a matter of how different curves propagate round-off noise in the computations, and higher precision algs would slow down the divergence of trajectories per iteration.
What we are examining here is essentially an optical scenario where gravity is bending the wave back on itself... With the parabola, every time it's reflected it passes back through the focus as if it was passing through a successive series of convex lenses. With a circle, there is no focal point, so it's as if the wave is passing it through a successive series of concave lenses. Thus you have periodic order over time (similar to a Time crystal) and you have periodic chaos over time.
@@joeg579 the paths are periodic. You can see that by how they all converge at different points in the parabola. Where they converge follows a pattern (or at least looks like one to my eyeballs), indicating that it will repeat at some point.
There are a number of mathematical proofs that can be used to quantitatively distinguish the difference in chaos between a parabolic bs semicircular container. The easiest (or at least most straightforward) would be the application of the scattering theory of asymptotic hyperbolic manifolds. This would actually allow one to predict with exact locational precision (x,y,z) any individual ball as a function of time (t)
Congratulations on another successful video! Before I repeat what you did - how did you check & solve collisions with the function? I mean if this is truly physical effect or some numerical issue with collision procedure which may vary from parabola to circle?
Thanks, fellow animator :). The algorithm has been kind to me, again. The collisions and paths are computed analytically by the SymPy lib in Python; thus accurate to machine precision. The only numerical issue I've encountered is that I must use a very fine sampling of the ball paths; plotted is just each 50th or 100th of the sampling points. Less fine sampling can be problematic for long simulations since errors may accumulate after collisions. That is, comparing two nearby balls, one may hit the function just before a sampling point, whereas the other may hit the function just after the sampling point... you get the point I guess. I've added a stripped version of the code used for the circle here: ua-cam.com/video/YoGmq0IdSxk/v-deo.html. Not my proudest work, but it works :).
@@animations_ag great job pal. Regarding the sampling problem - you could have used the simple fact that balls' paths are also parabolas (assuming they're moving with constant down acceleration) and solve their impact points analitically, as intersections of two parabolas, instead of sampling.
I would love for the 1D dots to have stayed as a 2D line or even to see this as 3D. I wonder if the order becomes more apparent the more dimensions we see.
In the parabola, I wonder if it is related to how trajectories behave in an ellipse. I remember seeing an elliptical pool table where shooting a ball from one foci would always result in a trajectory that intersected the hole, which was located at the other foci.
To make the parabola more real, extend an extra 1 unit above the plane. This will (in simulation compute it additionally) create an uncertain point, that will extend the range of possibilities like the irrational edge. Nice simulation and graphic!
I'm pretty hot on analytical geometry and trigonometry right now, so I'd like to mention that's NOT a circle, which would be 2 pi. It is a half circle, which is pi. Irrational and mystical, perhaps even godlike -- and a wonderfully impressive contrast with the conic section. Well done! Adding this to my favorites! 😎🖖
For those looking for clarification, the parabola is _not_ chaotic. The divergence over time in the parabola is constant (a linear development), whereas it is non-constant and non-periodic in the circle.
When you say the parabola's rate of divergence is linear, is it really fully linear? or just much closer to linear than usual for a curved structure? I just mean to ask if:
the rate of divergence = a*t + b , where t is time and a&b are some constants? Not even the tiniest exponential variable like a 0.0001t^2?
Not trying to be a stickler here for the sake of being a stickler lol. But, if its fully linear, then it would be even cooler to me and worth investigating further the properties of a parabola that make it so.
@@chasmurphy1227 That is a great point.
Suppose the parabola is modeled by f(x)=x^2 and the circle is modeled by g(x)=-(1-x^2)^0.5. Then at any point x=a where there is a collision of a “ball” bouncing inside the parabola, the normal vector points in the direction of a line with a slope of -1/(2a), since f’(x)=2x. In the circle, the normal vector “slope” is g(a)/a.
The important distinction here is that the way balls bounce inside the circle is governed by a recursive function. Chaos arises very easily from recursively defined dynamical systems.
I would venture to say that any polynomial curve would not produce chaotic bouncing behavior whereas almost any other type of function would.
@@JordanMetroidManiac You explained that very concisely!
@@HighTech636 Thank you!
I'm literally Carl Handricks from echo vn
If you notice, all the balls in the parabola unite at the same point, which I am pretty sure is the focus.
