Description
Return the first-order active Rodrigues rotation v+angle*(unitAxis cross v), with angle in radians.
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 first-order active Rodrigues rotation v+angle*(unitAxis cross v), with angle in radians.
When to use this tool
Use Vector Axis Angle Infinitesimal Step 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 firstOrderVector. Use the output convention below.
Require three-component vector and nonzero axis. Normalize the axis and return firstOrderVector=vector+angle*(unitAxis cross vector).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 | Small signed angle in radians for a first-order approximation. | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| firstOrderVector | First Order Vector | Required. Array of Finite number.; maximum items 256. |
{
"input": {
"vector": [
1,
0,
0
],
"axis": [
0,
0,
1
],
"angle": 0.1
},
"output": {
"firstOrderVector": [
1,
0.1,
0
]
}
}Limitations and assumptions
- Angle is in radians and the expression is only a first-order small-angle approximation. It does not preserve length exactly, bound approximation error, or replace a full finite-angle rotation.
- 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
- Vector Rodrigues Axis Angle Sensitivity
Calculate the angle derivative of a Rodrigues rotation for a fixed supplied unit axis.
- Vector Angle Bisector Equal Norm Case
Normalize the sum of equal-norm vectors to obtain a bisector direction, rejecting a zero sum.