Spherical Coordinates Library

A pure C# library for working with spherical coordinates. No external dependencies except for Microsoft's unit testing, built for .NET 9.0.

This code is free to use under the MIT.

What Are Spherical Coordinates?

Spherical coordinates provide a way to represent points in three-dimensional space using a radius, an azimuthal angle, and a polar angle. Unlike Cartesian coordinates (x, y, z), which define a point by horizontal and vertical displacement, spherical coordinates define a point as:

• Radius (r): The distance from the origin to the point.
• Theta (θ): The azimuthal angle (in radians) measured counterclockwise from the positive x-axis in the xy-plane.
• Phi (φ): The polar angle (in radians) measured from the positive z-axis.

Direction of Theta (θ) and Phi (φ)

• Theta (θ) is measured in radians and represents the angle around the z-axis.
• Phi (φ) is measured in radians from the positive z-axis downward.
• The angle ranges are typically:
  • θ in [0, 2π)
  • φ in [0, π]

Functionality of the Spherical Class

This Spherical class provides methods to work with spherical coordinates, including conversions, measurements, and calculations.

Constructor

public Spherical(double radius, double theta, double phi)

• Initializes a Spherical object with a specified radius, azimuthal angle, and polar angle.

Conversion Methods

FromCartesian(Vector3 point)

public static Spherical FromCartesian(Vector3 point)

• Converts a Cartesian coordinate (x, y, z) into spherical coordinates.
• Returns NaN values if the input contains NaN values.
• Uses the formula:
  • r = sqrt(x² + y² + z²)
  • θ = atan2(y, x)
  • φ = acos(z / r)

ToCartesian()

public Vector3 ToCartesian()

• Converts a spherical coordinate to Cartesian form (x, y, z).
• Uses the formulas:
  • x = r * sin(φ) * cos(θ)
  • y = r * sin(φ) * sin(θ)
  • z = r * cos(φ)

Measurement Functions

SurfaceArea(double radius)

public static double SurfaceArea(double radius)

• Computes the surface area of a sphere.
• Formula: Surface Area = 4πr².
• Returns NaN if the radius is negative.

Volume(double radius)

public static double Volume(double radius)

• Computes the volume of a sphere.
• Formula: Volume = (4/3)πr³.
• Returns NaN if the radius is negative.

GreatCircleDistance(double radius, double phi1, double theta1, double phi2, double theta2)

public static double GreatCircleDistance(double radius, double phi1, double theta1, double phi2, double theta2)

• Computes the shortest distance between two points on a sphere.
• Uses the Haversine formula.
• Returns NaN if any input is NaN.

Spherical Segments and Zones

SphericalCap(double radius, double height)

public static double SphericalCap(double radius, double height)

• Computes the surface area of a spherical cap.
• Formula: Cap Area = 2πr h.
• Returns NaN if any input is NaN.

SphericalZone(double radius, double height1, double height2)

public static double SphericalZone(double radius, double height1, double height2)

• Computes the surface area of a spherical zone.
• Formula: Zone Area = 2πr |h₂ - h₁|.
• Returns NaN if any input is NaN.

SphericalSegmentVolume(double radius, double height)

public static double SphericalSegmentVolume(double radius, double height)

• Computes the volume of a spherical segment.
• Formula: Volume = (π h² (3r - h)) / 3.
• Returns NaN if any input is NaN.

SphericalSectorVolume(double radius, double phi)

public static double SphericalSectorVolume(double radius, double phi)

• Computes the volume of a spherical sector.
• Formula: Volume = (2/3)πr³(1 - cos(φ)).
• Returns NaN if any input is NaN.

Usage Example

Vector3 point = new Vector3(3, 4, 5);
Spherical spherical = Spherical.FromCartesian(point);
Console.WriteLine($"Radius: {spherical.Radius}, Theta: {spherical.Theta}, Phi: {spherical.Phi}");
Vector3 cartesian = spherical.ToCartesian();
Console.WriteLine($"Converted back: ({cartesian.X}, {cartesian.Y}, {cartesian.Z})");

This library provides robust mathematical operations for handling spherical coordinates, including conversions, distance calculations, and curvature analysis.

Copyright TranscendAI.tech 2025.
Authored by Warren Harding. AI assisted.