axis angle to rotor

Daniel Parker

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17-03-2025

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The conversion from axis-angle representation to rotor representation is a fundamental operation in 3D rotations used extensively in robotics, computer graphics, and aerospace engineering. This article will delve into the mathematical details of this transformation and provide practical examples. Understanding this process is crucial for anyone working with quaternions and rotational kinematics.

Understanding the Representations

Before diving into the conversion, let's briefly define the two representations:

Axis-Angle Representation: This representation describes a rotation using a unit vector v (the axis of rotation) and an angle θ (the angle of rotation about that axis). The axis points in the direction of rotation according to the right-hand rule.

Rotor Representation (Quaternion): A rotor is a unit quaternion, a four-dimensional hypercomplex number often used to represent rotations in 3D space. A rotor q can be expressed as:

q = cos(θ/2) + vsin(θ/2)

where:

• θ is the angle of rotation
• v = (v_x, v_y, v_z) is the unit vector representing the axis of rotation.

The Conversion Process: Axis-Angle to Rotor

The conversion from axis-angle to rotor is relatively straightforward. Given the axis-angle representation (v, θ), we can directly substitute the values into the rotor equation:

1. Normalize the Axis Vector: Ensure that the axis vector v is a unit vector (||v|| = 1). If not, normalize it by dividing by its magnitude:

   v_normalized = v / ||v||

2. Calculate Half-Angle: Divide the angle of rotation θ by 2:

   θ' = θ / 2

3. Construct the Rotor: Substitute the normalized axis vector and half-angle into the rotor equation:

   q = cos(θ') + v_xsin(θ')i + v_ysin(θ')j + v_zsin(θ')k

   Where i, j, k are the quaternion basis vectors. This represents the rotor in its scalar-vector form.

Example

Let's consider a rotation of 90 degrees about the z-axis.

• Axis-Angle: v = (0, 0, 1), θ = 90 degrees (π/2 radians)

1. Normalization: v is already normalized.

2. Half-Angle: θ' = (π/2) / 2 = π/4 radians

3. Rotor Construction:

   q = cos(π/4) + 0sin(π/4)i + 0sin(π/4)j + 1sin(π/4)k

   q = cos(π/4) + ksin(π/4)

   q = 1/√2 + k(1/√2)

This gives us the rotor representation of the rotation. This rotor can then be used for various applications, including rotating vectors or other rotors.

Practical Applications

The axis-angle to rotor conversion is critical in various fields:

• Robotics: Representing and manipulating robot arm orientations.
• Computer Graphics: Efficiently performing 3D rotations in game engines and animation software.
• Aerospace Engineering: Modeling and simulating aircraft or spacecraft attitude control.
• 3D Modeling: Transforming objects within 3D modeling software.

Further Considerations

While this conversion is straightforward, remember that there are multiple ways to represent rotations. Understanding the strengths and weaknesses of each representation is crucial for selecting the most suitable approach for a specific application. For instance, while rotors avoid gimbal lock, axis-angle representation can sometimes be more intuitive for visualization.

This article provides a foundational understanding of the axis-angle to rotor conversion. Further exploration into quaternion algebra and its applications will broaden your understanding of 3D rotations. Remember to utilize libraries like Eigen or similar for efficient numerical computations in your implementation.