Hm
uh, nice point
That is what I was thinking as well. I think that is spot on so to say. On point if you get what I mean.
@@roygalaasen Iv'e got to dot this down!
@@Xayuap uh, nice pun
Please compare all conic curves! Especially in all borderline transitions between; circle/elipse/parabola/hiperbola. This is really interesting
@Repent and believe in Jesus Christ Why did you think this was appropriate on a maths video? Is maths against Jesus or something?
@@Dalendrion AI
@@dariusfilip4695 Hardly. A bot at the most. But probably just manual copy/paste.
Quelle 200 palle mi sembrano palle virtuali con elasticità infinita, senza attriti, giusto? ma io vorrei capire come le ha lasciate: aggregate in un grumo compatto? o una morula vuota dentro? o allineate in un segmento?
@dejuren agreed. It spelled "hyperbola" not "hiperbola". You can rest easy now that I've defended the correct spelling of these shapes that I'm sure you're very familiar with
This makes sense. Contact lenses and telescope use parabolic lenses to focus light. Better to be a parabola than a circle.
I thought they used a hyperbola, but just approximated it with a semicircle because that’s easier to manufacture
@@cara-seyun it's a parabola, but a circle can be a really good approximation to a parabola up to certain point
I think it makes slightly less sense than you let on. You speak in terms of optics, but the simulation here is kinematic. It's not obvious to me that a parabolic shape would play nice in this context.
@@joeg579 you can approximate how light behaves using individual balls bouncing off the object.
In this case, the ball is affected by gravity, but since energy is conserved, the same concepts should apply, at least in terms of angles.
I was thinking about this 👆👆
Ooo that's really interesting. Parabolas are certainly an interesting shape! I've been doing a few of my own over the past few days, thank you for inspiring me!
Oh, interesting. I wonder if the parabola will eventually get to chaos comparable to the circle or if it just ends up back where it started without much variation.
I guess that it doesn't. The parabola has a number of interesting properties, and I wouldn't be surprised if this were among them.
@Peter J. but .. we see the points diverge?
@@Hlebuw3k limits of the numerical simulation.
no, it wouldn't
@Peter J. So in other words the ball will go through chaos, then order, then chaos, then order, etc. The reason we only see the chaos part of the balls is because the video is not long enough to contain the full cycle.
The parabola has a ‘focal point’ meaning that all lines bouncing off the side of a parabola (obeying that entrance angle equals exit angle) will go through the same point. This is also why they are used to pick i.e. television signals - the parabolic antennas you see on the sides of apartment complexes. They can pick up signals from all sides because of this focal point mathematical quality.
I assume that is the reason for this.
The circle too though?
@@distrologic2925 a circle has a focal point at the centre, but since the balls are dropped below the centre, they can never pass through it. that’s my guess but idk for sure
Parabola: How the kids in the commercial play with the toy
Circle: How kids in real life play with it
I’m amazed that the behavior is more predictable in a parabola than a circle
Can we just apprisiate how those circles just agreed to make a beautiful snake and then fall apart? 😮
Appreciate * 😁👍
WE NEED TO SEE THE PARABOLA FOR LONGER!!
AND WE NEED MORE WINE AND NIBBLES!!
@@drewbewho Huh?
Wouldn’t change anything
@@daniyillin3171 too much wine for you.
@@daniyillin3171 sadly not on this occasion.
I think the reason is that gravity tends to create parabolic trajectories, so the parabola keeps everything in "sync" by maintaining the stability of motion.
This reminds me of the interference between sines, with same frequency, but different phase
That was my initial thought as well!
That the imposed boundary condition interplay with the nature of the PDE.
i think its also cause a lot of things are divided by square numbers ? like gravity ?
absolutely right, people claimed this movement is related to lensing and such but you pointed the correct reason.
@@vera-whatsurdiscord true, that's the mathematical explanation. The equation for a parabola is y=x^2
@@sharos404 What do you mean by "PDE"?
Makes me curious about hyperbolas now
So now we need to see bouncing in an ellipse with vertical major axis.
While the balls in the parabola did keep oscilating around a single point, their oscillations were getting bigger and bigger.
You'll notice that the balls all regularly meet up in the same place again. Their behaviour in the parabola continually loops back over again, following the same pattern.
