# 5 Best Ways to Plot Points on the Surface of a Sphere in Python’s Matplotlib

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π‘ Problem Formulation: Visualizing data in three dimensions is a common challenge in computational fields. When plotting on a sphere’s surface, the input includes spherical coordinates or Cartesian coordinates, and the desired output is a graphical representation of those points on the sphere. This article guides you through different methods to achieve this using Python and Matplotlib.

## Method 1: Using Scatter in 3D Plot

Matplotlib’s `mplot3d` toolkit extends the framework to include 3D plotting capabilities. We can plot points on a sphereβs surface using the `Axes3D.scatter()` function. This method requires the conversion of spherical coordinates to Cartesian coordinates, which are then passed to the scatter function to render the 3D plot.

Here’s an example:

```import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import numpy as np

# Define sphere parameters
r = 1
phi, theta = np.mgrid[0:2*np.pi:100j, 0:np.pi:50j]

# Convert to Cartesian coordinates
x = r * np.sin(theta) * np.cos(phi)
y = r * np.sin(theta) * np.sin(phi)
z = r * np.cos(theta)

fig = plt.figure()
ax.scatter(x, y, z)

plt.show()```

The output is a 3D graph showing points distributed across the sphere’s surface.

This snippet first defines the sphere’s parameters and generates a grid of points in spherical coordinates (phi, theta). These are converted to Cartesian coordinates (x, y, z) and plotted using a 3D scatter plot utilising Matplotlib functions.

## Method 2: Plotting with a Parametric Surface

The parametric surface method involves explicitly mapping spherical coordinates to their Cartesian counterparts and using Matplotlibβs `plot_surface` function. This method excels in creating a continuous surface instead of discrete points and offers the ability to customize transparency and colormap for improved aesthetics.

Here’s an example:

```import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import numpy as np

phi, theta = np.linspace(0, 2 * np.pi, 100), np.linspace(0, np.pi, 100)
phi, theta = np.meshgrid(phi, theta)
x = np.sin(theta) * np.cos(phi)
y = np.sin(theta) * np.sin(phi)
z = np.cos(theta)

fig = plt.figure()
ax.plot_surface(x, y, z, color='b', alpha=0.5)

plt.show()```

The output displays a semitransparent sphere with a bluish tint.

After setting up the meshgrid of spherical coordinates, this code converts them into Cartesian and plots the points using a surface plot. The sphere’s properties, such as color and transparency, are customizable.

## Method 3: Wireframe Sphere

To visualize the lattice structure of a sphere, the `wireframe` method is most suited. It creates a 3D wireframe representation which emphasizes the globe-like structure without filling the surface. Matplotlib’s `plot_wireframe` is used to realize this visualization.

Here’s an example:

```import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

theta, phi = np.linspace(0, 2 * np.pi, 100), np.linspace(0, np.pi, 100)
theta, phi = np.meshgrid(theta, phi)
r = 1
x = r * np.sin(phi) * np.cos(theta)
y = r * np.sin(phi) * np.sin(theta)
z = r * np.cos(phi)

fig = plt.figure()
ax.plot_wireframe(x, y, z, color='r')

plt.show()```

The resulting output is a 3D red wireframe sphere.

Similarly to Method 2, this wireframe sphere uses meshgrid coordinates in spherical form converted to Cartesian and plotted with `plot_wireframe`. The wireframe presents a skeletal view of the sphere.

## Method 4: Custom Point Distribution on Sphere

For cases where specific distribution of points on a sphere is required, such as a Fibonacci lattice or uniformly distributed points, custom algorithms can be created to produce these patterns. Afterward, points are plotted using the `scatter` function.

Here’s an example:

```import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import numpy as np

def fibonacci_sphere(samples=1):

points = []
phi = np.pi * (3. - np.sqrt(5.))  # golden angle in radians

for i in range(samples):
y = 1 - (i / float(samples - 1)) * 2
radius = np.sqrt(1 - y * y)

theta = phi * i

points.append((x, y, z))

return points

fig = plt.figure()
points = fibonacci_sphere(200)
x, y, z = zip(*points)
ax.scatter(x, y, z)

plt.show()```

The plot reveals points evenly distributed across the sphere’s surface.

Using the concept of the golden angle, this code snip generates a Fibonacci lattice of points on the sphere, which is theorized to be one of the most uniform distributions possible. These points are then plotted on a sphere in 3D.

## Bonus One-Liner Method 5: Using the Quiver Function

The `quiver` function offers a quick one-liner for adding vector representations to the surface of a sphere. While not strictly plotting points, it provides a unique perspective on depicting directional data on a spherical surface.

Here’s an example:

```import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import numpy as np

fig = plt.figure()
x, y, z = np.meshgrid(np.array([0]), np.array([0]), np.array([0]))
u, v, w = np.array([1]), np.array([1]), np.array([1])
ax.quiver(x, y, z, u, v, w)

plt.show()```

An arrow originates from the center and touches the sphere’s surface, indicating direction.

This concise code uses `quiver` to add a single vector, visually striking for representing a specific point or direction on the sphere.

## Summary/Discussion

• Method 1: Scatter in 3D Plot. Ideal for plotting discrete data points. Each point can be styled individually, but can become cluttered with many points.
• Method 2: Parametric Surface. Best for a continuous sphere surface. Provides a solid aesthetic but may obscure inner structures.
• Method 3: Wireframe Sphere. Useful for a structural perspective. Does not represent surface continuity and can become visually complex.
• Method 4: Custom Point Distribution. Suitable for specialized point distributions like Fibonacci lattices. May require additional code to generate the specific patterns.
• Method 5: Using Quiver Function. Quick and effective for directional data. Represents vectors rather than points, offering different visual insights.