title
Coding Challenge #93: Double Pendulum
description
In this coding challenge, I create a double pendulum simulation in Processing. Code: https://thecodingtrain.com/challenges/93-double-pendulum
đšī¸ p5.js Web Editor Sketch: https://editor.p5js.org/codingtrain/sketches/jaH7XdzMK
đĨ Previous video: https://youtu.be/188B6k_F9jU?list=PLRqwX-V7Uu6ZiZxtDDRCi6uhfTH4FilpH
đĨ Next video: https://youtu.be/JSn-DJU8qf0?list=PLRqwX-V7Uu6ZiZxtDDRCi6uhfTH4FilpH
đĨ All videos: https://www.youtube.com/playlist?list=PLRqwX-V7Uu6ZiZxtDDRCi6uhfTH4FilpH
References:
đ´ Physics Lab Double Pendulum: https://www.myphysicslab.com/pendulum/double-pendulum-en.html
Videos:
đ Pendulum Simulation: https://www.youtube.com/watch?v=9iaEqGOh5WM
đ´ Coding Train Live 120: https://youtu.be/KhogNPC24eI?t=10979s
Related Coding Challenges:
đ #86 Cube Wave by Bees and Bombs: https://youtu.be/H81Tdrmz2LA
đ #159 Simple Pendulum Simulation: https://youtu.be/NBWMtlbbOag
Timestamps:
0:00:00 Double Pendulum Simulation
0:03:45 Creating The Double Pendulum
0:08:35 Adding The Tracing of The Path
0:12:51 Implementing The Double Pendulum Formulas
0:24:45 Visualizing It
0:28:21 Adding Damping
0:29:16 Conclusions And Suggestions
Editing by Mathieu Blanchette
Animations by Jason Heglund
Music from Epidemic Sound
đ Website: http://thecodingtrain.com/
đž Share Your Creation! https://thecodingtrain.com/guides/passenger-showcase-guide
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đĨ Coding Challenges: https://www.youtube.com/playlist?list=PLRqwX-V7Uu6ZiZxtDDRCi6uhfTH4FilpH
đĨ Intro to Programming: https://www.youtube.com/playlist?list=PLRqwX-V7Uu6Zy51Q-x9tMWIv9cueOFTFA
đ p5.js: https://p5js.org
đ p5.js Web Editor: https://editor.p5js.org/
đ Processing: https://processing.org
đ Code of Conduct: https://github.com/CodingTrain/Code-of-Conduct
This description was auto-generated. If you see a problem, please open an issue: https://github.com/CodingTrain/thecodingtrain.com/issues/new
#doublependulum #pendulumharmonicmotion #processing
detail
{'title': 'Coding Challenge #93: Double Pendulum', 'heatmap': [{'end': 618.38, 'start': 602.25, 'weight': 1}, {'end': 1086.362, 'start': 1067.115, 'weight': 0.711}, {'end': 1482.501, 'start': 1432.395, 'weight': 0.945}, {'end': 1673.287, 'start': 1626.108, 'weight': 0.864}], 'summary': 'A coding challenge to create a double pendulum simulation, involving formulas, motion, off-screen graphics, implementation challenges, visual enhancements, math implementation, and coding adjustments for achieving specific patterns.', 'chapters': [{'end': 69.929, 'segs': [{'end': 46.478, 'src': 'embed', 'start': 2.108, 'weight': 0, 'content': [{'end': 3.709, 'text': 'Hello, and welcome to a coding challenge.', 'start': 2.108, 'duration': 1.601}, {'end': 8.653, 'text': "Today, I'm going to attempt to make a double pendulum simulation.", 'start': 3.809, 'duration': 4.844}, {'end': 14.457, 'text': "Now, I actually don't have a coding challenge that's a single pendulum, so I should double back.", 'start': 9.113, 'duration': 5.344}, {'end': 17.039, 'text': 'Double back and make the single pendulum challenge.', 'start': 15.178, 'duration': 1.861}, {'end': 25.145, 'text': 'I do have some videos about pendulum simulations in my Nature of Code playlist, which I will refer you to,', 'start': 17.319, 'duration': 7.826}, {'end': 28.687, 'text': "but I really always wanted to make the double pendulum, because it's like this crazy.", 'start': 25.145, 'duration': 3.542}, {'end': 36.732, 'text': "it creates a double pendulum which, if you haven't figured out already, is having a pendulum attached to another pendulum.", 'start': 28.687, 'duration': 8.045}, {'end': 46.478, 'text': 'As you swing a double pendulum and you trace the path of the bottom point, it makes these beautiful, chaotic kind of insane patterns,', 'start': 38.153, 'duration': 8.325}], 'summary': 'Coding challenge aims to create a double pendulum simulation.', 'duration': 44.37, 'max_score': 2.108, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE2108.jpg'}], 'start': 2.108, 'title': 'Creating a double pendulum simulation', 'summary': 'Involves the creation of a double pendulum simulation, a coding challenge on the list since june 24th, 2016. the speaker is excited about attempting it and refers to videos in the nature of code playlist for single pendulum simulations.', 'chapters': [{'end': 69.929, 'start': 2.108, 'title': 'Double pendulum simulation', 'summary': 'Involves the creation of a double pendulum simulation, which is a coding challenge that has been on the list since june 24th, 2016, and the speaker expresses excitement about finally attempting it, referring to videos in the nature of code playlist for single pendulum simulations.', 'duration': 67.821, 'highlights': ['The speaker expresses excitement about attempting to create a double pendulum simulation, which has been on the coding challenge list since June 24th, 2016, and refers to videos in the Nature of Code playlist for single pendulum simulations.', 'The double pendulum creates chaotic and beautiful patterns as it swings, and the speaker mentions finding a reference back in December, expressing enthusiasm about the project.', "A brief mention of not having a coding challenge for a single pendulum and considering 'double back' to make the single pendulum challenge is made, indicating the speaker's intention to cover both single and double pendulum simulations."]}], 'duration': 67.821, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE2108.jpg', 'highlights': ['The speaker expresses excitement about attempting to create a double pendulum simulation, which has been on the coding challenge list since June 24th, 2016, and refers to videos in the Nature of Code playlist for single pendulum simulations.', 'The double pendulum creates chaotic and beautiful patterns as it swings, and the speaker mentions finding a reference back in December, expressing enthusiasm about the project.', "A brief mention of not having a coding challenge for a single pendulum and considering 'double back' to make the single pendulum challenge is made, indicating the speaker's intention to cover both single and double pendulum simulations."]