Quaternion Unit Relative Rotation

Description

Quaternion order is scalar-first [w,x,y,z], Hamilton product and right-handed active rotations; compute the relative active rotation to * conjugate(from), normalized to remove numerical drift.

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; compute the relative active rotation to * conjugate(from), normalized to remove numerical drift.

When to use this tool

Use this tool to compute the relative active rotation to * conjugate(from), normalized to remove numerical drift.

How to use the tool and how it works

Enter from, to 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.

Require unit from and to records. Compute to × conjugate(from) using the Hamilton product, then normalize the result.^[1]

Input field	Meaning and units	Accepted input
from	Initial unit quaternion in [w, x, y, z] order.	Dense list of at most 128 finite numbers; integer/shape constraints above also apply.
to	Target unit 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.

json

{
  "input": {
    "from": [
      1,
      0,
      0,
      0
    ],
    "to": [
      0,
      1,
      0,
      0
    ]
  },
  "output": {
    "result": {
      "quaternion": [
        0,
        1,
        0,
        0
      ]
    }
  }
}

Limitations and assumptions

Multiplication order is significant; this is the stated scalar-first rotation convention.
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

Tool behavior and example: Quaternion Unit Relative Rotation — tool.io

Similar or alternative tools

Unit Quaternion Vector Rotation
Quaternion order is scalar-first [w,x,y,z], Hamilton product and right-handed active rotations; rotate a finite three-vector by a validated unit quaternion without applying translation.
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.