Crate axiolid_nurbs

General NURBS algorithms over Axiolid’s format-neutral B-spline values.

This crate builds on the portable scalar oracle: evaluation comes from axiolid-evaluate and is not reimplemented here. It owns differential geometry and exact shape-preserving transformations; importers and tessellators are consumers, not the capability boundary.

Every tolerance-sensitive solver takes explicit, bounded options (work ceilings, resolutions, tolerances) and reports an unresolved outcome or refuses when they run out. A shape-preserving transform is tested against independent evaluation samples of the original, not its own recurrence.

Modules

extrema

Extrema between points, curves and surfaces (#119, B7): the smallest distance between two pieces of geometry, certified.

Structs

BoundedResult

Outcome of a lossy operation: the curve, and the error it introduced.

BoundedSurface

A surface changed within a stated bound.

CertifiedCurveIntersectionOptions

Accuracy and work policy for certified planar root isolation.

CertifiedCurveSurfaceIntersectionOptions

Bounded policy for certified curve/surface root isolation.

CertifiedProjectionOptions

Explicit work and accuracy policy for globally certified projection.

CertifiedRegion

One classified leaf of the certified subdivision.

CertifiedSurfaceArcs3

Result of certified curved surface/surface analysis.

CertifiedSurfaceArcsOptions

Explicit policy for certified curved analysis.

CertifiedSurfaceProjectionOptions

Explicit accuracy and work policy for globally certified surface projection.

CertifiedSurfaceSurfaceIntersectionOptions

Bounded policy for certified surface/surface intersection and affine tracing.

ClassifiedCurveContact2

One classified non-transverse contact, owned by exactly one parameter box.

ConstructedCurveSurfacePoint3

A certified curve/surface crossing, reported as a point rather than a curve.

ConstructedIntersectionCurve3

An intersection curve constructed from a certified trace.

CurveCurveHit

One point where two curves meet, with its parameter on each.

CurveDifferential2

Parameter-invariant differential properties of a planar curve.

CurveDifferential3

Parameter-invariant differential properties of a spatial curve.

CurveDistanceCertificate2

Globally bounded minimum-distance result for two planar curves.

CurveDistanceCertificate3

Globally bounded minimum-distance result for two spatial curves.

CurvePairParameterBox

Product-domain box that may contain a globally closest curve pair.

CurveProjection2

Best planar-curve projection candidate found within the supplied budget.

CurveProjection3

Best spatial-curve projection candidate found within the supplied budget.

CurveProjectionCertificate2

Globally bounded closest-point result for a planar curve.

CurveProjectionCertificate3

Globally bounded closest-point result for a spatial curve.

CurveSurfaceParameterBox

Native parameter box for a curve and a tensor-product surface patch.

ExactCurveHit

One intersection point.

ExactIntersectionCurve

The exact intersection curve of two elementary surfaces, when one exists in closed form.

FundamentalForm

Coefficients (e, f, g) of a quadratic fundamental form.

Isolated

A parameter isolated by certified subdivision, held as the exact double it was refined to (so parameters compare exactly, bit for bit).

ParameterInterval

Closed native-parameter interval that may contain a global minimizer.

PeriodicBSplineSurface

An owned tensor-product B-spline surface with validated cyclic axes.

PeriodicCurve2

Verified closed-seam extension of a planar B-spline curve.

PeriodicCurve3

Verified closed-seam extension of a spatial B-spline curve.

ProjectionOptions

Explicit work and convergence policy for bounded projection.

SurfaceDifferential

First/second-order differential properties of an oriented surface.

SurfaceInversionCertificate3

Unique native parameters proven to name a point on the surface.

SurfaceKnotRemoval

The outcome of removing one knot value from a surface up to some number of times.

SurfaceParameterBox

Closed native (u, v) box that may contain a global minimizer.

SurfaceProjection

Best surface projection candidate found within the supplied budget.

SurfaceProjectionCertificate3

Globally bounded closest-point certificate for a spatial surface.