}, {'end': 289.783, 'segs': [{'end': 152.57, 'src': 'embed', 'start': 69.989, 'weight': 0, 'content': [{'end': 71.87, 'text': "So we can actually just, let's look at this here.", 'start': 69.989, 'duration': 1.881}, {'end': 73.631, 'text': "We're basically, I want to make this.", 'start': 72.37, 'duration': 1.261}, {'end': 77.7, 'text': 'Okay, so this is what I want to make.', 'start': 75.659, 'duration': 2.041}, {'end': 80.242, 'text': "Right there, there's the double pendulum.", 'start': 79.001, 'duration': 1.241}, {'end': 83.784, 'text': 'But really what I need is the formulas on this page.', 'start': 80.782, 'duration': 3.002}, {'end': 89.128, 'text': "So let's first establish, let's establish for ourself this diagram.", 'start': 84.024, 'duration': 5.104}, {'end': 97.013, 'text': "So what I need for a double pendulum is this I'm going to consider to be .", 'start': 90.409, 'duration': 6.604}, {'end': 101.056, 'text': "00 Then this I'm going to consider to be .", 'start': 97.013, 'duration': 4.043}, {'end': 101.136, 'text': 'X1, Y1.', 'start': 101.056, 'duration': 0.08}, {'end': 109.156, 'text': 'And this here, and let me be more thoughtful about how I draw these lines.', 'start': 104.751, 'duration': 4.405}, {'end': 119.808, 'text': "This here, I'm going to consider to be point x2, y2.", 'start': 115.603, 'duration': 4.205}, {'end': 122.171, 'text': 'So there are important variables here.', 'start': 120.449, 'duration': 1.722}, {'end': 124.067, 'text': "Well, there's a bunch of things.", 'start': 123.266, 'duration': 0.801}, {'end': 130.152, 'text': 'In theory, there are two what you might call bobs, the bob being this thing at the end of a pendulum.', 'start': 124.307, 'duration': 5.845}, {'end': 133.717, 'text': "So there's the mass of this one, which is mass one.", 'start': 130.674, 'duration': 3.043}, {'end': 136.46, 'text': "There's the mass of this one, which is mass two.", 'start': 134.117, 'duration': 2.343}, {'end': 141.185, 'text': 'Okay, I also need to keep track of this length, arm length.', 'start': 136.48, 'duration': 4.705}, {'end': 142.706, 'text': 'the arm of the pendulum.', 'start': 141.625, 'duration': 1.081}, {'end': 145.607, 'text': "So I'm going to call this one R1 and R2.", 'start': 142.726, 'duration': 2.881}, {'end': 152.57, 'text': "Now, I just realized that if you look at the web page I'm looking at, they called it L1 and L2, but I'm going to call it R1 because variable name L,", 'start': 145.887, 'duration': 6.683}], 'summary': 'Analyzing double pendulum variables: mass, length, and positions for mathematical formulation.', 'duration': 82.581, 'max_score': 69.989, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE69989.jpg'}, {'end': 289.783, 'src': 'embed', 'start': 186.274, 'weight': 3, 'content': [{'end': 188.915, 'text': "Okay, I think I've got this pretty much right.", 'start': 186.274, 'duration': 2.641}, {'end': 191.156, 'text': 'These are all the variables of my system.', 'start': 189.376, 'duration': 1.78}, {'end': 196.419, 'text': 'The only thing that I really need to keep track of is theta one and theta two.', 'start': 191.376, 'duration': 5.043}, {'end': 199.576, 'text': 'R1 and R2.', 'start': 197.834, 'duration': 1.742}, {'end': 207.082, 'text': "I'm going to be calculating where these points are based on those angles, okay? So this is the diagram.", 'start': 199.956, 'duration': 7.126}, {'end': 209.985, 'text': "So that means I'm going to do this in processing.", 'start': 207.763, 'duration': 2.222}, {'end': 216.796, 'text': "that means I can start by having, oh yeah, I guess, Yeah, I kind of that's.", 'start': 209.985, 'duration': 6.811}, {'end': 221.379, 'text': "this is, I might have been backwards about the theta two angle, but I no, it's fine.", 'start': 216.796, 'duration': 4.583}, {'end': 222.1, 'text': "it's fine, it's fine.", 'start': 221.379, 'duration': 0.721}, {'end': 222.86, 'text': "I'm good, I'm good.", 'start': 222.18, 'duration': 0.68}, {'end': 225.222, 'text': "We'll find out if we made a mistake later.", 'start': 223.841, 'duration': 1.381}, {'end': 228.444, 'text': "So I'm going to say I need R1, this is going to be 100.", 'start': 225.242, 'duration': 3.202}, {'end': 230.245, 'text': 'Oh, I forgot I was in processing.', 'start': 228.444, 'duration': 1.801}, {'end': 235.609, 'text': 'R1 is 100, R2 is also 100.', 'start': 230.265, 'duration': 5.344}, {'end': 237.831, 'text': "Let's just have mass one be 10.", 'start': 235.609, 'duration': 2.222}, {'end': 238.592, 'text': "I'm making these up.", 'start': 237.831, 'duration': 0.761}, {'end': 240.574, 'text': 'Mass two be 10.', 'start': 239.193, 'duration': 1.381}, {'end': 247.18, 'text': "You know, there's also, so I have the length, the mass, and I need this .", 'start': 240.574, 'duration': 6.606}, {'end': 249.883, 'text': "00, so I'm going to need, okay, so then I need setup.", 'start': 247.18, 'duration': 2.703}, {'end': 252.425, 'text': 'Then I need draw.', 'start': 251.725, 'duration': 0.7}, {'end': 260.346, 'text': "And I'm going to create a window that is, let's make it 600 by 600.", 'start': 252.445, 'duration': 7.901}, {'end': 264.971, 'text': "And I'm going to say background 0, and then stroke.", 'start': 260.346, 'duration': 4.625}, {'end': 269.815, 'text': "Actually, let's make it background 255, stroke 0.", 'start': 265.331, 'duration': 4.484}, {'end': 272.096, 'text': "Let's make everything a little bit thicker so we can see it.", 'start': 269.815, 'duration': 2.281}, {'end': 274.138, 'text': 'Stroke weight 2.', 'start': 272.156, 'duration': 1.982}, {'end': 277.341, 'text': "And now, what do I need to do? Let's run this.", 'start': 274.138, 'duration': 3.203}, {'end': 281.404, 'text': "The first thing I want to do, and I'm going to move this over a little bit here.", 'start': 278.001, 'duration': 3.403}, {'end': 287.222, 'text': 'The first thing that I want to do is I want the 0, 0 point to be like here in the middle.', 'start': 282.779, 'duration': 4.443}, {'end': 289.783, 'text': "So the way to do that is I'm just going to use translate.", 'start': 287.562, 'duration': 2.221}], 'summary': 'Developing a system with variables theta one and theta two, calculating points based on angles, and setting up parameters for processing.', 'duration': 103.509, 'max_score': 186.274, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE186274.jpg'}], 'start': 69.989, 'title': 'Double pendulum formulas, calculating points, and visual simulation', 'summary': 'Explores the establishment of double pendulum diagrams, calculation of points based on angles theta 1 and theta 2 with r1 and r2 values of 100, and creation of a visual simulation with specific dimensions and colors.', 'chapters': [{'end': 152.57, 'start': 69.989, 'title': 'Double pendulum formulas', 'summary': 'Discusses the process of establishing the diagram for a double pendulum and the variables involved, including the masses and arm lengths, necessary for the formulas.', 'duration': 82.581, 'highlights': ['The chapter discusses the process of establishing the diagram for a double pendulum and the variables involved, including the masses and arm lengths, necessary for the formulas.', 'The variables involved in the double pendulum include mass one, mass two, arm length R1, and arm length R2.', 'The chapter also mentions the consideration of the bob as the thing at the end of a pendulum.']