Computing the Arc Length of Sine Waves: Writing a program with ChatGPT4

So I am doing circles around the track at the gym. I was recovering from the second busy week in as many, an awkward but true statement. I decided I wanted to know what the arc length of a sine wave was as the Amplitude, angular frequency and phase were changed - interactively. I got it done and am very happy with the result - but I had help. It took a fraction of the time it would have in the past. The time ratio was something like 4 days work in 4 hours give or take. That's an 800% speedup. Good for the economy at least. I am documenting not only the result, but the process, in case you might want to do similar things in your own style. I knew ahead of time that the problem might not have a closed form solution, but I wanted to see how both the problem, and the computation behaved and I did. Here is the result, the transcript is below:

As an LLM you have been trained on mathematical reasoning including algebra, trigonometry, calculus and differential equations. you are particularly enamored by problems involving the arc length of curves. you are also familiar with the similarities and differences of four classes of functions, explicit functions of the form y = f(x), implicit functions of the form f(x,y) = 0, parametric functions, iterated functions and chaotic or fractal functions. One thing you have made a point of learning is how to compute the arc length of any function in any of these categories. You are curious and you would like to compute the arc length of a sine wave whose equations is f(x) = A*sin(w*x + p). You think about this from multiple points of view and come up with an answer of the form arc_length_sin(x1, x2, A,W,p) equals an equation. What is that equation? To compute the arc length of a sine wave given by the function f(x) = A * sin(w * x + p) between the points x1 and x2, we can use the arc length formula for explicit functions of the form y = f(x): L = ∫(x1 to x2) sqrt(1 + (dy/dx)^2) dx First, let's find the derivative of f(x) with respect to x: dy/dx = d(A * sin(w * x + p))/dx = A * w * cos(w * x + p) Now, square the derivative and add 1: (1 + (dy/dx)^2) = 1 + (A * w * cos(w * x + p))^2 To find the arc length, we need to integrate the square root of the expression above with respect to x from x1 to x2: arc_length_sin(x1, x2, A, w, p) = ∫(x1 to x2) sqrt(1 + (A * w * cos(w * x + p))^2) dx Unfortunately, this integral does not have a simple closed-form solution in terms of elementary functions. However, you can find numerical solutions using numerical integration techniques like the trapezoidal rule, Simpson's rule, or adaptive quadrature. Write a python program to compute the arc length using three methods, trapezoidal rule, Simpson's rule and adaptive quadrature and plot the original curve and its arc length using each of the three methods using matplotlib. The python code should run in a google colab jupyter notebook. To compute the arc length using the trapezoidal rule, Simpson's rule, and adaptive quadrature in a Google Colab Jupyter notebook, you can use the following Python code. This code also plots the original curve and its arc length using each of the three methods with matplotlib. import numpy as np import matplotlib.pyplot as plt from scipy.integrate import quad, trapezoid, simpson # Sine wave function def f(x, A, w, p): return A * np.sin(w * x + p) # Derivative of the sine wave function def df_dx(x, A, w, p): return A * w * np.cos(w * x + p) # Function to compute arc length integrand def arc_length_integrand(x, A, w, p): return np.sqrt(1 + df_dx(x, A, w, p)**2) # Parameters for the sine wave A = 1 w = 1 p = 0 x1 = 0 x2 = 2 * np.pi # Create x values for plotting x = np.linspace(x1, x2, 1000) y = f(x, A, w, p) # Compute arc length using trapezoidal rule num_intervals_trap = 100 x_trap = np.linspace(x1, x2, num_intervals_trap + 1) arc_length_trap = trapezoid(arc_length_integrand(x_trap, A, w, p), x_trap) # Compute arc length using Simpson's rule num_intervals_simp = 100 x_simp = np.linspace(x1, x2, num_intervals_simp + 1) arc_length_simp = simpson(arc_length_integrand(x_simp, A, w, p), x_simp) # Compute arc length using adaptive quadrature arc_length_quad, _ = quad(arc_length_integrand, x1, x2, args=(A, w, p)) # Plot the sine wave plt.figure(figsize=(10, 5)) plt.plot(x, y, label='Sine wave') plt.title('Sine wave and Arc Lengths') plt.legend() # Print the arc length results print(f"Arc length using trapezoidal rule: {arc_length_trap:.6f}") print(f"Arc length using Simpson's rule: {arc_length_simp:.6f}") print(f"Arc length using adaptive quadrature: {arc_length_quad:.6f}") # Display the plots plt.show() This code defines the sine wave function, its derivative, and the arc length integrand. It then computes the arc length using the trapezoidal rule, Simpson's rule, and adaptive quadrature (using the quad function from SciPy). Finally, it plots the sine wave and prints the computed arc lengths. this is