Description
Quaternion order is scalar-first [w,x,y,z], Hamilton product and right-handed active rotations; convert a unit rotation to canonical shortest axis-angle; zero rotation has an unknown/null axis, not an invented direction.
A quaternion has one scalar and three vector components. This family uses [w, x, y, z] unless a layout converter says otherwise. Unit quaternions can represent rotations, and q and −q describe the same rotation under this convention; component arithmetic and physical rotation interpretation are separate operations.
Quaternion order is scalar-first [w,x,y,z], Hamilton product and right-handed active rotations; convert a unit rotation to canonical shortest axis-angle; zero rotation has an unknown/null axis, not an invented direction.
When to use this tool
Use this tool to convert a unit rotation to canonical shortest axis-angle; zero rotation has an unknown/null axis, not an invented direction.
How to use the tool and how it works
- Enter quaternion using the field definitions below.
- Run the operation. Correct input errors before interpreting an output.
- Read result.radians, result.axis. Use the ordering, boundary rules and absence conditions described below.
Choose the canonical sign of a unit quaternion. Return angle 2 × atan2(vector-part norm, max(0, scalar)) and the normalized vector part as axis.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| quaternion | Four scalar-first components [w, x, y, z], unit length when required above. | Dense list of at most 128 finite numbers; integer/shape constraints above also apply. |
| Output field | Interpretation |
|---|---|
| result | Object containing the named fields below. |
| result.radians | Rotation angle or shorter rotation separation, in radians. |
| result.axis | Normalized three-component rotation axis, or null for identity. |
{
"input": {
"quaternion": [
1,
0,
0,
0
]
},
"output": {
"result": {
"radians": 0,
"axis": null
}
}
}Limitations and assumptions
- Angle is in radians; the identity rotation has a null axis rather than an arbitrary direction.
- When an operation requires unit input, the fixed acceptance condition is |norm − 1| ≤ 1e−10. The norm audit instead uses its supplied tolerance. Angles use radians; vector units are preserved by a unit rotation. Floating-point roundoff is possible.
- Every declared top-level input field is required and unexpected fields are rejected. Ordinary strings are limited to 8,192 UTF-16 code units and lists to 128 entries, with tighter operation-specific bounds where stated. JSON text has bounded nesting and node counts.
- Numeric outputs must be finite and no larger in absolute magnitude than the safe-integer limit; operations returning exact large integers use decimal text. Positions are zero-based and spans end exclusively unless the operation explicitly states a different convention.
Alternative or Complementary methods
Confirm component layout, multiplication order and coordinate-frame convention before using another library’s records. Normalize only when appropriate, and distinguish component-space tolerances from angular tolerances.
References
Similar or alternative tools
- Axis Angle to Unit Quaternion Record
Quaternion order is scalar-first [w,x,y,z], Hamilton product and right-handed active rotations; convert an explicit nonzero axis and angle to a unit quaternion; axis is normalized, radians are not silently clamped.
- Unit Quaternion Basis Vector Images
Quaternion order is scalar-first [w,x,y,z], Hamilton product and right-handed active rotations; return the three active basis-vector images as columns-as-vectors, not a transposed row-major matrix convention.