}, {'end': 235.609, 'start': 152.57, 'title': 'Calculating points based on angles', 'summary': 'Discusses the process of calculating points in a system based on the angles theta 1 and theta 2, along with the variables r1 and r2, with the values of r1 and r2 being 100.', 'duration': 83.039, 'highlights': ['I have two angles, theta 1 and theta 2, and the variables R1 and R2 are both set to 100.', "I'm going to be calculating where these points are based on those angles in the system.", 'The chapter discusses the process of calculating points in a system based on the angles theta 1 and theta 2, along with the variables R1 and R2.']}, {'end': 289.783, 'start': 235.609, 'title': 'Creating a visual simulation', 'summary': 'Covers the process of setting up and drawing a visual simulation with specific dimensions and color specifications.', 'duration': 54.174, 'highlights': ['The author sets up a visual simulation with a window size of 600 by 600, background color of 255, and stroke color of 0.', 'The author uses a stroke weight of 2 to make the visual elements thicker and more visible.', 'The author uses the translate function to position the 0, 0 point in the middle of the visual simulation.']}], 'duration': 219.794, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE69989.jpg', 'highlights': ['The chapter discusses the process of establishing the diagram for a double pendulum and the variables involved, including the masses and arm lengths, necessary for the formulas.', 'The chapter also mentions the consideration of the bob as the thing at the end of a pendulum.', 'The variables involved in the double pendulum include mass one, mass two, arm length R1, and arm length R2.', 'The author sets up a visual simulation with a window size of 600 by 600, background color of 255, and stroke color of 0.', 'The author uses a stroke weight of 2 to make the visual elements thicker and more visible.', 'The chapter discusses the process of calculating points in a system based on the angles theta 1 and theta 2, along with the variables R1 and R2.', 'I have two angles, theta 1 and theta 2, and the variables R1 and R2 are both set to 100.', "I'm going to be calculating where these points are based on those angles in the system.", 'The author uses the translate function to position the 0, 0 point in the middle of the visual simulation.']}, {'end': 636.222, 'segs': [{'end': 353.486, 'src': 'embed', 'start': 319.558, 'weight': 0, 'content': [{'end': 323.801, 'text': "It's a little bit of a different scenario because x is actually across from the angle.", 'start': 319.558, 'duration': 4.243}, {'end': 330.606, 'text': 'Sine of that angle, theta 1, equals x divided by r1.', 'start': 324.642, 'duration': 5.964}, {'end': 336.624, 'text': 'So x actually equals r1 times sine of theta 1.', 'start': 331.027, 'duration': 5.597}, {'end': 344.534, 'text': "Okay, so now I need to, I can calculate x1 equals r1 times, what did I say, sine of, oh, I don't have the angle.", 'start': 336.624, 'duration': 7.91}, {'end': 351.123, 'text': 'So let me create two variables, a1 is zero, a2, angle one and angle two.', 'start': 344.815, 'duration': 6.308}, {'end': 353.486, 'text': 'Sine of a1.', 'start': 352.405, 'duration': 1.081}], 'summary': 'Calculation of x using trigonometric functions and variables a1 and a2.', 'duration': 33.928, 'max_score': 319.558, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE319558.jpg'}, {'end': 411.011, 'src': 'embed', 'start': 381.049, 'weight': 1, 'content': [{'end': 384.112, 'text': "And then let's make sure I say fill zero.", 'start': 381.049, 'duration': 3.063}, {'end': 385.473, 'text': "Let's run this.", 'start': 384.172, 'duration': 1.301}, {'end': 391.218, 'text': "there we go, so we can see this is the top of the pendulum and let's make the.", 'start': 385.473, 'duration': 5.745}, {'end': 396.642, 'text': "Let's make it a little bit longer and a little bit bigger.", 'start': 391.218, 'duration': 5.424}, {'end': 397.383, 'text': "Okay, so that's good.", 'start': 396.642, 'duration': 0.741}, {'end': 401.626, 'text': "That's the top of the pendulum that looks right now.", 'start': 397.403, 'duration': 4.223}, {'end': 411.011, 'text': "what I need to do is is calculate X2, which is R2 times sine of A2, the same exact math that's going to apply.", 'start': 401.626, 'duration': 9.385}], 'summary': 'Adjust pendulum to calculate x2 using r2 and a2.', 'duration': 29.962, 'max_score': 381.049, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE381049.jpg'}, {'end': 490.923, 'src': 'embed', 'start': 466.227, 'weight': 2, 'content': [{'end': 472.729, 'text': "So like, for example, there's no animation happening yet, but if I were to say something like you know, pi divided by two, that's 90 degrees.", 'start': 466.227, 'duration': 6.502}, {'end': 475.639, 'text': "Yeah, okay, so that's right.", 'start': 474.099, 'duration': 1.54}, {'end': 482.321, 'text': 'And if I were to say like pi divided by eight and pi divided by four, that makes sense, right?', 'start': 476.039, 'duration': 6.282}, {'end': 489.062, 'text': "So the angle is the angle and the length is what's controlling where everything is positioned?", 'start': 482.561, 'duration': 6.501}, {'end': 490.923, 'text': 'And I could start doing weird stuff.', 'start': 489.342, 'duration': 1.581}], 'summary': 'Discussion on angles and their values in degrees.', 'duration': 24.696, 'max_score': 466.227, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE466227.jpg'}, {'end': 