Polar Coordinates Library

A pure C# library for working with polar 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 Polar Coordinates?

Polar coordinates provide a way to represent points in a plane using a distance from a reference point (the radius) and an angle from a reference direction (theta). Unlike Cartesian coordinates (x, y), which define a point by horizontal and vertical displacement, polar coordinates define a point as:

• Radius (r): The distance from the origin to the point.
• Theta (θ): The angle (in radians) measured counterclockwise from the positive x-axis.

Direction of Theta (θ)

• Theta (θ) is measured in radians.
• Positive angles rotate counterclockwise from the positive x-axis.
• Negative angles rotate clockwise from the positive x-axis.
• The angle is typically normalized to the range [0, 2π).

Functionality of the Polar Class

This Polar class provides methods to work with polar coordinates, including conversions, measurements, and calculations.

Constructor

public Polar(double radius, double theta)

• Initializes a Polar object with a specified radius and angle.

Conversion Methods

FromCartesian(Vector2 point)

public static Polar? FromCartesian(Vector2 point)

• Converts a Cartesian coordinate (x, y) into polar coordinates.
• Returns null if the input contains NaN values.
• Uses the formula:
  • r = sqrt(x² + y²)
  • θ = atan2(y, x)

ToCartesian()

public Vector2 ToCartesian()

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

Measurement Functions

ArcLength(double radius, double startTheta, double endTheta)

public static double ArcLength(double radius, double startTheta, double endTheta)

• Computes the arc length of a circular segment between two angles.
• Formula: Arc Length = r * |θ₂ - θ₁|.
• Returns NaN if the radius is negative.

SectorArea(double radius, double startTheta, double endTheta)

public static double SectorArea(double radius, double startTheta, double endTheta)

• Calculates the area of a sector of a circle.
• Formula: Sector Area = (1/2) * r² * |θ₂ - θ₁|.
• Returns NaN if the radius is negative.

Distance(Polar? p1, Polar? p2)

public static double Distance(Polar? p1, Polar? p2)

• Computes the Euclidean distance between two points given in polar form.
• Uses Cartesian conversion for distance calculation:
  • distance = sqrt((x₁ - x₂)² + (y₁ - y₂)²)
• Returns NaN if either point is null.

Angle Normalization

NormalizeAngle(double theta)

public static double NormalizeAngle(double theta)

• Ensures that an angle is in the range [0, 2π).
• Uses modular arithmetic to adjust negative angles.

Calculus and Curvature

TangentSlope()

public double TangentSlope()

• Computes the slope of the tangent line at the given polar coordinate.
• Uses Cartesian form: slope = -x / y.
• Returns NaN if the denominator is zero.

Curvature()

public double Curvature()

• Computes the curvature at the given point.
• Formula: Curvature = 1 / Radius.
• Returns NaN if the radius is zero.

PolarSlope()

public double PolarSlope()

• Computes the radial slope dr/dθ at the given point.
• Returns the radius as the slope.

Curve Fitting

IsOnCurve(Func<double, double> radiusFunction, double tolerance = 1e-10)

public bool IsOnCurve(Func<double, double> radiusFunction, double tolerance = 1e-10)

• Checks if the point lies on a given polar curve r = f(θ).
• Uses absolute difference to determine if Radius ≈ f(Theta).

Usage Example

Vector2 point = new Vector2(3, 4);
Polar? polar = Polar.FromCartesian(point);
if (polar != null)
{
    Console.WriteLine($"Radius: {polar.Radius}, Theta: {polar.Theta}");
    Vector2 cartesian = polar.ToCartesian();
    Console.WriteLine($"Converted back: ({cartesian.X}, {cartesian.Y})");
}

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

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