Description
Compare oriented planar determinants before and after a supplied x-by-y shear.
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.
Compare oriented planar determinants before and after a supplied x-by-y shear.
When to use this tool
Use Vector 2d Shear Area Preservation Residual 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 first, second, shear. 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 before, after, residual. Use the output convention below.
Require two two-component vectors. Calculate their oriented determinant before and after replacing x by x+shear*y in both vectors, then return residual=after-before.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| first | First | Required. Array of Finite number.; maximum items 256. |
| second | Second | Required. Array of Finite number.; maximum items 256. |
| shear | Shear | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| before | Before | Required. Finite number. |
| after | After | Required. Finite number. |
| residual | Residual | Required. Finite number. |
{
"input": {
"first": [
2,
3
],
"second": [
4,
5
],
"shear": 2
},
"output": {
"before": -2,
"after": -2,
"residual": 0
}
}Limitations and assumptions
- The shear coefficient is supplied and no tolerance is used. The determinants are algebraically equal, but floating-point cancellation can leave a residual; this is not a geometric area measurement.
- 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 2d Complex Multiplication Pair
Multiply two real/imaginary component pairs as complex numbers.
- Vector Affine Combination Weight Residual
Sum weights and return residual=sum-1.