Description
Quaternion order is scalar-first [w,x,y,z], Hamilton product and right-handed active rotations; multiply left then right algebraically without commuting operands or assuming unit rotations.
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; multiply left then right algebraically without commuting operands or assuming unit rotations.
When to use this tool
Use this tool to multiply left then right algebraically without commuting operands or assuming unit rotations.
How to use the tool and how it works
- Enter left, right 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.
Compute the ordered Hamilton product left × right in [w, x, y, z] layout.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| left | Left quaternion in [w, x, y, z] order. | Dense list of at most 128 finite numbers; integer/shape constraints above also apply. |
| right | Right quaternion in [w, x, y, z] order. | 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": {
"left": [
0,
1,
0,
0
],
"right": [
0,
0,
1,
0
]
},
"output": {
"result": {
"quaternion": [
0,
0,
0,
1
]
}
}
}Limitations and assumptions
- The product is generally noncommutative; inputs need not be unit quaternions.
- 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
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