I know that objects fall in a parabola curve. I never considered that bouncing inside an inverted parabola curve would prevent chaotic movement.
It isn’t preventing it…. You literally see it just takes a bit longer.
I am honestly confused by people. You see that chaos is happening and then boldly assert that it isn’t.
@@pyropulseIXXI it comes back into 1 point again for the non chaotic. chaotic wont return to its first point state
@@pyropulseIXXI Chaos means that slight changes in starting conditions eventually lead to wildly different behaviors in the long term (and so, if you have a limit to the precision in your measurements of the initial conditions, it is unpredictable). For the parabola, there is a periodic expansion and contraction of the distances between the points, but this is not chaotic behavior.
@@tracyh5751 stating something is chaotic alone is not meaningful. these kind of kinematic systems are deterministic so they cannot be truly chaotic. bottom one is more chatoic compared to the upper one. capability of predicting the outcome determines how chaotic we assume it to be. but ideally, with all the capabilites, no kinematic system can be chaotic since we have the capability to predict.
Circle represent choices made in real life. Parabola represent choices made in Telltale Games.
The parabola is diverging slightly, but the reason it is not chaotic is that it is diverging *along the color gradient* and still is essentially a line. So you could project the ending location of any arbitrary precision because it would be along the same designated spot on that line. However in chaotic systems, an arbitrarily small change in the starting position will result in completely unpredictable ending position.
Maybe, gravity give the ball a falling trajectory of a parabola over time. This might be something that related with the shape
Call an ambulance! but not for the parabola
I'm interested in the video but I gotta say the music really steals the show. The drums are SO tight I love it
Hence, the parabolic microphone 🎤
This is why mirrors use parabolic surface
Kinda neat to think about this in the context of optics. The spread could be similar to chromatic (or spherical?) aberration, best in the center near the focal point, worse at the edges. Of course the mechanism is completely different. But consider the shape of reflector telescope lenses, radar dishes, etc.
Pretty neat.
Oh man I wanna see that go longer! The parabola was starting to diverge but they kept meeting up at the focal points!
Here's an idea: Create a shape that is on one axis a circle, and at each point on the circle affix the apex of a parabola, and then run this simulation in 3d.
For reference, graphing this as a function might look like
z=(y^2 /4)-sqrt(1- x^2)+1{x^2≤1}
[Clarification]: This function is constructed as follows;
z=p(y,s) - c(x,r){in bounds for non-imaginary numbers}+r
Where r represents the circle's radius, and s represents the parabola's scale-- note that both functions place the shape on the z=0 plane at f(0)
c(g,r) is the function for the circle
= sqrt(r²-g²)
p(g,s) is the function for the parabola
= sg²
Reduced form:
z=r+(s*y²)-sqrt(r²-x²) {x²≤r²}
English please.
And what is the expected application of this,?
@@prsimoibn2710 getting likes
@@prsimoibn2710maybe it'll be chaotic in one axis but not in the other?
@@prsimoibn2710 well there's multiple possibilities. One could simply be experimenting to see how it would work for the sake of doing so. Depending on how it functioned, could be a pretty interesting thing to know and be applicable in... Somewhere I guess haha
That's really interesting, it would be good to see if the same non-chaotic behaviour holds for a 3d revolution parabolloid; my hypothesis is that it will do exhibit chaotic behaviour. My intuition comes as of my knowledge of when you run a 2d simulation of a brownian motion, you almost surely reach the origin point, but the same doesn't hold true for higher dimensions.
And this is the actual reason the parabolic satellite dishes have that weird form
The music sounds like the menu music in NFS MW 2012 lmao
I wonder if this is why spherical lense in telescopes cause defects. The parabola seemed to keep a preferred direction on landing, but the circle seems to be unbiased in terms of what particular angle it lands at. Very interesting
Yes, this is exactly why we use parabolas for lenses; telescopes, contact lenses, microscopes etc.
For the content, you make you really need more subscribers
Lol everyone is living in their own little parabola world and I'm over here in a crazy circle.
The circle is not special. It's the paraboloid that is special. There are very few curves that do not give a chaotic bounce. Am I right?
Great animation, also holy fk, gpcbass went hard on the music here.
