Cd Matrix Image Axis Angles

Infer both image-axis angles in the WCS tangent plane from CD columns, exposing skew rather than assuming an orthogonal scalar rotation.

Description

A CD matrix can encode rotation, scale and skew.

A CD matrix can encode rotation, scale and skew. Examining both column directions exposes nonorthogonal image axes rather than assuming one scalar image rotation describes the matrix.^[1]

Infer both image-axis angles in the WCS tangent plane from CD columns, exposing skew rather than assuming an orthogonal scalar rotation.

When to use Cd Matrix Image Axis Angles

Use this tool to infer both image-axis angles in the WCS tangent plane from CD columns, exposing skew rather than assuming an orthogonal scalar rotation.

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 Cd Matrix Image Axis Angles 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 cdMatrix using the field table and calculation rules, then run the tool.
Read firstAxisAngleDegrees, secondAxisAngleDegrees, axisSeparationDegrees. 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
cdMatrix	Cd matrix	list of list of number	Required

Enter a finite nonsingular 2×2 CD matrix. The tool treats each column as an image-axis direction in the WCS tangent plane, returns their angles wrapped to [0,360), and their absolute shortest separation from 0 through 180 degrees. Angles are measured from the first tangent-plane axis.

axis1=atan2(CD21,CD11); axis2=atan2(CD22,CD12)

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

Output field	Meaning	Value type
firstAxisAngleDegrees	First axis angle degrees	number
secondAxisAngleDegrees	Second axis angle degrees	number
axisSeparationDegrees	Axis separation degrees	number

json

{
  "input": {
    "cdMatrix": [
      [
        0,
        -1
      ],
      [
        1,
        0
      ]
    ]
  },
  "output": {
    "firstAxisAngleDegrees": 90,
    "secondAxisAngleDegrees": 180,
    "axisSeparationDegrees": 90
  }
}

Limitations and assumptions

This does not report sky position angle at every pixel or decompose all scales and distortions. A nonsingular matrix is required, and axis angles must not be assumed to follow the east-of-north bearing convention.

Numeric inputs must be finite. Follow the calculation-specific rules for overflow, underflow and exceptional cases. Matrices are limited to 128 rows and 128 entries per row; other lists are limited to 100,000 entries. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Tan Wcs Pixel To Sky: Apply inverse gnomonic TAN projection after a CD transform; pixel and reference-pixel values must use the same indexing convention, and SIP is not applied here. For a complementary task, use Sip Distortion Evaluation: Validate and evaluate forward SIP distortion terms for reference-pixel offsets; linear and constant terms belong in WCS, not SIP.

References

Cd Matrix Image Axis Angles — tool.io

Similar or alternative tools

Tan Wcs Pixel To Sky
Apply inverse gnomonic TAN projection after a CD transform; pixel and reference-pixel values must use the same indexing convention, and SIP is not applied here.
Sip Distortion Evaluation
Validate and evaluate forward SIP distortion terms for reference-pixel offsets; linear and constant terms belong in WCS, not SIP.