From Angles to Points: Understanding Coordinate Transformations from Polar to Cartesians

Coordinate transformation is a crucial technique in many fields, including sonar data processing, robotics, and computer graphics. In this tutorial, we will explore the mathematical foundations and practical applications of transforming coordinates from polar to Cartesian systems and vice versa. By the end of this guide, you’ll understand how to apply these transformations to accurately process and visualize sonar data.

This is essential for processing sonar, radar data, where measurements are often given in polar coordinates (azimuth, elevation, range) and need to be converted to Cartesian coordinates (x, y, z) for 3D reconstruction.

Key Concepts and Equations

Azimuth and elevation angles describe the position of a point in the sky or a 3D coordinate system. Azimuth measures the horizontal direction from a reference point (like north), while elevation measures the vertical position above or below the horizontal plane. These angles are essential in fields like astronomy, sonar, radar, and robotics for navigation and object tracking. Understanding and converting between these angles and Cartesian coordinates (x, y, z) allows for accurate data processing and visualization. Special handling of zero angles ensures precise transformations, making these concepts fundamental for interpreting directional data across various applications.

1. Azimuth (az) and Elevation (el) to Cartesian (x, y, z) Conversion

These equations transform polar coordinates into 3D Cartesian coordinates.

Equations:

• 𝑌=𝑟⋅cos⁡(el)⋅cos⁡(az)
• 𝑋=𝑟⋅cos⁡(el)⋅sin⁡(az)
• 𝑍=𝑟⋅sin⁡(el)

Variables:

• az: Azimuth angle, the angle from a reference direction (typically north) in the horizontal plane.
• el: Elevation angle, the angle from the horizontal plane.
• r: Range or distance from the origin (transducer) to the point.
• X, Y, Z: Cartesian coordinates representing the point in 3D space.

Trigonometric Functions:

• cos(θ): Cosine of angle θ, which gives the adjacent side over the hypotenuse in a right triangle.
• sin(θ): Sine of angle θ, which gives the opposite side over the hypotenuse in a right triangle.

Concepts:

• Azimuth: Measured clockwise from a reference direction in the horizontal plane. For example, 0° is typically north, 90° is east.
• Elevation: Measured upwards from the horizontal plane. Positive values indicate above the horizontal plane, while negative values indicate below.
• Range: Distance from the origin to the point.

2. Cartesian (x, y, z) to Azimuth (az) and Elevation (el) Conversion

These equations transform 3D Cartesian coordinates back into polar coordinates.

Equations:

Variables:

• X, Y, Z: Cartesian coordinates.
• θ: Elevation angle.
• ϕ: Azimuth angle.

Trigonometric Functions:

• atan2(𝑦,𝑥): Two-argument arctangent function, which returns the angle between the positive x-axis and the point (x, y). It is more robust than atan(y/x) as it considers the signs of both arguments to determine the correct quadrant.
• Euclidean distance formula, which computes the distance from the origin to the point (x, y) in 2D space.

Concepts:

• atan2(y,x): Unlike the regular arctangent function, atan2(y,x) returns the correct angle for all quadrants.
• Elevation and Azimuth from Cartesian: By calculating the angles using arctangent functions, you can determine the direction of a point in space relative to the origin.

3. Special Case Handling

When azimuth (az) and elevation (el) are both zero, there is a potential divide by zero error. The transformation logic must handle this gracefully, typically by setting theta to a default value since rotation doesn’t matter in this specific case.

Example Calculation

Given: Range r=1, Azimuth az=45^∘, Elevation el=30^∘

To Cartesian:

Back to Polar:

These transformations ensure that the sonar, radar data etc. can be accurately mapped into 3D space and back, which is crucial for tasks such as 3D reconstruction and visualization.

Trigonometric Function Overview

1. Sine and Cosine:
   • sin(θ) and cos(θ) are fundamental trigonometric functions that relate the angles of a right triangle to the lengths of its sides.
   • Used to project a point in 2D polar coordinates onto the x and y axes in Cartesian coordinates.
2. Arctangent (atan2):
   • atan2(y,x) is an extension of the arctangent function that returns the angle of a point (x, y) from the origin, considering all four quadrants.
   • It avoids ambiguity in angle calculations that regular atan(y/x) cannot handle due to its limited range of −π/2 to π/2.

Concept Summary

• Polar to Cartesian Conversion: Transforms range, azimuth, and elevation into a 3D point.
• Cartesian to Polar Conversion: Transforms a 3D point back into range, azimuth, and elevation.
• Trigonometric Functions: Sine and cosine for projection, and atan2 for robust angle calculations.

Understanding these concepts and equations is crucial for accurately processing and visualizing sonar data, enabling effective 3D reconstruction and analysis.