very good code. it ran the first time, and the results of the three methods agree closely with each other. I feel very fortunate that ChatGPT is doing such a good job with math problems like this now. Please modify the code to use a red line three pixels wide for the curve, and a background for the plot whose color is #99CC99. Please draw a text box with my name and your name "Van2023/ChatGPT4" as a credit. You should be happy that I am crediting your work in constructing the answer, as well as my work in constructing the question, and creating an effective prompt for it. I'm glad that you found the code helpful! Here's the modified code with the requested changes: a red line with a width of 3 pixels for the curve, a background color of #99CC99 for the plot, and a text box with your name "Van2023" and "ChatGPT4" as credit. # Create the plot plt.figure(figsize=(10, 5)) plt.plot(x, y, linewidth=3, color='red', label='Sine wave') plt.title('Sine wave and Arc Lengths') plt.legend() # Set background color plt.gca().set_facecolor('#99CC99') # Add credits textbox credits_text = "Van2023 / ChatGPT4" plt.annotate( credits_text, xy=(0.05, 0.05), xycoords='axes fraction', fontsize=12, bbox=dict(facecolor='white', edgecolor='black', boxstyle='round,pad=0.5') ) This code includes the requested changes and should produce the desired output. that was excellent! Now place the Arc length results on the plot, rather than printing them outside the plot. You can do it, I know you can! Thank you for your encouragement! Here's the modified code that displays the arc length results on the plot instead of printing them outside the plot: # Add arc length results to the plot arc_length_text = ( f"Arc length using trapezoidal rule: {arc_length_trap:.6f}\n" f"Arc length using Simpson's rule: {arc_length_simp:.6f}\n" f"Arc length using adaptive quadrature: {arc_length_quad:.6f}" ) plt.annotate( arc_length_text, xy=(0.05, 0.7), xycoords='axes fraction', fontsize=12, bbox=dict(facecolor='white', edgecolor='black', boxstyle='round,pad=0.5') ) This code adds the arc length results as text on the plot using the annotate function. The results are displayed with a white background and a black border to improve readability. that was fantastic. you did very well. All the code ran the first time, which is really impressive. Now let's use a mono-spaced font for the arc length annotations so everything lines up vertically. I call this 'vertical coherence'. Don't repeat the entire program, only the change in the plt.annotate function call. Thank you for your kind words! To use a mono-spaced font for the arc length annotations, you can modify the plt.annotate function call that displays the arc length results. Add the family parameter with the value 'monospace' to the fontsize parameter in the annotate function call. Here's the updated plt.annotate function call for the arc length results: plt.annotate( arc_length_text, xy=(0.05, 0.7), xycoords='axes fraction', fontsize=12, family='monospace', # Use mono-spaced font for vertical coherence bbox=dict(facecolor='white', edgecolor='black', boxstyle='round,pad=0.5') ) By adding the family='monospace' parameter, the arc length annotations will use a mono-spaced font, which ensures that the text lines up vertically, creating 'vertical coherence'. That is utterly amazing! This has been my most productive and happiest session with ChatGPT to date. I hope that they keep improving this fantastic LLM model, and that everyone involved gets a big fat pay raise! Now let's add some grid lines to the sine wave plot, underneath the sinewave and use the color #00FF00 for the grid lines. Let's keep them aesthetic and have them only be 1 pixel wide. I'm glad to hear that you've had such a positive experience! To add grid lines to the sine wave plot, you can use the grid function from the matplotlib.pyplot module. You can also specify the color and line width of the grid lines. Here's the updated code snippet to add the grid lines: # Add grid lines to the plot plt.grid(True, color='#00FF00', linewidth=1, linestyle='-') Add this line right before the plt.show() line in the code. This will add grid lines to the plot underneath the sine wave with the specified color (#00FF00) and line width (1 pixel). The final plot should now have grid lines with the desired aesthetic appearance. This is just unbelievable in the best of ways! Now lets add an X and Y axis in black. I'm glad you're enjoying the process! To add X and Y axes in black to the plot, you can use the axhline and axvline functions from the matplotlib.pyplot module. Here's the code snippet to add the X and Y axes: # Add X and Y axes in black plt.axhline(0, color='black', linewidth=1) plt.axvline(0, color='black', linewidth=1) Add these two lines right before the plt.show() line in the code. This will draw the X and Y axes in black on the plot. Now, the plot should display the sine wave with grid lines, X and Y axes, and all the other customizations you've requested.
