geometry · PointCloud

14 accepted declarations. Python and Rust forms, contracts, evidence, failure boundaries, and checked usages are kept together below.

Worked example

The complete checked program Enumerate and certify rational points on a finite-field hyperbola uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Enumerate and certify rational points on a finite-field hyperbola."""

from noether.commalg import AffineSpace
from noether.numthy import FiniteField


def main() -> None:
    field = FiniteField.prime(5)
    plane = AffineSpace(field, 2, names=["x", "y"])
    x, y = plane.coordinate_ring.generators()
    hyperbola = plane.closed_subscheme(plane.coordinate_ring.ideal([x * y - 1]))
    count, certificate = hyperbola.point_count(shards=3)
    certificate.verify(hyperbola)
    assert count == 4
    assert len(list(hyperbola.rational_points())) == count
    print(f"xy=1 over F_5 has {count} rational points")


if __name__ == "__main__":
    main()

Run it with .venv/bin/python examples/18_finite_field_schemes.py.

geometry.PointCloud

A checked finite point cloud in Euclidean space.

PythonRust

PointCloud(points: Sequence[Sequence[float]])

noether::geometry::PointCloud · Direct Rust declaration

pub struct PointCloud { /* private fields */ }

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, geom-core, convex-hull
GIL	attached / bounded
Semantic ID	noether/geometry/PointCloud@1

Executable usage: validation/unit/geometry/test_acceptance_workload.py:14 · validation/unit/geometry/test_acceptance_workload.py:29 · validation/unit/geometry/test_acceptance_workload.py:44 · validation/unit/geometry/test_acceptance_workload.py:9 · validation/unit/geometry/test_geometry_facade.py:12 · validation/unit/geometry/test_geometry_facade.py:13

geometry.PointCloud.__getitem__

Return self[key].

PythonRust

(self, index: int) -> list[float]

noether::geometry::PointCloud · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, geom-core, convex-hull
GIL	attached / bounded
Semantic ID	noether/geometry/PointCloud@1

geometry.PointCloud.__len__

Return len(self).

PythonRust

(self) -> int

noether::geometry::PointCloud · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, geom-core, convex-hull
GIL	attached / bounded
Semantic ID	noether/geometry/PointCloud@1

Executable usage: validation/unit/topology/test_filtration_builders.py:109 · validation/unit/topology/test_filtration_builders.py:14

geometry.PointCloud.affine_rank

Dimension of the affine hull occupied by the sites.

Detailed contract

Dimension of the affine hull of the cloud.

PythonRust

(self) -> int

noether::geometry::PointCloud::affine_rank · Direct Rust declaration

pub fn affine_rank(&self) -> usize

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, geom-core, convex-hull
GIL	attached / bounded
Semantic ID	noether/geometry/PointCloud@1

Executable usage: validation/unit/geometry/test_acceptance_workload.py:24 · validation/unit/geometry/test_geometry_facade.py:24 · validation/unit/geometry/test_geometry_facade.py:47

geometry.PointCloud.ambient_dimension

Coordinate count of each site.

PythonRust

(self) -> int

noether::geometry::PointCloud · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, geom-core, convex-hull
GIL	attached / bounded
Semantic ID	noether/geometry/PointCloud@1

Executable usage: validation/unit/commalg/test_kahler_differentials.py:98 · validation/unit/geometry/test_geometry_facade.py:23 · validation/unit/geometry/test_geometry_facade.py:46 · validation/unit/geometry/test_polytope_facade.py:30 · validation/unit/linalg/test_exact_linear_algebra_facade.py:152 · validation/unit/linalg/test_exact_linear_algebra_facade.py:153

geometry.PointCloud.delaunay

Exact-predicate Delaunay triangulation.

PythonRust

(self) -> Triangulation

noether::geometry::PointCloud · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, geom-core, convex-hull
GIL	released
Semantic ID	noether/geometry/PointCloud@1

Executable usage: validation/unit/geometry/test_acceptance_workload.py:25 · validation/unit/geometry/test_acceptance_workload.py:44 · validation/unit/geometry/test_geometry_facade.py:53 · validation/unit/geometry/test_geometry_facade.py:58 · validation/unit/geometry/test_geometry_facade.py:79 · validation/unit/geometry/test_geometry_facade.py:9

geometry.PointCloud.distance

Euclidean distance between two indexed sites.

