Frame Small Angle Rotation

Compare exact Rodrigues and first-order active frame rotations for a supplied small rotation vector; passive coordinate transformations require the opposite sign.

Description

A small active rotation can be approximated by adding a cross-product term to a vector.

A small active rotation can be approximated by adding a cross-product term to a vector. Comparing that result with an exact rotation exposes the approximation's higher-order difference.^[1]

Compare exact Rodrigues and first-order active frame rotations for a supplied small rotation vector; passive coordinate transformations require the opposite sign.

When to use Frame Small Angle Rotation

Use this tool to compare exact Rodrigues and first-order active frame rotations for a supplied small rotation vector; passive coordinate transformations require the opposite sign.

Prepare the source, detector or coordinate measurements. Confirm frame, epoch, angular units, axis directions and the stated RA proper-motion convention before entering values.

How to use Frame Small Angle Rotation and understand its results

Prepare the source, detector or coordinate measurements. Confirm frame, epoch, angular units, axis directions and the stated RA proper-motion convention before entering values.
Enter inputVector, rotationVectorRadians using the field table and calculation rules, then run the tool.
Read exactRotatedVector, linearizedRotatedVector. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
inputVector	Input vector	list of number	Required
rotationVectorRadians	Rotation vector radians	list of number	Required

Enter finite three-component input and rotation vectors, the latter in radians with norm at most 0.01. The tool returns exact and first-order rotated vectors. A zero rotation copies the input directly; nonzero rotation normalizes its axis for Rodrigues evaluation.

linearized=v+rotationVector×v; exact=Rodrigues(v,rotationAxis,|rotationVector|)

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
exactRotatedVector	Exact rotated vector	list of number
linearizedRotatedVector	Linearized rotated vector	list of number

json

{
  "input": {
    "inputVector": [
      1,
      0,
      0
    ],
    "rotationVectorRadians": [
      0,
      0,
      0.001
    ]
  },
  "output": {
    "exactRotatedVector": [
      0.9999995000000417,
      0.0009999998333333417,
      0
    ],
    "linearizedRotatedVector": [
      1,
      0.001,
      0
    ]
  }
}

Limitations and assumptions

This uses active-vector rotation; passive coordinate-axis transformations require the opposite sign convention. It does not fit a frame rotation, propagate uncertainty or allow arbitrary large rotations.

Numeric inputs must be finite. Follow the calculation-specific rules for overflow, underflow and exceptional cases. Lists are limited to 100,000 entries; the rules above may require smaller limits. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Equatorial Ecliptic Axis Rotation: Rotate a sky direction between equatorial and ecliptic axes using supplied obliquity; the same epoch must be used for both frames. For a complementary task, use Icrs To Galactic Coordinates: Apply the fixed ICRS/J2000 Galactic rotation to a sky direction; this is not an epoch propagation.

References

Frame Small Angle Rotation — tool.io

Similar or alternative tools

Equatorial Ecliptic Axis Rotation
Rotate a sky direction between equatorial and ecliptic axes using supplied obliquity; the same epoch must be used for both frames.
Icrs To Galactic Coordinates
Apply the fixed ICRS/J2000 Galactic rotation to a sky direction; this is not an epoch propagation.