Spherical Polygon Area Calculation

This Python program calculates the area of a polygon defined by vertices specified in spherical coordinates on a sphere. It also visualizes the polygon and the sphere in a 3D plot.

Features

• Converts spherical coordinates (r, θ, φ) to Cartesian coordinates (x, y, z).
• Computes the angles between polygon edges using vector mathematics.
• Calculates the area of spherical polygons based on interior angles.
• Visualizes the sphere and the defined polygon in 3D.

Requirements

• Python 3.x
• numpy
• matplotlib

You can install the required libraries using:

pip install numpy matplotlib

Usage

1. Define Polygon Vertices: Specify the vertices of the polygon in spherical coordinates. Each vertex is defined by a tuple of the form (r, θ, φ), where:

   • r is the radius of the sphere,
   • θ is the azimuthal angle (longitude),
   • φ is the polar angle (colatitude).

   radius = 10
   polygon_vertices = [
       (radius, math.radians(0), math.radians(0)),       # Vertex 1
       (radius, math.radians(0), math.radians(90)),      # Vertex 2
       (radius, math.radians(90), math.radians(90))      # Vertex 3
   ]

2. Calculate Area: Ensure that there are at least three vertices, and then calculate the area of the polygon using the area_of_the_polygon function. This function requires the interior angles of the polygon and the radius.

   if len(polygon_vertices) < 3:
       print("At least 3 points are needed to form a polygon!")
   else:
       area = area_of_the_polygon(spherical_polygon_angles(polygon_vertices), radius)
       print('Area of the Polygon:', area)

3. Visualize: Visualize the polygon and the sphere in 3D using the visualize_sphere_and_polygon function. This will create a 3D plot showing the polygon on the sphere.

   visualize_sphere_and_polygon(polygon_vertices, radius)

Code Structure

• spherical_to_cartesian(r, theta, phi): Converts spherical coordinates to Cartesian coordinates.
• dot_product(v1, v2): Computes the dot product of two vectors.
• magnitude(v): Returns the magnitude of a vector.
• angle_between(v1, v2): Calculates the angle between two vectors.
• points_to_normal(p1, p2, p3): Computes the normal vector to the plane defined by three points.
• spherical_polygon_angles(vertices): Calculates the interior angles of the polygon.
• area_of_the_polygon(interior_angles, radius): Calculates the area of the polygon based on its angles.
• visualize_sphere_and_polygon(vertices, radius): Visualizes the sphere and polygon in 3D.

Example Output

When you run the program, you will see output similar to the following:

Area of the Polygon: 157.07963267948966

The visualization of the spherical polygon is displayed as a 3D plot. You can find the output image in the project directory after running the script.

License

This project is licensed under the MIT License.

Acknowledgments

• Matplotlib - A plotting library for Python that makes it easy to create visualizations.
• NumPy - A fundamental package for scientific computing with Python.

Contact

For any inquiries or feedback, feel free to reach out:

• Name: Nurpeissov Alisher
• Email: nurpeissov.a0818@gmail.com

Disclaimer

This project is for educational purposes only. It is recommended to verify the accuracy of the calculations and results for critical applications.