583.304, 'src': 'embed', 'start': 553.838, 'weight': 3, 'content': [{'end': 555.9, 'text': 'Oh, canvas.begindraw.', 'start': 553.838, 'duration': 2.062}, {'end': 563.268, 'text': "So this is an off-screen graphics context, the same as the thing that you're seeing in the window,", 'start': 557.842, 'duration': 5.426}, {'end': 573.699, 'text': 'but the reason why I want to use it is because I want to draw the background on it only once, so I can have nice trails.', 'start': 563.268, 'duration': 10.431}, {'end': 580.302, 'text': "And then what I'm going to do is at the end here, I probably should have waited this till later.", 'start': 575.299, 'duration': 5.003}, {'end': 583.304, 'text': "I don't know why I'm doing this now, but it's the order that I'm doing it in.", 'start': 580.322, 'duration': 2.982}], 'summary': 'Using off-screen graphics context for drawing background for nice trails.', 'duration': 29.466, 'max_score': 553.838, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE553838.jpg'}, {'end': 631.075, 'src': 'heatmap', 'start': 602.25, 'weight': 1, 'content': [{'end': 605.072, 'text': "So I'll make it a line or something later, but I just want to see.", 'start': 602.25, 'duration': 2.822}, {'end': 618.38, 'text': 'okay, and now, if at the beginning I actually draw that as the background image canvas zero zero.', 'start': 605.072, 'duration': 13.308}, {'end': 624.064, 'text': 'canvas dot stroke weight four.', 'start': 618.38, 'duration': 5.684}, {'end': 626.205, 'text': 'canvas dot stroke zero.', 'start': 624.064, 'duration': 2.141}, {'end': 628.151, 'text': 'There we go.', 'start': 627.81, 'duration': 0.341}, {'end': 631.075, 'text': "So you can see this pattern that it's doing.", 'start': 628.571, 'duration': 2.504}], 'summary': 'Experimenting with drawing a pattern using canvas and stroke weight of four.', 'duration': 28.825, 'max_score': 602.25, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE602250.jpg'}], 'start': 290.183, 'title': 'Double pendulum motion and off-screen graphics', 'summary': 'Covers the calculation of x1 and x2 using trigonometry for double pendulum motion, along with creating an off-screen graphics context to draw background once and have nice trails, illustrated with specific code and output.', 'chapters': [{'end': 508.193, 'start': 290.183, 'title': 'Double pendulum motion', 'summary': 'Discusses the calculation of x1 and x2 using trigonometry, visualizing top and bottom pendulum points, and confirming the correctness of the calculations through manual testing.', 'duration': 218.01, 'highlights': ['The chapter discusses the calculation of x1 and x2 using trigonometry, presenting the equations x1 = r1 * sin(theta1) and x2 = r2 * sin(theta2, and explaining the offset calculations for y1 and y2 in the context of a double pendulum motion.', 'The speaker visualizes the top of the pendulum by drawing a line and ellipse using the calculated x1 and y1 coordinates, demonstrating the application of the derived equations in a visual context.', 'The chapter confirms the correctness of the calculations by manually testing different angles, such as pi/2 and pi/8, and observing the corresponding positions, providing a practical validation of the trigonometric calculations.']}, {'end': 636.222, 'start': 509.073, 'title': 'Creating off-screen graphics', 'summary': 'Discusses creating an off-screen graphics context to draw background once and have nice trails, illustrating the process with specific code and output.', 'duration': 127.149, 'highlights': ['The speaker explains the process of creating an off-screen graphics context to draw the background only once for nice trails, utilizing specific code and output to illustrate the concept.', "The speaker demonstrates the use of 'createGraphics' to create an off-screen graphics context with dimensions 600x600, and explains the purpose behind this approach.", "The speaker utilizes the 'canvas.beginDraw' function to initialize the off-screen graphics context, aiming to draw the background only once for achieving desired visual effects."]}], 'duration': 346.039, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE290183.jpg', 'highlights': ['The chapter discusses the calculation of x1 and x2 using trigonometry, presenting the equations x1 = r1 * sin(theta1) and x2 = r2 * sin(theta2, and explaining the offset calculations for y1 and y2 in the context of a double pendulum motion.', 'The speaker visualizes the top of the pendulum by drawing a line and ellipse using the calculated x1 and y1 coordinates, demonstrating the application of the derived equations in a visual context.', 'The chapter confirms the correctness of the calculations by manually testing different angles, such as pi/2 and pi/8, and observing the corresponding positions, providing a practical validation of the trigonometric calculations.', 'The speaker explains the process of creating an off-screen graphics context to draw the background only once for nice trails, utilizing specific code and output to illustrate the concept.', "The speaker demonstrates the use of 'createGraphics' to create an off-screen graphics context with dimensions 600x600, and explains the purpose behind this approach.", "The speaker utilizes the 'canvas.beginDraw' function to initialize the off-screen graphics context, aiming to draw the background only once for achieving desired visual effects."]