This is why lenses and dishes are parabolic, lots of nice convergence
You can very clearly see that the one in a parabola takes a lot longer while to start up its chaotic dismembered motion than one in the circle.
The balls will always reunite at some point, and furthermore it is easy to predict.
I see as long as we square ourselves and square life, the square is events will never be chaotic
I REALLY need this song on spotify it's so good
I have absolutely no idea what I just watched, but it sure was pretty
I just stumbled across the genius idea to design parabolic toilets!
from this video we learn that a stronger curvature reflects more dispersion, and as a consequence, more chaos during a time period . in the parabolic shape the strongest curvature is located in the lowest point and decreases as we move away from it. it is that zone around that point responsible for introducin that much chaos or variety.
in the case of the circular shape the curvature is constant and evenly distributed along all the line which means it will introduce more chaos faster (with every bump or reflection)
That is why some antenna are parabolic and not circular. Parabola has properties that make points to converge to the center, no matter the direction, as well as the electromagnetic wave that carries the signals.
Love the song and all but I'm watching balls bouncing not exploring an abandoned tomb in Egypt
it just takes a *lot* longer before the chaos sets in.
Please do this with a cycloid curve
That's the visual representation of what's going on inside my head! My thoughts a trapped in a circle....
No nonsense, straight to the point 👍
The parabola is also stretching over time. Simply it will get longer to turn into chaos.
This is actually why the most expensive telescopes and dishes use a parabolic shape as a paraboloid is a curve that no matter where you intersect the reflection will always pass through the focus. The focus is thus where we put the measuring device that receives the signal
As a time traveller from the future i can confirm this a study piece of time ergonomics.
This music has got me on a trip
This is a good conceptual visualization of spherical aberration too!
I’m pretty sure the parabola having a focus is part of why this happens
Insane that the most complicated shape in the universe is the simplest to draw.
Does the parabola work differently for point masses vs balls? If the balls bounced against their centers instead of their surfaces, world they diverge more slowly, or perhaps not at all?
Haven't done the maths but intuitively it shouldn't diverge at all since all vertical motion reflects through the focus
"And this is why ladies and gentlemen, the earth is round and not flat..."
This is why they use parabolic shapes in dish antennas (and not spherical)
Now add interactions between the dots 🤣
The parabola shape is focusing. If a ball goes a little bit more left or right than an other, it will be directed back a little bit to the middle, counteracting little differences. Wile the circle doesn't, so any little differences can accumulate and let it run more and more different paths.
Nicely shows how parabolic reflectors eliminate chromatic aberation
Can you explain why the 200 balls that are identical except for color dropped at the same exact time and velocity and direction seem to differentiate in their paths at all. Every ball would follow the same physics and therefore the same path, therefore the balls would never differentiate in their paths. The only way to get the bottom pattern is if each ball had a slight difference in elasticity or mass. Then it wouldn't matter what the shape was you dropped them into they would take different paths eventually. So I'm just not sure what the shape of the half-pipe matters.
They all have a very little difference in initial conditions (i.e. starting position). And the property(by definition actually) of chaotic systems is that a tiny difference in initial conditions will lead to huge difference in the behavior.
man, dat rearange itself,
😳 because of gravity parable symmetry.
I feel like there's something intuitively profound in this visual example that tells us something about the nature of gravity and reality
It has to be a matter of how different curves propagate round-off noise in the computations, and higher precision algs would slow down the divergence of trajectories per iteration.
What we are examining here is essentially an optical scenario where gravity is bending the wave back on itself... With the parabola, every time it's reflected it passes back through the focus as if it was passing through a successive series of convex lenses. With a circle, there is no focal point, so it's as if the wave is passing it through a successive series of concave lenses. Thus you have periodic order over time (similar to a Time crystal) and you have periodic chaos over time.
Poincaré recurrence challenge: Run a simulation of balls bouncing chaotically in a circle until they all converge again
We'd be watching forever.
It seems to me like if you wait long enough it will become more chaotic. Maybe a parabola minimizes the amount of chaos but does not bring it to zero.
No, if the simulation ran for much longer, you’d see all the points returning to the origin, in a cycle
@@cara-seyun why is that, how do you know?
@@joeg579 the paths are periodic. You can see that by how they all converge at different points in the parabola. Where they converge follows a pattern (or at least looks like one to my eyeballs), indicating that it will repeat at some point.
