Axis-Angle Calculator

An axis-angle calculator handles the three most common representations of 3D rotations: axis-angle (a unit vector plus a rotation angle), quaternions (a four-component number w + xi + yj + zk), and rotation vectors (the axis scaled by the angle). These representations are equivalent — a 90° rotation around the Z-axis can be expressed as axis (0,0,1) with angle 90°, as the quaternion (0.7071, 0, 0, 0.7071), or as the rotation vector (0, 0, 1.5708).

The calculator converts any of the three input types into all three output types simultaneously and also computes the 3×3 rotation matrix using Rodrigues' rotation formula. This is essential in computer graphics, robotics, aerospace engineering, and game development — anywhere that objects rotate in 3D space. Quaternions avoid gimbal lock, rotation vectors are compact (three numbers), and axis-angle is the most intuitive to reason about.

The conversion formulas are bidirectional. Axis-angle to quaternion: q = (cos(θ/2), ǔ sin(θ/2)). Quaternion to axis-angle: θ = 2·arccos(w), then axis = (x,y,z)/sin(θ/2). The rotation matrix uses Rodrigues' formula.

1. Choose input type: Select Axis-Angle, Quaternion, or Rotation Vector.
2. Enter values: For axis-angle, enter the three axis components and the angle in degrees. For quaternion, enter w, x, y, z. For rotation vector, enter the three vector components (axis scaled by angle in radians).
3. Read all equivalent forms: The calculator outputs the normalized axis-angle pair, the unit quaternion, the rotation vector, and the full 3×3 rotation matrix — all instantly.

Axis-angle representations are fundamental in robotics, where robot arm joints rotate about specific axes. Forward and inverse kinematics calculations use quaternions or rotation matrices for composing multiple joint rotations. In aerospace, spacecraft attitude is controlled using quaternions because they avoid the gimbal lock problem that plagues Euler angles — a spacecraft can rotate through any orientation without singularities.

In computer animation and game engines, quaternions are the standard for interpolating between orientations (slerp — spherical linear interpolation). A character turning from facing north to facing northeast involves a smooth rotation that quaternions handle with uniform angular velocity. In computer vision, camera pose estimation and bundle adjustment use rotation vectors for compact optimization because they have only three parameters with no constraints.