}, {'end': 804.344, 'segs': [{'end': 782.325, 'src': 'embed', 'start': 718.765, 'weight': 0, 'content': [{'end': 725.067, 'text': "I'm just thinking of one frame as one unit of time, so I'm going to do one adjustment to the angle,", 'start': 718.765, 'duration': 6.302}, {'end': 730.469, 'text': "whereas if I'm really trying to be more accurate about this, I might try to do several iterations of the math per frame.", 'start': 725.067, 'duration': 5.402}, {'end': 733.01, 'text': "That'll make more sense later as I get a little further here.", 'start': 730.97, 'duration': 2.04}, {'end': 738.172, 'text': 'But this should give me the structure from which to do this.', 'start': 733.25, 'duration': 4.922}, {'end': 745.543, 'text': "So if I run this now, Once again, it's not moving, but let's just give a little, let's give it a little bit of acceleration.", 'start': 738.392, 'duration': 7.151}, {'end': 752.637, 'text': "0.01, let's make this one negative 0.001.", 'start': 747.167, 'duration': 5.47}, {'end': 756.798, 'text': "Because now it's accelerating out of control really fast because those numbers are accumulating on each other.", 'start': 752.637, 'duration': 4.161}, {'end': 759.159, 'text': 'But you can see this is kind of the idea.', 'start': 757.038, 'duration': 2.121}, {'end': 763, 'text': 'I need to figure out what are those acceleration values of the angles.', 'start': 759.259, 'duration': 3.741}, {'end': 769.461, 'text': "And if you go and watch my simple pendulum video, you'll see it's kind of an easy thing to calculate based on gravity.", 'start': 763.46, 'duration': 6.001}, {'end': 770.562, 'text': 'Really simple formula.', 'start': 769.682, 'duration': 0.88}, {'end': 776.383, 'text': "The thing about the double pendulum is, and I'm not going to derive these formulas, but as I scroll through this page,", 'start': 770.882, 'duration': 5.501}, {'end': 779.884, 'text': "it's going to actually attempt to derive those formulas of motion.", 'start': 776.383, 'duration': 3.501}, {'end': 781.545, 'text': "I'm just going to scroll down and get to the end.", 'start': 779.924, 'duration': 1.621}, {'end': 782.325, 'text': 'Here we go.', 'start': 781.725, 'duration': 0.6}], 'summary': 'Adjusting angle iterations for accurate motion in double pendulum simulation.', 'duration': 63.56, 'max_score': 718.765, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE718765.jpg'}], 'start': 636.382, 'title': 'Implementing double pendulum formulas', 'summary': 'Discusses implementing the formulas for a double pendulum, addressing concepts of velocity, acceleration, and angular acceleration, while also highlighting the need for more accurate time iterations and the challenges of accumulating acceleration values.', 'chapters': [{'end': 804.344, 'start': 636.382, 'title': 'Implementing double pendulum formulas', 'summary': 'Discusses implementing the formulas for a double pendulum, including the concepts of velocity, acceleration, and angular acceleration, while also addressing the need for more accurate time iterations and the challenges of accumulating acceleration values on each other.', 'duration': 167.962, 'highlights': ['Implementing formulas for a double pendulum, including concepts of velocity, acceleration, and angular acceleration The chapter delves into the implementation of formulas for a double pendulum, covering concepts of velocity, acceleration, and angular acceleration.', 'Addressing the need for more accurate time iterations The need for more accurate time iterations is mentioned, where the author considers doing several iterations of the math per frame for improved accuracy.', 'Challenges of accumulating acceleration values on each other The challenges of accumulating acceleration values on each other are discussed, highlighting the potential issue of numbers accumulating rapidly and leading to out-of-control acceleration.']}], 'duration': 167.962, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE636382.jpg', 'highlights': ['Implementing formulas for a double pendulum, including concepts of velocity, acceleration, and angular acceleration', 'Addressing the need for more accurate time iterations', 'Challenges of accumulating acceleration values on each other']}, {'end': 1108.934, 'segs': [{'end': 866.474, 'src': 'embed', 'start': 818.774, 'weight': 1, 'content': [{'end': 826.119, 'text': "I'm going to actually just take a screenshot of this, absurdly, because I want to be able to see it and look at my code at the same time.", 'start': 818.774, 'duration': 7.345}, {'end': 834.464, 'text': "So I should be able to go now to the, I don't need this open, should be able to go to the desktop where I have that screenshot.", 'start': 827.3, 'duration': 7.164}, {'end': 837.379, 'text': "And so I've got that up there now.", 'start': 836.098, 'duration': 1.281}, {'end': 844.326, 'text': "And let's lower my processing window to down here.", 'start': 838.48, 'duration': 5.846}, {'end': 847.749, 'text': "Okay, so let's find a place where I can actually implement this.", 'start': 844.346, 'duration': 3.403}, {'end': 851.193, 'text': "So I'll just do it at the top of draw.", 'start': 849.471, 'duration': 1.722}, {'end': 855.724, 'text': 'Theta one, angle one A equals.', 'start': 853.462, 'duration': 2.262}, {'end': 858.747, 'text': 'Now I just need to put all that stuff in my code.', 'start': 855.864, 'duration': 2.883}, {'end': 864.952, 'text': "So one time when I see a formula like this, it's kind of like, I don't want to write this all in one line of code, it's going to be crazy.", 'start': 858.807, 'duration': 6.145}, {'end': 866.474, 'text': 'I like to break it up into parts.', 'start': 865.153, 'duration': 1.321}], 'summary': 'Implementing code, breaking down formula into parts for easier coding.', 'duration': 47.7, 'max_score': 818.774, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE818774.jpg'}, {'end': 1097.05, 'src': 'heatmap', 'start': 1067.115, 'weight': 0, 'content': [{'end': 1069.417, 'text': 'Float, angle one, oh, I forgot a semicolon.', 'start': 1067.115, 'duration': 2.302}, {'end': 1074.721, 'text': 'Okay, so this is promising.', 'start': 1073, 'duration': 1.721}, {'end': 1077.003, 'text': 'Why? Because that looks like a single pendulum.', 'start': 1075.181, 'duration': 1.822}, {'end': 1079.125, 'text': "Now it's kind of spinning a little bit out of control.", 'start': 1077.043, 'duration': 2.082}, {'end': 1083.375, 'text': 'So like probably going to want to add some dampening, but this is right.', 'start': 1080.73, 'duration': 2.645}, {'end': 1086.362, 'text': "It's going in a perfect circle So I'm not totally insane.", 'start': 1083.435, 'duration': 2.927}, {'end': 1087.785, 'text': "I've got something that's working here.", 'start': 1086.402, 'duration': 1.383}, {'end': 1092.649, 'text': "That's kind of nice okay now Time for formula number two.", 'start': 1087.945, 'duration': 4.704}, {'end': 1097.05, 'text': 'And by the way, somebody in the chat asked what crazy person thought up this formula.', 'start': 1092.849, 'duration': 4.201}], 'summary': 'Testing a single pendulum, trying to add dampening, and working on a second formula.', 'duration': 29.935, 'max_score': 1067.115, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE1067115.jpg'}], 'start': 806.567, 'title': 'Enhancing