Description
Return the component along a supplied nonzero axis, which is invariant under rotation about that axis.
A vector is an ordered set of components in a chosen basis. Its norm measures magnitude, while dot products, projections, and coordinate transformations describe different relationships. Component order, angle units, normalization, and numerical tolerances determine the meaning of a computed result.
Return the component along a supplied nonzero axis, which is invariant under rotation about that axis.
When to use this tool
Use Vector Rodrigues Axial Component when you need the particular coordinate operation, derivative, residual, or domain check under a known basis convention.
How to use the tool and how it works
- Enter vector, axis, angle. Keep parallel arrays aligned. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read axial, invariant. Inspect invariant before interpreting the associated values.
Require aligned vector and axis dimensions and a positive axis squared norm. Return axial=axis*dot(vector,axis)/dot(axis,axis) and invariant true.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| vector | Vector | Required. Array of Finite number.; maximum items 256. |
| axis | Axis | Required. Array of Finite number.; maximum items 256. |
| angle | Finite supplied angle; does not affect the returned axial projection. | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| axial | Axial | Required. Array of Finite number.; maximum items 256. |
| invariant | Statement of the axial-component property, not a numerical rotation test. | Required. Boolean. |
{
"input": {
"vector": [
2,
3,
4
],
"axis": [
0,
0,
2
],
"angle": 1
},
"output": {
"axial": [
0,
0,
4
],
"invariant": true
}
}Limitations and assumptions
- Angle is finite but does not enter the returned projection. Invariant states the axial-component property rather than numerically testing a performed rotation; the tool does not rotate the vector.
- The field tables state structural bounds; the mechanics also specify cross-field constraints. Malformed inputs and unsupported domains cause errors. A successful false flag, empty branch value, or zero placeholder has the meaning stated above and is distinct from an input error.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Signed zero is exported as zero.
Alternative or Complementary methods
Choose a complete vector or matrix operation when a single component or residual is insufficient. For sensitive near-zero or near-singular cases, inspect numerical conditioning and use an appropriate precision and tolerance.
References
Similar or alternative tools
- Signed Component Ratio
Calculate a signed scalar component ratio and report whether both nonzero components share a sign.
- Vector Component Cancellation Ratio
Sum components for signedSum and their magnitudes for absoluteMass.