No, because parabolas have foci. The points will converge at the foci at given intervals based on observation in the video.
Most of the comments here are just people throwing words around.
This video should also be titled..."How to hypnotize a stoner for 1.5 minutes"
There are a number of mathematical proofs that can be used to quantitatively distinguish the difference in chaos between a parabolic bs semicircular container.
The easiest (or at least most straightforward) would be the application of the scattering theory of asymptotic hyperbolic manifolds. This would actually allow one to predict with exact locational precision (x,y,z) any individual ball as a function of time (t)
Can you also try a cosh(x) curve? I’m curious about whether that would be more or less chaotic as it minimises kinetic energy
I'm guessing this is related to why satellite dishes are parabolic rather than bowls?
Congratulations on another successful video! Before I repeat what you did - how did you check & solve collisions with the function? I mean if this is truly physical effect or some numerical issue with collision procedure which may vary from parabola to circle?
Thanks, fellow animator :). The algorithm has been kind to me, again. The collisions and paths are computed analytically by the SymPy lib in Python; thus accurate to machine precision. The only numerical issue I've encountered is that I must use a very fine sampling of the ball paths; plotted is just each 50th or 100th of the sampling points. Less fine sampling can be problematic for long simulations since errors may accumulate after collisions. That is, comparing two nearby balls, one may hit the function just before a sampling point, whereas the other may hit the function just after the sampling point... you get the point I guess. I've added a stripped version of the code used for the circle here: ua-cam.com/video/YoGmq0IdSxk/v-deo.html. Not my proudest work, but it works :).
@@animations_ag great job pal. Regarding the sampling problem - you could have used the simple fact that balls' paths are also parabolas (assuming they're moving with constant down acceleration) and solve their impact points analitically, as intersections of two parabolas, instead of sampling.
Imagine what will happen to you in 10 years if you don't drink coffee today..
This guy beat the hell out of Jon bones Jones then put together one of the coolest simulations I've ever seen
Obrigado sr. UA-cam por me recomendar tal obra de arte
Wave vs no wave. Particle being in existence vs no existence. 1 possibility vs every possibility
That music sounds amazing on my Xbox headphones
At the bottom we have "ball pit"
at the top we have "breakdancing worm"
This is the Big Bang in a nutshell
this is beautiful
video: colorful bouncing balls
music: 🌴 🪘🗿🌴🎸 🪘 🌴 🎵 🌴 🗿🪘 🌴
Ppl: "ohh , its amazing!"
Me: "its a simple geometry knowledge from school. Circle has no focus , parabola has focus. "
I would love for the 1D dots to have stayed as a 2D line or even to see this as 3D. I wonder if the order becomes more apparent the more dimensions we see.
When you use one less toilet paper square than usual and it destroys reality as you know it
The dots are the possible locations in the universe where the eraser you dropped can be at
Do you think that perhaps the parabola is different bc it directs incoming rays towards the focus? So it'll self correct in a sense?
In the parabola, I wonder if it is related to how trajectories behave in an ellipse. I remember seeing an elliptical pool table where shooting a ball from one foci would always result in a trajectory that intersected the hole, which was located at the other foci.
Intresting and probably very useful to solve something in math or physics.
To make the parabola more real, extend an extra 1 unit above the plane. This will (in simulation compute it additionally) create an uncertain point, that will extend the range of possibilities like the irrational edge. Nice simulation and graphic!
I don't understand. Could you please elaborate further?
I'm pretty hot on analytical geometry and trigonometry right now, so I'd like to mention that's NOT a circle, which would be 2 pi. It is a half circle, which is pi. Irrational and mystical, perhaps even godlike -- and a wonderfully impressive contrast with the conic section. Well done! Adding this to my favorites! 😎🖖
I'd love to see the same video but with a tracer on for each of the balls so we can see the history of the path.
Next hyperbolic as well
This is so mesmerising ..
That song is a banger
Wouldn't this still be butterfly effect, just much slower in the parabola
No, the balls return to a focal point without being farther apart, a property impossible for the circle.
We can only approximate Pi. Collision detection in the simulation is therefore a little divergent, introducing accumulation of error over time.
Am I the only one watching just because its satisfying?
That's the point of a parabola, it focus to one point everything that hits it
make me miss my geometry/algebra classes