visuals and implementing pendulum formulas', 'summary': 'Discusses taking screenshots to enable simultaneous code and content viewing, enhancing productivity, and implementing a complex pendulum formula in code, breaking it down into parts and variables, and testing its functionality for a single pendulum.', 'chapters': [{'end': 844.326, 'start': 806.567, 'title': 'Taking screenshot for better visuals', 'summary': 'Discusses the process of taking a screenshot to enable simultaneous viewing of code and the content, enhancing productivity and efficiency.', 'duration': 37.759, 'highlights': ['The speaker discusses the need for more space and decides to take a screenshot for better visibility and productivity.', 'The speaker wants to be able to see the screenshot and look at the code at the same time, indicating the need for efficient multitasking.', 'The speaker moves the screenshot to the desktop for easy access, demonstrating a practical approach to file management.']}, {'end': 1108.934, 'start': 844.346, 'title': 'Implementing pendulum formulas', 'summary': 'Discusses the process of implementing a complex pendulum formula in code, breaking it down into parts and variables, and testing its functionality for a single pendulum.', 'duration': 264.588, 'highlights': ['The process of implementing a complex pendulum formula in code, breaking it down into parts and variables, and testing its functionality for a single pendulum.', 'Breaking down the formula into parts and variables to simplify its implementation in code.', 'Testing the functionality of the implemented formula for a single pendulum.']}], 'duration': 302.367, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE806567.jpg', 'highlights': ['The process of implementing a complex pendulum formula in code, breaking it down into parts and variables, and testing its functionality for a single pendulum.', 'Breaking down the formula into parts and variables to simplify its implementation in code.', 'Testing the functionality of the implemented formula for a single pendulum.', 'The speaker discusses the need for more space and decides to take a screenshot for better visibility and productivity.', 'The speaker wants to be able to see the screenshot and look at the code at the same time, indicating the need for efficient multitasking.', 'The speaker moves the screenshot to the desktop for easy access, demonstrating a practical approach to file management.']}, {'end': 1475.512, 'segs': [{'end': 1260.107, 'src': 'embed', 'start': 1183.616, 'weight': 0, 'content': [{'end': 1189.28, 'text': '2 times M2 times cosine of angle 1 minus angle 2.', 'start': 1183.616, 'duration': 5.664}, {'end': 1194.044, 'text': 'And if I look here at this, these are the same, right? The denominators are the same.', 'start': 1189.28, 'duration': 4.764}, {'end': 1195.765, 'text': "The only thing that's different is L1 and L2.", 'start': 1194.084, 'duration': 1.681}, {'end': 1197.466, 'text': 'Double check, double check, double check.', 'start': 1196.266, 'duration': 1.2}, {'end': 1208.154, 'text': 'Yes So I could copy paste this code here and bring it down here and just say R2.', 'start': 1197.486, 'duration': 10.668}, {'end': 1230.43, 'text': "Now, this now equals It's numerator one times numerator two plus numerator three plus numerator four divided by denominator.", 'start': 1208.875, 'duration': 21.555}, {'end': 1245.056, 'text': 'What is the chance that this is actually correct? What is the chance that this is correct.', 'start': 1232.631, 'duration': 12.425}, {'end': 1250.06, 'text': "Very, very, oh, uh-oh, uh-oh, I'm being told that I have some transcription errors.", 'start': 1245.416, 'duration': 4.644}, {'end': 1254.043, 'text': 'Make it denominator two, I did that.', 'start': 1251.2, 'duration': 2.843}, {'end': 1260.107, 'text': 'And A2A equals num1 times, okay, hold on, hold on.', 'start': 1255.124, 'duration': 4.983}], 'summary': 'Deriving a formula involving trigonometric functions and checking for transcription errors.', 'duration': 76.491, 'max_score': 1183.616, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE1183616.jpg'}, {'end': 1443.806, 'src': 'embed', 'start': 1380.262, 'weight': 2, 'content': [{'end': 1388.108, 'text': 'There we go, right? Because when I put this all together, I need this thing times these three things added together with parentheses around it.', 'start': 1380.262, 'duration': 7.846}, {'end': 1391.791, 'text': "And if I want to be really safe about this, I'll put another parentheses around it.", 'start': 1388.349, 'duration': 3.442}, {'end': 1393.933, 'text': 'Okay Okay, now.', 'start': 1391.992, 'duration': 1.941}, {'end': 1395.975, 'text': 'Okay, okay.', 'start': 1395.054, 'duration': 0.921}, {'end': 1408.704, 'text': 'All right.', 'start': 1407.703, 'duration': 1.001}, {'end': 1410.885, 'text': 'Hey, that looks kind of like a double pendulum.', 'start': 1408.744, 'duration': 2.141}, {'end': 1416.007, 'text': "Whoa Now, it's got quite a bounce to it.", 'start': 1412.546, 'duration': 3.461}, {'end': 1420.309, 'text': "I don't know if I've actually done it right.", 'start': 1418.649, 'duration': 1.66}, {'end': 1426.412, 'text': 'It is kind of spiraling out of control, which I think maybe I might have an issue.', 'start': 1421.53, 'duration': 4.882}, {'end': 1430.675, 'text': "So let's think about what could be the problem here.", 'start': 1426.433, 'duration': 4.242}, {'end': 1441.544, 'text': "I don't think the order matters, because putting it here is the same as you know, just not doing it the first frame.", 'start': 1432.395, 'duration': 9.149}, {'end': 1443.806, 'text': 'Oh, yeah.', 'start': 1443.466, 'duration': 0.34}], 'summary': 'Experimenting with mathematical operations and observing oscillating behavior in a double pendulum.', 'duration': 63.544, 'max_score': 1380.262, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE1380262.jpg'}], 'start': 1109.174, 'title': 'Implementing math in coding train', 'summary': 'Involves the process of implementing math in the coding train, with the speaker working through a series of equations and expressions, encountering transcription errors and seeking assistance from the audience.', 'chapters': [{'end': 1475.512, 'start': 1109.174, 'title': 'Implementing math in coding train', 'summary': 'Involves the process of implementing math in the coding train, with the speaker working through a series of equations and expressions, encountering transcription errors and seeking assistance from the audience.', 'duration': 366.338, 'highlights': ['The speaker works through a series of equations and expressions, including 2 times sine of angle one minus angle two and various numerators and denominators.', 'The speaker encounters transcription errors and seeks assistance from the audience to correct the errors.', 'The double pendulum being implemented and the speaker expressing uncertainty about whether it has been done correctly.', 'The speaker discusses the order of operations and potential issues, seeking input from the audience to resolve them.']