Description
Return a two-dimensional lensing Jacobian determinant and parity.
Gravitational lensing changes a local angular mapping through its Jacobian. Its determinant controls signed magnification and image parity, while a zero determinant marks a singular local mapping; those terms require a lens model rather than establishing one from a single supplied point.
Return a two-dimensional lensing Jacobian determinant and parity.
When to use this tool
Use Lensing Parity Determinant Sign when checking a supplied local lensing determinant, parity or magnification convention.
How to use the tool and how it works
- Enter jacobian. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read determinant, parity, critical. Inspect critical before interpreting the associated values.
Return a two-dimensional lensing Jacobian determinant and parity. Require exactly four row-major entries; determinant = a*d-b*c, parity = sign(determinant), critical uses exact equality a*d === b*c.1
Lensing magnification and parity depend on the local mapping Jacobian; a signed reciprocal determinant is distinct from an unsigned flux magnification.2
| Input field | Meaning and units | Accepted input |
|---|---|---|
| jacobian | Four row-major entries [a,b,c,d] of a dimensionless local 2x2 lensing Jacobian. | Required. Array of finite number; maximum items 256. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| determinant | Determinant | Required. Finite number. |
| parity | Sign of determinant: -1, 0 or +1; zero marks exact local singularity. | Required. Safe integer. |
| critical | Critical | Required. Boolean. |
{
"input": {
"jacobian": [
2,
0,
0,
-1
]
},
"output": {
"determinant": -2,
"parity": -1,
"critical": false
}
}Limitations and assumptions
- Zero parity marks this exact local singularity; no tolerance, image reconstruction or complete lens model is inferred.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- 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. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Use a complete lens model and observational uncertainty analysis for source reconstruction or physical lensing inference.
References
-
Calculation and input contract: cosmology/lensing-parity-determinant-sign — tool.io
-
Weak Gravitational Lensing — Matthias Bartelmann and Peter Schneider
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