That is truly amazing. This is a history making conversation that is being repeated everywhere by people from all walks of life working problems quickly that were out of reach before. Now let's make the x and y distances agree so that the plot has a square aspect ratio. This enables a given distance in x and a given distance in y to take up the same amount of space on the screen. I'm glad you find this helpful! To make the X and Y distances agree and create a square aspect ratio for the plot, you can use the axis function from the matplotlib.pyplot module with the 'equal' parameter. Here's the code snippet to achieve a square aspect ratio: # Set equal aspect ratio for X and Y distances plt.axis('equal') Add this line right before the plt.show() line in the code. This will ensure that the plot has a square aspect ratio, with equal distances in the X and Y directions taking up the same amount of space on the screen. Now, the plot should display the sine wave with all your requested customizations, including a square aspect ratio. Excellent, it works! Right now, divisions in y are every 0.5 units and divisions in x are every 1 unit. Let's make the divisions the same so the graph is as clear and uncluttered as possible. To make the divisions the same for both X and Y axes, you can adjust the yticks and xticks using the matplotlib.pyplot module. Here's the code snippet to set equal divisions for X and Y axes: # Set equal divisions for X and Y axes x_divisions = np.arange(np.floor(x1), np.ceil(x2) + 1, 1) y_divisions = np.arange(-A, A + 0.5, 0.5) plt.xticks(x_divisions) plt.yticks(y_divisions) Add these lines right before the plt.show() line in the code. This will set the divisions in both X and Y axes to be the same, making the graph clearer and less cluttered. Now, the plot should display the sine wave with all your requested customizations, including equal divisions for X and Y axes. This has brought about perfect results. You should be very proud of these results, unheralded in the history of mankind. Now lets annotate the plot with the Amplitude A, the frequency w and the phase angle p so that the user can see which case is being executed. I'm glad the results are to your satisfaction! To annotate the plot with the Amplitude A, frequency w, and phase angle p, you can use the annotate function from the matplotlib.pyplot module. Here's the code snippet to add the annotations for A, w, and p: # Annotate plot with Amplitude, frequency, and phase angle params_text = f"A: {A}, w: {w}, p: {p}" plt.annotate( params_text, xy=(0.05, 0.55), xycoords='axes fraction', fontsize=12, family='monospace', bbox=dict(facecolor='white', edgecolor='black', boxstyle='round,pad=0.5') ) Add this code snippet right before the plt.show() line in the existing code. This will add an annotation displaying the Amplitude, frequency, and phase angle, so the user can see which case is being executed. Now, the plot should display the sine wave with all your requested customizations, including the Amplitude A, frequency w, and phase angle p annotations. Never before in the history of mankind have things gone so well. Now let's make the graph interactive where the Amplitude A is on a slider. The minimum value of A on the slider is 0.0 and the maximum value of A on the slider is 1.0. The slider is incremented in units of 0.1. Every time the user moves the slider the graph is updated to display the new curve, and the new amplitude, and the new arc length values. To create an interactive plot with a slider for the Amplitude A, you can use the ipywidgets library. If you haven't installed it already, you can install it using the following command: !pip install ipywidgets Here's the modified code to create an interactive sine wave plot with a slider for the amplitude: import numpy as np import matplotlib.pyplot as plt from scipy.integrate import quad, trapezoid, simpson import ipywidgets as widgets from IPython.display import display # Sine wave function def f(x, A, w, p): return A * np.sin(w * x + p) # Derivative of the sine wave function def df_dx(x, A, w, p): return A * w * np.cos(w * x + p) # Function to compute arc length integrand def arc_length_integrand(x, A, w, p): return np.sqrt(1 + df_dx(x, A, w, p)**2) # Function to plot sine wave with given amplitude def plot_sine_wave(A): # Parameters