Spherical2

 Below, patch1:

                                                                                       

Asked Gemini to check if these coordinates actually produced a spherical triangle:

I am working with patches on a sphere. Can you tell me if the area listed for these points is a proper spherical triangle?

                                                                               

Conclusion

Yes, the area listed (7.32586) is correct. It accurately represents the area of a spherical triangle formed by those three vertices on a sphere with a radius of 4.

Would you like me to calculate the side lengths (arc lengths) between these points, or perhaps help you calculate the area for a different patch?

                                                                    *     *     *

                                                                                 

import numpy as np
import matplotlib.pyplot as plt
import mpl_toolkits
from mpl_toolkits.mplot3d import Axes3D, art3d

# Define the vertices from your image
vertices = np.array([
    [1.43328777, 3.5667342, 1.10638752],
    [-2.76614052, -1.52383864, 2.45486912],
    [0.43402442, 3.94186737, 0.52278524]
])


def get_arc(p1, p2, points=50):
    """Generates points along the great circle arc between p1 and p2."""
    # Normalize vectors to ensure they are on the sphere surface
    p1_norm = p1 / np.linalg.norm(p1)
    p2_norm = p2 / np.linalg.norm(p2)

    # Calculate the angle between vectors
    dot = np.dot(p1_norm, p2_norm)
    angle = np.arccos(np.clip(dot, -1.0, 1.0))

    # Interpolate between vectors
    t = np.linspace(0, 1, points)
    # Slerp formula (Spherical Linear Interpolation)
    sin_angle = np.sin(angle)
    arc = (np.sin((1 - t) * angle)[:, None] * p1 +
           np.sin(t * angle)[:, None] * p2) / sin_angle
    return arc


# Setup Plot
fig = plt.figure(figsize=(10, 8))
ax = fig.add_subplot(111, projection='3d')

# 1. Draw a wireframe sphere for context (Radius = 4)
u, v = np.mgrid[0:2 * np.pi:30j, 0:np.pi:20j]
x = 4 * np.cos(u) * np.sin(v)
y = 4 * np.sin(u) * np.sin(v)
z = 4 * np.cos(v)
ax.plot_wireframe(x, y, z, color="gray", alpha=0.1, linewidth=0.5)

# 2. Draw the arcs (the sides of the triangle)
edges = [(0, 1), (1, 2), (2, 0)]
for i, j in edges:
    arc = get_arc(vertices[i], vertices[j])
    ax.plot(arc[:, 0], arc[:, 1], arc[:, 2], color='red', linewidth=3)

# 3. Plot the vertices
ax.scatter(vertices[:, 0], vertices[:, 1], vertices[:, 2], color='black', s=50)

# Formatting
ax.set_title("Spherical Triangle Patch (R=4)")
ax.set_xlabel("X")
ax.set_ylabel("Y")
ax.set_zlabel("Z")
ax.set_box_aspect([1, 1, 1])  # Equal aspect ratio

plt.show()

How the code works

The Sphere: Since we verified the radius is R=4, the code creates a wireframe sphere to act as a reference grid.SLERP (Spherical Linear Interpolation): Standard linear interpolation would draw a straight line through the sphere. 

To stay on the surface, we use SLERP to calculate the curved "Great Circle" path between your coordinates.

The Patch: The red lines represent the boundaries of your area (7.32586).Would you like me to modify the code to calculate the center point (centroid) of this patch as well?