}], 'duration': 366.338, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE1109174.jpg', 'highlights': ['The speaker works through a series of equations and expressions, including 2 times sine of angle one minus angle two and various numerators and denominators.', 'The speaker encounters transcription errors and seeks assistance from the audience to correct the errors.', 'The double pendulum being implemented and the speaker expressing uncertainty about whether it has been done correctly.', 'The speaker discusses the order of operations and potential issues, seeking input from the audience to resolve them.']}, {'end': 1859.291, 'segs': [{'end': 1673.287, 'src': 'heatmap', 'start': 1626.108, 'weight': 0.864, 'content': [{'end': 1628.069, 'text': "Now, let's actually have it start higher up.", 'start': 1626.108, 'duration': 1.961}, {'end': 1632.35, 'text': "Let's have it start pi divided by two, pi divided by two.", 'start': 1628.969, 'duration': 3.381}, {'end': 1636.092, 'text': "Let's see what happens here.", 'start': 1634.991, 'duration': 1.101}, {'end': 1641.314, 'text': 'So this should be, and I think this is the right pattern.', 'start': 1636.452, 'duration': 4.862}, {'end': 1652.589, 'text': "I don't know what initial settings to give it, but it kind of looks pretty similar to what I started with when I did that Google image search.", 'start': 1643.103, 'duration': 9.486}, {'end': 1660.982, 'text': 'If you remember my Google image search at the beginning of this video, this one, That looks pretty similar to that.', 'start': 1652.829, 'duration': 8.153}, {'end': 1664.363, 'text': 'So I think I have now done the double pendulum.', 'start': 1661.302, 'duration': 3.061}, {'end': 1673.287, 'text': 'Now the thing is playing with the mass, playing with the g value, playing with the different r values, playing with the color.', 'start': 1664.663, 'duration': 8.624}], 'summary': 'Experimented with double pendulum settings, resembling initial google image search.', 'duration': 47.179, 'max_score': 1626.108, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE1626108.jpg'}, {'end': 1664.363, 'src': 'embed', 'start': 1643.103, 'weight': 4, 'content': [{'end': 1652.589, 'text': "I don't know what initial settings to give it, but it kind of looks pretty similar to what I started with when I did that Google image search.", 'start': 1643.103, 'duration': 9.486}, {'end': 1660.982, 'text': 'If you remember my Google image search at the beginning of this video, this one, That looks pretty similar to that.', 'start': 1652.829, 'duration': 8.153}, {'end': 1664.363, 'text': 'So I think I have now done the double pendulum.', 'start': 1661.302, 'duration': 3.061}], 'summary': 'The speaker achieved a similar result to a previous google image search, completing the double pendulum.', 'duration': 21.26, 'max_score': 1643.103, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE1643103.jpg'}, {'end': 1718.657, 'src': 'embed', 'start': 1690.768, 'weight': 1, 'content': [{'end': 1702.946, 'text': 'A couple things that might be worth noting here is that, what was I going to say? Oh yeah, I could add some dampening.', 'start': 1690.768, 'duration': 12.178}, {'end': 1712.454, 'text': 'So one thing I could do is say like, I could simulate this idea of like friction or air resistance by saying a1 that times equals 0.999.', 'start': 1703.266, 'duration': 9.188}, {'end': 1716.178, 'text': "So it's always losing a little angular velocity every frame.", 'start': 1712.454, 'duration': 3.724}, {'end': 1718.657, 'text': 'So we could do this.', 'start': 1717.697, 'duration': 0.96}], 'summary': 'Simulating friction by reducing angular velocity in each frame.', 'duration': 27.889, 'max_score': 1690.768, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE1690768.jpg'}, {'end': 1798.698, 'src': 'embed', 'start': 1739.746, 'weight': 0, 'content': [{'end': 1742.887, 'text': "So that's one thing that we can kind of play with.", 'start': 1739.746, 'duration': 3.141}, {'end': 1745.968, 'text': "I can't stop.", 'start': 1742.907, 'duration': 3.061}, {'end': 1750.349, 'text': 'When this gets ported to p5.js as a JavaScript version, we could have sliders to play with all this stuff.', 'start': 1746.088, 'duration': 4.261}, {'end': 1751.789, 'text': 'But you can see what that really does.', 'start': 1750.509, 'duration': 1.28}, {'end': 1754.07, 'text': 'The dampening really causes it to not be able to swing.', 'start': 1752.009, 'duration': 2.061}, {'end': 1756.871, 'text': "So I'm going to comment this out though and leave this in the code.", 'start': 1754.71, 'duration': 2.161}, {'end': 1762.714, 'text': 'The other thing that I would do is what does it mean if the mass of these is 20?', 'start': 1757.211, 'duration': 5.503}, {'end': 1765.54, 'text': "How does that affect the pattern that I'm going to get?", 'start': 1762.714, 'duration': 2.826}, {'end': 1775.492, 'text': 'What if I make the arm length of the first one like 50? And the second one like 400?', 'start': 1765.981, 'duration': 9.511}, {'end': 1776.712, 'text': 'You know, this is sort of interesting.', 'start': 1775.492, 'duration': 1.22}, {'end': 1781.154, 'text': "What do I get? What kind of pattern do I get here? So there's so many possibilities.", 'start': 1776.752, 'duration': 4.402}, {'end': 1785.355, 'text': 'So this is the end of this coding challenge, the double pendulum coding challenge.', 'start': 1781.234, 'duration': 4.121}, {'end': 1791.316, 'text': 'This, I think, is one which is really ripe for your variations of, for your own, do your own twist on this.', 'start': 1785.555, 'duration': 5.761}, {'end': 1792.417, 'text': 'How