Can Plotly Draw a Klein Bottle with rotate, zoom, and pan

brombo · March 30, 2025, 12:05am

I like to know if Plotly can draw a Klein bottle like the following link using Asymptote with interactive rotate, zoom, and pan?

Klein Bottle Asymptote Webgl

DaveG · April 3, 2025, 4:15am

Did you try chatgpt?

provide the python code to a) create a digital Klein bottle 3D model and b) plot it using Plotly charts

ChatGPT said:

To create a 3D model of a digital Klein bottle and plot it using Plotly, we’ll approach the problem in two parts:

Part (a): Create a digital Klein bottle 3D model

The Klein bottle is a non-orientable surface with no boundary. In mathematical terms, it can be represented parametrically, but its construction requires identifying opposite sides of a square (i.e., “gluing” opposite sides together in a specific way).

One common parameterization of a Klein bottle is as follows:

r(u,v)=[(cos⁡(u)⋅(1+sin⁡(v)))(sin⁡(u)⋅(1+sin⁡(v)))(sin⁡(v)⋅cos⁡(u/2))]\mathbf{r}(u, v) = \begin{bmatrix} \left( \cos(u) \cdot (1 + \sin(v)) \right) \ \left( \sin(u) \cdot (1 + \sin(v)) \right) \ \left( \sin(v) \cdot \cos(u / 2) \right) \end{bmatrix}r(u,v)=(cos(u)⋅(1+sin(v)))(sin(u)⋅(1+sin(v)))(sin(v)⋅cos(u/2))

Where u,v∈[0,2π]u, v \in [0, 2\pi]u,v∈[0,2π]. This creates the surface of the Klein bottle with two circular parameters: uuu for the horizontal loop and vvv for the vertical loop.

Part (b): Plot the Klein bottle using Plotly

We can use Plotly’s plotly.graph_objects to create a 3D surface plot.

Below is the Python code that generates the Klein bottle and visualizes it using Plotly.

python

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import numpy as np
import plotly.graph_objects as go

# Parameters for the Klein bottle
u = np.linspace(0, 2 * np.pi, 100)
v = np.linspace(0, 2 * np.pi, 100)

# Create a meshgrid for u and v
u, v = np.meshgrid(u, v)

# Parametric equations for Klein bottle
x = (np.cos(u) * (1 + np.sin(v)))
y = (np.sin(u) * (1 + np.sin(v)))
z = np.sin(v) * np.cos(u / 2)

# Create 3D plot with Plotly
fig = go.Figure(data=[go.Surface(z=z, x=x, y=y, colorscale='Viridis')])

# Set the layout of the plot
fig.update_layout(
    title='Klein Bottle 3D',
    scene=dict(
        xaxis_title='X',
        yaxis_title='Y',
        zaxis_title='Z',
        aspectmode='cube'
    ),
    autosize=True
)

# Show the plot
fig.show()

Explanation:

Parameterization of the Klein Bottle:

uuu and vvv are used as parameters for generating the surface.
The parametric equations for xxx, yyy, and zzz are based on a mathematical representation of the Klein bottle surface.

Meshgrid:

np.meshgrid(u, v) creates a grid of points on the uuu-vvv plane. This grid is used to evaluate the parametric equations.

Plotly 3D Surface Plot:

go.Surface() is used to plot the 3D surface defined by xxx, yyy, and zzz.
The colorscale='Viridis' argument adds a color gradient to the surface.
The scene layout sets the labels and aspect ratio of the plot.

Plot:

fig.show() renders the plot interactively.

Dependencies:

You’ll need numpy and plotly for the code to work:

bash