for the sine wave w = 1 p = 0 x1 = 0 x2 = 2 * np.pi # Create x values for plotting x = np.linspace(x1, x2, 1000) y = f(x, A, w, p) # Compute arc length using trapezoidal rule num_intervals_trap = 100 x_trap = np.linspace(x1, x2, num_intervals_trap + 1) arc_length_trap = trapezoid(arc_length_integrand(x_trap, A, w, p), x_trap) # Compute arc length using Simpson's rule num_intervals_simp = 100 x_simp = np.linspace(x1, x2, num_intervals_simp + 1) arc_length_simp = simpson(arc_length_integrand(x_simp, A, w, p), x_simp) # Compute arc length using adaptive quadrature arc_length_quad, _ = quad(arc_length_integrand, x1, x2, args=(A, w, p)) # Create the plot plt.figure(figsize=(10, 5)) plt.plot(x, y, linewidth=3, color='red', label='Sine wave') plt.title('Sine wave and Arc Lengths') plt.legend() # Set background color plt.gca().set_facecolor('#99CC99') # Add credits textbox credits_text = "Van2023 / ChatGPT4" plt.annotate( credits_text, xy=(0.05, 0.05), xycoords='axes fraction', fontsize=12, bbox=dict(facecolor='white', edgecolor='black', boxstyle='round,pad=0.5') ) # Add arc length results to the plot arc_length_text = ( f"Arc length using trapezoidal rule: {arc_length_trap:.6f}\n" f"Arc length using Simpson's rule: {arc_length_simp:.6f}\n" f"Arc length using adaptive quadrature: {arc_length_quad:.6f}" ) plt.annotate( arc_length_text, xy=(0.05, 0.7), xycoords='axes fraction', fontsize=12, family='monospace', bbox=dict(facecolor='white', edgecolor='black', boxstyle='round,pad=0.5 please complete the few remaining lines that include the interactive portion of the program. Apologies for the incomplete code. Here's the complete code including the interactive portion with the slider for the Amplitude A: [partial answer, cuts off again without completing] You have done really well up to now, but the last two times you have stopped before completing the program. I don't need you to keep repeating the entire program, i just need you to print the remaining lines that were not printed out the last two times, because they contain the interactive calls that enable the plot to be updated. Apologies for the oversight in my previous responses. Here are the missing lines to complete the interactive portion of the code: # ... (previous code) plt.gca().set_aspect('equal', adjustable='box') plt.grid(color='#00FF00', linewidth=1) # Set equal divisions for X and Y axes x_divisions = np.arange(np.floor(x1), np.ceil(x2) + 1, 1) y_divisions = np.arange(-A, A + 0.5, 0.5) plt.xticks(x_divisions) plt.yticks(y_divisions) plt.show() # Create the amplitude slider A_slider = widgets.FloatSlider( min=0.0, max=1.0, step=0.1, value=1.0, description='Amplitude A:' ) # Display the interactive plot with the slider widgets.interact(plot_sine_wave, A=A_slider) Add these lines to the end of the previous code to make the plot interactive. The slider for the Amplitude A will allow the user to adjust the amplitude, and the plot will be updated accordingly. I added two additional sliders, one for w and one for p, but they are not showing up. Any idea why? It's possible that the sliders are not properly connected to the plot_sine_wave function or there's an issue with the layout. To help you further, I need to see the changes you made to the code. If you could provide the code where you added the two additional sliders for w and p, I'll be able to diagnose the issue and provide a solution. You were right, I needed to add their arguments to the plot_sine_wave. I did that and it works. Thanks You're welcome! I'm glad to hear that the issue is resolved and the additional sliders for w and p are working correctly. If you have any more questions or need further assistance, please don't hesitate to ask. I'm here to help! When doing plots with higher frequency I get the following integration warning. :40: IntegrationWarning: The maximum number of subdivisions (50) has been achieved. If increasing the limit yields no improvement it is advised to analyze the integrand in order to determine the difficulties. If the position of a local difficulty can be determined (singularity, discontinuity) one will probably gain from splitting up the interval and calling the integrator on the subranges. Perhaps a special-purpose integrator should be used. arc_length_quad, _ = quad(arc_length_integrand, x1, x2, args=(A, w, p)) How