E4 ยท Publication Volume 26

Numerical Robustness and Testing

degeneracy, tolerances, coordinate magnitude and golden cases

Learning objectives

  • Explain the decision and evidence boundary for degeneracy, tolerances, coordinate magnitude and golden cases.
  • Select and implement the relevant representation or algorithm without hidden coordinate, support or topology assumptions.
  • Separate exact predicates, approximation error, source uncertainty and visual delivery.
  • Produce a spatial-computing assurance suite with predicate, property and metamorphic tests from synthetic evidence.

The lesson is complete only when the learner can defend the representation, transform, predicates, tests and release decision. A visually clean map or 3D scene without executable invariants and provenance remains unverified.

This is a general, institution-neutral tutorial with no relationship to any company or individual. All coordinates, geometries, grids, points, surfaces, volumes, attributes and review events in the lesson are synthetic and must not be used for an operational decision.

Decision context

The decision is whether an implementation gives stable, explainable results across normal, boundary, degenerate and numerically difficult inputs. Robustness does not mean forcing every input to produce geometry. It means classifying admissible states consistently, detecting invalid or indeterminate cases, using a documented numerical strategy and preserving evidence needed to reproduce the result. The assurance plan identifies exact combinatorial rules, robust predicates, metric tolerances, construction precision, deterministic ordering and the operations whose outputs require independent reference fixtures.

Write the intended use, consequence of error, required evidence, spatial support and release authority before selecting a representation or transformation. Fitness is evaluated against a versioned contract and use, not attached permanently to a file extension.

Core concept

Predicates decide topology; constructions create coordinates. A tiny sign error in orientation can change connectivity even when coordinate error is visually invisible. Adaptive or exact predicates improve classification, while constructed intersections may still be approximate and need residual checks. Degeneracies such as coincident points, collinearity, coplanarity, zero-area faces and tangency are first-class input classes. Tolerances apply to declared metric equivalence or uncertainty, not to replace a failed topological predicate. Stable total ordering uses exact identity tie-breakers after spatial keys.

Keep received evidence, accepted analytical views and derived representations as distinct objects. This allows corrected evidence, a changed transform or a new level of detail to generate a new result without rewriting history. Every coordinate and primitive therefore answers both a spatial question and a provenance question.

Algorithm and data model

Layer tests from smallest to broadest. Unit tests cover transforms, predicates and typed boundary cases. Golden fixtures contain analytic points, grids, meshes and solids with independently known results. Property tests generate valid inputs and check invariants such as containment of bounds. Metamorphic tests apply transformations that should preserve or predictably change results: translation, rotation, uniform scaling, vertex permutation with orientation adjustment and equivalent chunking. Differential tests compare independent implementations. Round-trip tests cover formats, while end-to-end tests verify manifests and release gates.

Define parsing, semantic validation, canonicalisation, indexing, exact or approximate calculation, quality evaluation and encoding as separate stages. Each stage emits structured output and does not depend on interface state, file order, graphics-driver behaviour or undocumented defaults.

Constraints and invariants

| Invariant | Executable or review test | | --- | --- | | Predicate strategy, construction precision and metric tolerance are separate policies. | Reject or quarantine the exact affected object and preserve the received representation. | | Degenerate and indeterminate inputs have explicit expected outcomes. | Evaluate this condition before creating a derived geometry, grid, surface or volume. | | Golden results are independently derived and versioned with fixtures. | Record the predicate, tolerance policy, observed values and coordinate frame. | | Test output records seed, platform, versions, manifests and exact failures. | Make every repair a new version and rerun all dependent golden cases. |

An invariant must survive import, transformation, processing, export and rerun. A failed hard invariant produces no apparently valid substitute. Diagnostics remain visible with predicate, threshold, coordinate frame, scope and evidence, and require a reviewed rule before they can trigger repair.

Quantitative reasoning

For metric comparison, a common declared form is |a-b|\le a_{tol}+r_{tol}\max(|a|,|b|), with units and rationale for both terms. This formula is not used for integer identity, topology or category equality. Report fixture counts by case class, exact pass and fail counts, maximum residual, transformation invariance error, random seed, generated-case shrink result, platform and implementation version. Golden checks include orientation sign, affine inverse, grid alignment, volume identities and index recall. Test coordinates near zero and large offsets, subnormal and non-finite values, near-degenerate triangles, repeated primitives and changes in input ordering.

Every metric includes units, support, numerator and denominator where applicable, exclusions, comparison policy and evaluation version. Aggregate metrics are stratified when pooling can hide local geometry failure. A performance gain cannot overrule invalid topology, missing reference metadata or broken lineage.

