Interval Complement Within Domain

Compute half-open uncovered domain cells after clipping supplied intervals; no extrapolation outside the explicit nonempty domain.

Description

Compute half-open uncovered domain cells after clipping supplied intervals; no extrapolation outside the explicit nonempty domain.

A complement within a finite domain describes the gaps left after coverage is removed. Declaring the domain avoids an unbounded outside result.

When to use this tool

Compute half-open uncovered domain cells after clipping supplied intervals; no extrapolation outside the explicit nonempty domain.

Prepare the supplied intervals, domainStart, domainEnd fields for this operation.
Compare the complete example with the representation used by your own input before carrying the result into another step.

How to use the tool and how it works

Enter intervals, domainStart, domainEnd. Use the exact text syntax, units and index conventions explained below.
Run the tool. If input validation fails, correct the supplied fields before interpreting any result.
Read result.complement. Interpret false, null and empty lists using the distinctions below; they can be valid outputs.

Require a positive-length domain, clip supplied intervals to it, discard empty intersections, and partition at all remaining endpoints plus domain boundaries. Return cells covered only by the domain itself.^[1]

Input field	Meaning and representation	Accepted input
intervals	intervals.	Numeric rows; at most 128 rows and 128 values per row
domainStart	domain start.	Finite number
domainEnd	domain end.	Finite number

Output field	Interpretation
result	Object containing the named outputs below.
result.complement	The complement list in the ordering and representation described above.

json

{
  "input": {
    "intervals": [
      [
        1,
        3
      ]
    ],
    "domainStart": 0,
    "domainEnd": 4
  },
  "output": {
    "result": {
      "complement": [
        [
          0,
          1
        ],
        [
          3,
          4
        ]
      ]
    }
  }
}

Limitations and assumptions

Adjacent uncovered cells may remain separate. This works on continuous span interiors without open/closed flags; it does not represent isolated included or excluded points.
Interval-array rows contain exactly two finite endpoints in start/end order, with at most 128 intervals per collection. Individual operations state when positive length, integer endpoints or explicit inclusion flags are required.
Every declared top-level field is required and unexpected fields are rejected. Numeric inputs must be finite; ordinary text fields are bounded to 8,192 UTF-16 code units, with tighter operation-specific limits where stated.
Ordinary string lists have at most 128 entries of at most 128 code units each; numeric lists and matrices have separate bounded size checks. Nonfinite results and excessively large serialized outputs are rejected.
Indexes are zero-based and spans have an exclusive end unless explicitly stated otherwise. A valid false, null or empty-list result is distinct from malformed input.

Alternative or Complementary methods

Retain the endpoint convention with every result. Continuous length, integer point counts and open/closed membership differ; merge returned cells separately when a canonical minimal interval collection is required.

References

Tool behavior and example: Interval Complement Within Domain — tool.io

Similar or alternative tools

Interval Affine Transform
Affine-transform interval endpoints, reorder the image bounds and retain negative-scale orientation reversal.
Interval Clamp and Empty Audit
Audit raw clipped interval bounds and emptiness; disjoint intersections retain start>end as evidence, not a fabricated nonempty interval.