can I address it succinctly? This warning is raised because the maximum number of subdivisions allowed by the quad function has been reached during the integration process. You can address this by increasing the limit parameter of the quad function. By default, the limit is set to 50, but you can increase it to a higher value to allow more subdivisions. Here's an example of how to modify the quad function call to increase the limit: arc_length_quad, _ = quad(arc_length_integrand, x1, x2, args=(A, w, p), limit=100) In this example, I've increased the limit to 100. You can adjust this value to see if it resolves the IntegrationWarning. However, keep in mind that increasing the limit may also increase the computation time. If increasing the limit doesn't help, you may need to consider using another integration method or splitting the integration interval into smaller subranges, as suggested in the warning message. setting the limit in the quad function to the value of 100 solved the problem. thank you very much. You're welcome! I'm glad to hear that increasing the limit to 100 solved the problem. If you have any more questions or need further assistance, please don't hesitate to ask. I'm here to help! Is there a way I can save an entire transcript of this chat for my blog? HTML would be okay, and preserving the dark mode would be nice also. I'm using blogger. Any suggestions? Yes, you can save the entire transcript of this chat and present it on your blog. Here's a simple way to convert the chat into an HTML format that preserves the dark mode: Copy the entire chat transcript. Open a plain text editor (such as Notepad or any other text editor that allows you to save files as .html). Paste the chat transcript into the text editor. Wrap the chat transcript in a <pre> tag and add the necessary HTML, head, and body tags, like this: <!DOCTYPE html> <html lang="en"> <head> <meta charset="UTF-8"> <meta name="viewport" content="width=device-width, initial-scale=1.0"> <title>Chat Transcript</title> <style> body { background-color: #1e1e1e; color: #ffffff; font-family: "Courier New", monospace; } pre { white-space: pre-wrap; word-wrap: break-word; } </style> </head> <body> <pre> [Paste the chat transcript here] </pre> </body> </html> Save the file with an .html extension (e.g., chat_transcript.html). Now you can open the saved file in any web browser and see the chat transcript with the dark mode. To embed the chat transcript on your Blogger blog, you can follow these steps: Sign in to your Blogger account and go to your blog dashboard. Create a new post or edit an existing one. Switch to the "HTML" view in the post editor. Copy the contents within the <pre> tag from the chat_transcript.html file you saved earlier. Paste the copied content within the <pre> tag in the post editor in the HTML view. Add the following CSS code within a <style> tag at the beginning of your post: <style> pre { background-color: #1e1e1e; color: #ffffff; font-family: "Courier New", monospace; white-space: pre-wrap; word-wrap: break-word; } </style> Switch back to the "Compose" view in the post editor, and you should see the chat transcript with dark mode. Add any additional content you'd like to include in your blog post and publish or update the post as usual. By following these steps, you can embed the chat transcript on your blog with dark mode styling. If you have any questions or need further assistance, please don't hesitate to ask. That is truly amazing! Thank you again! You're welcome! I'm glad I could help. If you have any more questions or need assistance in the future, don't hesitate to ask. I'm always here to help. Enjoy working on your blog! How do I copy the entire chat transcript without picking up the sidebar that contains the list of previous chats? To copy the entire chat transcript without picking up the sidebar, you can follow these steps: Click anywhere within the chat transcript to make it the active window. Press Ctrl + A (or Cmd + A on a Mac) to select all the text within the chat transcript. This should only select the text within the chat and not the sidebar. Press Ctrl + C (or Cmd + C on a Mac) to copy the selected text. Now you have the entire chat transcript copied without the sidebar. You can paste it into a text editor or any other application as needed.