Evidence and uncertainty

Keep acquisition uncertainty, interpretation uncertainty, discretisation error, numeric round-off and delivery error separate. Increasing coordinate digits or triangle count does not improve the original evidence. A sampled surface may be smooth and watertight while remaining poorly constrained between observations. Report uncertainty in the quantity and support to which it belongs.

Build an evidence packet containing immutable received objects, semantic declarations, validation findings, transform inputs and outputs, measured errors, test results, reviewer decisions and fingerprints. Contradictory evidence remains available. When a required reference, topology state or classification cannot be resolved, return unknown, conflict or blocked rather than inventing geometry.

Interfaces and storage

Interfaces transmit identity, coordinate reference, units, axis order, support, topology expectations, attribute association, null state, version and lineage beside coordinates. Structured errors identify the object, primitive, predicate, observed value, expected condition and rule. An interface that carries vertices but drops the transform or face orientation has not preserved the object.

Store authoritative received evidence separately from reproducible analytical derivatives and disposable delivery artefacts. Indexes, caches, pyramids and render meshes improve access but cannot become the only copy of source attributes or coordinate metadata. Round-trip tests verify identity, precision, topology, ordering, missingness and association after encoding changes.

Governance and review

Assign responsibilities to roles rather than named organisations or people: evidence custodian, representation author, algorithm maintainer, independent validator and release reviewer. A role may propose a repair but cannot erase the received geometry. Transform, predicate and tolerance changes are versioned and evaluated against fixed regression fixtures before release.

Exceptions are explicit decisions with scope, rationale, evidence, approving role, affected versions and review trigger. They never turn invalid topology into valid topology by label. The host website has no ownership or scientific-authority role in this workflow; it only delivers the tutorial.

Integration checkpoint

a spatial-computing assurance suite with predicate, property and metamorphic tests
a spatial-computing assurance suite with predicate, property and metamorphic tests

Read the figure as a reasoning map from preserved evidence through declared support and coordinates, controlled transformation, validation and scoped release. Each arrow represents a declared relationship. Integrate a spatial-computing assurance suite with predicate, property and metamorphic tests into SYN-SPATIAL, rerun earlier fixtures and record every changed assumption.

Synthetic worked example

A synthetic orientation fixture uses three almost collinear points translated by a large world offset. One naive floating implementation changes sign after translation, while a robust predicate preserves the analytic classification. The test suite stores both cases, the exact expected sign and the construction residual. A second metamorphic test rotates a closed tetrahedron and verifies unchanged volume magnitude and inside classifications. When a deliberately invalid self-intersecting mesh is supplied, success means a stable failed predicate and blocked derivative, not a fabricated repair.

  1. Preserve the received object and state the intended decision without repair.
  2. Resolve identity, reference, units, support, topology and evidence eligibility.
  3. Run the versioned transform or predicate while retaining intermediate diagnostics.
  4. Issue accept, reject or quarantine and show how an independent reviewer reproduces it.

Practice task

Implement the chapter artefact against a synthetic fixture containing one normal case, one boundary case, one invalid case and one unresolved-evidence case. Preserve the received fixture. Produce canonical input, validation findings, derivative output, processing manifest, measured error and a short release decision.

Acceptance criteria:

  • Every required identity, coordinate reference, unit, support and convention is explicit.
  • The implementation is deterministic under stable ordering and the declared numerical policy.
  • No repair overwrites received evidence or converts unknown into a guessed value.
  • All hard failures block the affected derivative and remain machine-readable.
  • A second implementation or reviewer can reproduce the result from the package alone.

Submit a spatial-computing assurance suite with predicate, property and metamorphic tests, golden and adversarial fixtures, exact findings, measured error and a limitations note. A screenshot is not sufficient evidence because it does not identify input versions, transforms, algorithms or rule configuration.

Common failure modes

  • Adding one large epsilon until tests stop failing.
  • Using approximate coordinate equality to decide topology.
  • Generating golden results with the same implementation under test.
  • Treating a blocked invalid input as a test failure of robustness.

These failures share a pattern: implicit convenience is substituted for evidence. Diagnose the earliest boundary where the assumption entered, restore the source statement, make the transform or predicate explicit, rerun all dependent derivatives and supersede rather than overwrite the affected release.

Review questions

  1. How do predicates differ from constructions?
  2. Which comparisons may use absolute and relative tolerance?
  3. What transformations make useful metamorphic tests?
  4. Why can a correctly blocked result demonstrate robustness?

For every answer, identify the governing invariant, evidence needed to evaluate it, numerical or semantic policy involved and correct behaviour when the condition fails.

Sources and further reading