Description
Quaternion order is scalar-first [w,x,y,z], Hamilton product and right-handed active rotations; invert a nonzero quaternion as normalized conjugate divided by its norm; nonfinite/unsafe results reject.
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; invert a nonzero quaternion as normalized conjugate divided by its norm; nonfinite/unsafe results reject.
When to use this tool
Use this tool to invert a nonzero quaternion as normalized conjugate divided by its norm; nonfinite/unsafe results reject.
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.quaternion. Use the ordering, boundary rules and absence conditions described below.
Normalize the nonzero quaternion, conjugate it, then divide again by the original norm to obtain its algebraic inverse.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.quaternion | Four result components in [w, x, y, z] order. |
{
"input": {
"quaternion": [
2,
0,
0,
0
]
},
"output": {
"result": {
"quaternion": [
0.5,
0,
0,
0
]
}
}
}Limitations and assumptions
- This is not merely conjugation for a non-unit quaternion; zero norm is rejected.
- 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
- Quaternion Conjugate Components
Quaternion order is scalar-first [w,x,y,z], Hamilton product and right-handed active rotations; conjugate vector components without assuming a unit norm or taking an inverse.
- Quaternion Normalized Components
Quaternion order is scalar-first [w,x,y,z], Hamilton product and right-handed active rotations; normalize a nonzero finite quaternion with hypot-based norm, rejecting zero/overflowing norms.