3D Spherical Coordinates - Converting Cartesian to Spherical Coordinates for Plotting

3D spherical coordinates are a coordinate system commonly used to represent point positions in three-dimensional space, consisting of three parameters: radius, polar angle, and azimuthal angle. Compared to Cartesian coordinates (x,y,z), spherical coordinates can more intuitively describe point distribution patterns in certain scenarios (such as astronomy and physical field analysis).

The core conversion logic involves: - Radius (r) representing the distance from the point to the origin - Polar angle (θ) being the angle between the point and the z-axis - Azimuthal angle (φ) being the angle between the point's projection on the xy-plane and the x-axis

The conversion formulas are: x = r·sinθ·cosφ y = r·sinθ·sinφ z = r·cosθ

In 3D plotting, this conversion is commonly used for: - Direct input of distance and angle parameters when simulating celestial orbits - Visualizing radial distributions in electric/magnetic field visualizations - Creating parametric spherical models (such as globes)

Mainstream tools like Matplotlib's 3D module or Mayavi can implement mutual conversion plotting between Cartesian and spherical coordinates. The key technique lies in maintaining boundary handling for θ∈[0,π] and φ∈[0,2π] to avoid numerical jumps at surface seams. For scenarios requiring annotations, it's recommended to use both coordinate systems: use spherical coordinates for locating data points, and Cartesian coordinates for displaying grids and labels.

Code implementation typically involves creating conversion functions: def cartesian_to_spherical(x, y, z): r = np.sqrt(x**2 + y**2 + z**2) theta = np.arccos(z/r) phi = np.arctan2(y, x) return r, theta, phi def spherical_to_cartesian(r, theta, phi): x = r * np.sin(theta) * np.cos(phi) y = r * np.sin(theta) * np.sin(phi) z = r * np.cos(theta) return x, y, z

When plotting, ensure proper handling of angle ranges using numpy's arctan2 function for azimuthal angles and arccos for polar angles to maintain mathematical consistency across all quadrants.