are you drawing it?', 'start': 1791.536, 'duration': 0.881}, {'end': 1793.457, 'text': 'How are you visualizing it?', 'start': 1792.477, 'duration': 0.98}, {'end': 1796.178, 'text': 'How are you adding user interface controls?', 'start': 1793.677, 'duration': 2.501}, {'end': 1797.678, 'text': 'Is it the kind of thing that you make?', 'start': 1796.458, 'duration': 1.22}, {'end': 1798.698, 'text': 'how do you make it interactive??', 'start': 1797.678, 'duration': 1.02}], 'summary': 'Exploring variations and interactions in double pendulum coding challenge with potential for user interface controls and visualizations.', 'duration': 58.952, 'max_score': 1739.746, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE1739746.jpg'}], 'start': 1475.512, 'title': 'Double pendulum simulation and coding challenge', 'summary': 'Discusses coding adjustments in a double pendulum simulation aiming to achieve a pattern similar to a reference image. it also explores various parameters and encourages viewers to experiment and submit variations to thecodingtrain.com.', 'chapters': [{'end': 1664.363, 'start': 1475.512, 'title': 'Double pendulum simulation', 'summary': 'Discusses coding adjustments in a double pendulum simulation, including velocity, position, and initial settings, aiming to achieve a pattern similar to a reference image, with the goal of simulating a double pendulum.', 'duration': 188.851, 'highlights': ['The correct way to adjust the velocity based on force and then the position fixed the simulation.', 'Introducing variables centerX and centerY to reuse in the code and adjusting canvas size to 900x900.', 'Starting the pendulum simulation with initial settings similar to a reference image from a Google search.']}, {'end': 1859.291, 'start': 1664.663, 'title': 'Double pendulum coding challenge', 'summary': 'Explores the various parameters that can be adjusted in a double pendulum simulation, such as mass, gravity, arm length, and damping, allowing for numerous possibilities and visualizations. the chapter concludes by encouraging viewers to experiment with their own variations and submit them to thecodingtrain.com.', 'duration': 194.628, 'highlights': ['The chapter discusses adjusting parameters like mass, gravity, arm length, and damping in a double pendulum simulation, opening up numerous possibilities for visualizations and interactions.', "The concept of adding damping to the simulation is explored, with the value of angular velocity being reduced by a fraction in each frame, demonstrating the impact of dampening on the pendulum's motion.", 'Experimenting with different values for mass and arm length is highlighted, showcasing the influence of these parameters on the patterns and behavior of the double pendulum.', 'The potential for user interface controls and interactivity in the JavaScript version of the simulation is mentioned, indicating the possibility of incorporating sliders to manipulate parameters.', 'The chapter concludes by encouraging viewers to create their own variations of the double pendulum simulation and submit them to thecodingtrain.com, further emphasizing the exploration of different visualizations and interactions.']}], 'duration': 383.779, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/uWzPe_S-RVE/pics/uWzPe_S-RVE1475512.jpg', 'highlights': ['The chapter concludes by encouraging viewers to create their own variations of the double pendulum simulation and submit them to thecodingtrain.com, further emphasizing the exploration of different visualizations and interactions.', "The concept of adding damping to the simulation is explored, with the value of angular velocity being reduced by a fraction in each frame, demonstrating the impact of dampening on the pendulum's motion.", 'Experimenting with different values for mass and arm length is highlighted, showcasing the influence of these parameters on the patterns and behavior of the double pendulum.', 'The potential for user interface controls and interactivity in the JavaScript version of the simulation is mentioned, indicating the possibility of incorporating sliders to manipulate parameters.', 'Starting the pendulum simulation with initial settings similar to a reference image from a Google search.']}], 'highlights': ['The speaker expresses excitement about attempting to create a double pendulum simulation, which has been on the coding challenge list since June 24th, 2016, and refers to videos in the Nature of Code playlist for single pendulum simulations.', 'The double pendulum creates chaotic and beautiful patterns as it swings, and the speaker mentions finding a reference back in December, expressing enthusiasm about the project.', 'The chapter discusses the process of establishing the diagram for a double pendulum and the variables involved, including the masses and arm lengths, necessary for the formulas.', 'The chapter also mentions the consideration of the bob as the thing at the end of a pendulum.', 'The author sets up a visual simulation with a window size of 600 by 600, background color of 255, and stroke color of 0.', 'The chapter discusses the process of calculating points in a system based on the angles theta 1 and theta 2, along with the variables R1 and R2.', 'The chapter discusses the calculation of x1 and x2 using trigonometry, presenting the equations x1 = r1 * sin(theta1) and x2 = r2 * sin(theta2, and explaining the offset calculations for y1 and y2 in the context of a double pendulum motion.', 'The speaker visualizes the top of the pendulum by drawing a line and ellipse using the calculated x1 and y1 coordinates, demonstrating the application of the derived equations in a visual context.', 'Implementing formulas for a double pendulum, including concepts of velocity, acceleration, and angular acceleration', 'The process of implementing a complex pendulum formula in code, breaking it down into parts and variables, and testing its functionality for a single pendulum.', 'The speaker works through a series of equations and expressions, including 2 times sine of angle one minus angle two and various numerators and denominators.', 'The chapter concludes by encouraging viewers to create their own variations of the double pendulum simulation and submit them to thecodingtrain.com, further emphasizing the exploration of different visualizations and interactions.']}