SurfaceSurfaceParameterBox

Native four-parameter box for a pair of tensor-product surfaces.

SurfaceSurfaceTraceEndpoint3

Certified endpoint enclosure for an affine surface/surface trace.

TransverseCurveIntersection2

One isolated transverse planar intersection.

TransverseCurveSurfaceIntersection3

Existence-and-uniqueness certificate for one transverse intersection.

TransverseSurfaceSurfaceTrace3

Certificate for one complete regular affine intersection segment.

Enums

CertifiedCurveIntersection2

Planar curve/curve classification under the implemented proof paths.

CertifiedCurveSurfaceIntersection3

Certified curve/surface query outcome.

CertifiedSurfaceProjection3

Exhaustive bounded outcome for spatial surface projection.

CertifiedSurfaceSurfaceIntersection3

Certified surface/surface query outcome.

CoverageFault

Why a set of regions failed to account for the whole domain.

CurveIntersectionDegeneracy

Singular or not-yet-isolated curve relationship.

Derivation

The closed-form identity behind an exact intersection curve.

ExactCurveIntersection

The exact intersection of a curve with another operand.

ExactCurveParameter

Where on the first curve an intersection lies, exactly.

ExactCurveRefusal

Why an exact curve intersection was not computed.

ExactIntersectionRefusal

Why an elementary pair has no exact closed-form intersection curve here.

IntersectionCurveRefusal

Why a certified query produced no constructed intersection curve.

ProjectionStatus

Outcome of the selected bounded local solve.

RegionKind

What was certified about one leaf region of the parameter domain.

SeamContinuity

Highest endpoint continuity verified in the curve’s native parameter.

SurfaceInversionRefusal

Why a sound inversion query did not yield unique parameters.

SurfaceProjectionUnresolvedReason

Why a valid surface projection remains unresolved without exhausting work.

Constants

MAX_AUDIT_DEPTH

Largest subdivision depth the coverage audit can represent exactly.

MAX_CERTIFIED_SURFACE_PROJECTION_DEPTH

Maximum accepted binary subdivision depth for certified surface projection.

MAX_CERTIFIED_SURFACE_PROJECTION_WORK

Maximum accepted shared work budget for certified surface projection.

Functions

analyze_curve2

Analyze a regular planar curve at t.

analyze_curve3

Analyze a regular spatial curve at t.

analyze_surface

Analyze a regular oriented surface at (u, v).

audit_coverage

Verify that the regions tile the root parameter box exactly.

bezier_segments2

Decompose a planar B-spline into exact rational/polynomial Bézier segments.

bezier_segments3

Decompose a spatial B-spline into exact rational/polynomial Bézier segments.

certify_surface_arcs

Certify the transversality structure of a curved surface pair.

construct_curve_surface_points

Construct curve/surface intersection points for a clamped NURBS pair.

construct_surface_surface_curves

Construct intersection curves for two clamped NURBS surfaces.

curve2_seam_continuity

Verify native-parameter endpoint continuity of a planar B-spline curve.

curve3_seam_continuity

Verify native-parameter endpoint continuity of a spatial B-spline curve.

distance_curve2_certified

Certify the global minimum distance between two planar B-spline curves.

distance_curve3_certified

Certify the global minimum distance between two spatial B-spline curves.

elevate_degree2

Raise a planar curve’s degree by one without changing its image.

elevate_degree3

Raise a spatial curve’s degree by one without changing its image.

elevate_surface_degree_u

Raise the surface’s u degree by one without changing it.

elevate_surface_degree_v

Raise the surface’s v degree by one; as elevate_surface_degree_u.

exact_curve_curve_intersection2

The exact intersection of two analytic plane curves.

exact_curve_curve_intersection3

The exact intersection of two analytic space curves.

exact_curve_surface_intersection

The exact intersection of an analytic curve with an elementary surface.

exact_surface_intersection

Derive the exact intersection curve of two elementary surfaces.

extract_stretch

The stretch of one of curves that runs from start through first and then second to end (all in parameters, each possibly a whole period off), as a curve of its own running from start to end. closed asks for a whole loop from start back to itself, in the direction that meets first before second.

implicit_surface_intersection

Every component of the section of carrier by other in a window of carrier’s parameters, as implicit curves on carrier.

implicit_view

A section-family curve over span as an ImplicitSection3, with the same point set and direction; None for other families or where the trace is refused.

insert_knot2

Insert one interior knot into a planar curve in homogeneous coordinates.

insert_knot3

Insert one interior knot into a spatial curve in homogeneous coordinates.

insert_surface_knot_u

Insert one U-axis knot into every homogeneous control-net column.

insert_surface_knot_v

Insert one V-axis knot into every homogeneous control-net row.

interpolate_curve3

Interpolate a cubic B-spline through points in order.

intersect_curve2_certified

Classify intersections of two clamped planar B-spline curves.

intersect_curve_surface_certified

Certify all isolated transverse roots of curve(t) = surface(u, v).

intersect_surface_surface_certified

Certify bounded intersections of two clamped, internally continuous NURBS surfaces.

invert_periodic_surface_certified

Globally invert a point against one canonical period of a cyclic surface.

invert_surface_certified

Globally invert a point against a clamped B-spline surface.

iso_curve_at_u

The iso-curve u = parameter, parametrised by v.

iso_curve_at_v

The iso-curve v = parameter, parametrised by u; as iso_curve_at_u.

loft_surface

Loft a surface through ordered section curves.

project_curve2

Find the best planar-curve projection candidate within explicit budgets.

project_curve3

Find the best spatial-curve projection candidate within explicit budgets.

project_curve2_certified

Globally bound the closest point on a clamped planar B-spline curve.

project_curve3_certified

Globally bound the closest point on a clamped spatial B-spline curve.

project_periodic_surface_certified

Exhaustively bound the global point-to-surface minimum over one canonical period of an explicitly validated cyclic B-spline surface.

project_surface

Find the best tensor-product surface projection candidate within budgets.

project_surface_certified

Exhaustively bound the global point-to-surface minimum over the closed native domain of a finite clamped NURBS surface.

reduce_degree2

Lower a planar curve’s degree by one, or refuse.

reduce_degree3

Lower a spatial curve’s degree by one, or refuse.

reduce_surface_degree_u

Lower the surface’s u degree by one, within tolerance, or refuse.

reduce_surface_degree_v

Lower the surface’s v degree by one; as reduce_surface_degree_u.

remove_knot2

Remove one interior knot from a planar curve, or refuse.

remove_knot3

Remove one interior knot from a spatial curve, or refuse.

remove_surface_knot_u

Remove the u knot parameter up to times times, keeping the surface within tolerance of the input.

remove_surface_knot_v

Remove the v knot parameter up to times times; as remove_surface_knot_u.

reverse2

Reverse a planar B-spline curve without changing its image.

reverse3

Reverse a spatial B-spline curve without changing its image.

reverse_surface_u

Reverse the surface’s U parameter without changing its image.

reverse_surface_v

Reverse the surface’s V parameter without changing its image.

section_curve_curve_intersection2

Where two plane curves meet, each over its span.

section_curve_curve_intersection3

Where two space curves meet, each over its span, within tolerance.

section_curve_surface_intersection

Where a section-family curve meets a surface, with parameters on the curve itself.

section_field_of

other’s implicit equation read in carrier’s parameters, where both are analytic: zero exactly where carrier’s point lies on other (for a cone, on either nappe).

spline_pair_intersection

Every component of the section of two B-spline surfaces within windows of their parameters (their whole domains when None), each a PairSection3 whose first surface is first (ADR 0077).

split2

Split a planar curve exactly at an interior parameter.

split3

Split a spatial curve exactly at an interior parameter.

trace_section_pcurves

Every component of the section of surface by other in a window of surface’s parameters, as curves in those parameters.

wrap_curve2_parameter

Wrap a finite planar-curve parameter into its active domain.

wrap_curve3_parameter

Wrap a finite spatial-curve parameter into its active domain.