Detailed contract

Euclidean distance between two points.

PythonRust

(self, i: int, j: int) -> float

noether::geometry::PointCloud::distance · Direct Rust declaration

pub fn distance(&self, i: usize, j: usize) -> f64
fn distance(&self, i: usize, j: usize) -> f64

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, geom-core, convex-hull
GIL	attached / bounded
Semantic ID	noether/geometry/PointCloud@1

Executable usage: validation/unit/combin/test_graph_facade.py:105 · validation/unit/combin/test_graph_facade.py:48 · validation/unit/geometry/test_geometry_facade.py:25 · validation/unit/geometry/test_geometry_facade.py:48 · validation/unit/topology/test_filtration_builders.py:100 · validation/unit/topology/test_filtration_builders.py:112

geometry.PointCloud.enclosing_radius

Smallest radius about an input site containing the whole cloud.

PythonRust

(self) -> float

noether::geometry::PointCloud::enclosing_radius · Direct Rust declaration

fn enclosing_radius(&self) -> f64

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, geom-core, convex-hull
GIL	attached / bounded
Semantic ID	noether/geometry/PointCloud@1

Executable usage: validation/unit/geometry/test_geometry_facade.py:26 · validation/unit/topology/test_filtration_builders.py:115 · validation/unit/topology/test_filtration_builders.py:24

geometry.PointCloud.from_json

Deserialize and revalidate dimensions, finiteness, and signed zeros.

PythonRust

(value: str) -> PointCloud

noether::geometry::PointCloud · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, geom-core, convex-hull
GIL	attached / bounded
Semantic ID	noether/geometry/PointCloud@1

Executable usage: validation/unit/category/test_category.py:14 · validation/unit/combin/test_acceptance_workload.py:65 · validation/unit/combin/test_acceptance_workload.py:86 · validation/unit/combin/test_graph_facade.py:106 · validation/unit/combin/test_graph_facade.py:25 · validation/unit/combinatorics/test_kashiwara_crystal.py:32

geometry.PointCloud.is_empty

Whether the cloud has no sites.

Detailed contract

Is the cloud empty?

PythonRust

(self) -> bool

noether::geometry::PointCloud::is_empty · Direct Rust declaration

pub fn is_empty(&self) -> bool
fn is_empty(&self) -> bool

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, geom-core, convex-hull
GIL	attached / bounded
Semantic ID	noether/geometry/PointCloud@1

Executable usage: validation/unit/combinatorics/test_poset.py:113 · validation/unit/topology/test_filtration_builders.py:119 · validation/unit/topology/test_persistence_facade.py:45 · validation/unit/topology/test_persistence_facade.py:47 · validation/unit/topology/test_simplicial_complex_facade.py:47 · validation/unit/topology/test_simplicial_complex_facade.py:49

geometry.PointCloud.points

Sites in input order.

PythonRust

(self) -> list[list[float]]

noether::geometry::PointCloud · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, geom-core, convex-hull
GIL	attached / bounded
Semantic ID	noether/geometry/PointCloud@1

Executable usage: examples/18_finite_field_schemes.py:18 · examples/18_finite_field_schemes.py:3 · examples/39_kahler_differentials.py:3 · validation/unit/commalg/test_scheme_facade.py:103 · validation/unit/commalg/test_scheme_facade.py:104 · validation/unit/commalg/test_scheme_facade.py:122

geometry.PointCloud.project_to_affine_hull

Re-express the cloud isometrically in its affine-hull coordinates.

Detailed contract

Re-express the cloud isometrically in coordinates for its affine hull.

PythonRust

(self) -> PointCloud

noether::geometry::PointCloud::project_to_affine_hull · Direct Rust declaration

pub fn project_to_affine_hull(&self) -> Result<PointCloud, String>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, geom-core, convex-hull
GIL	released
Semantic ID	noether/geometry/PointCloud@1

Executable usage: validation/unit/geometry/test_geometry_facade.py:44

geometry.PointCloud.regular_triangulation

Exact-predicate regular triangulation with one finite weight per site.

PythonRust

(self, weights: Sequence[float]) -> Triangulation

noether::geometry::PointCloud · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, geom-core, convex-hull
GIL	released
Semantic ID	noether/geometry/PointCloud@1

Executable usage: validation/unit/geometry/test_acceptance_workload.py:26 · validation/unit/geometry/test_geometry_facade.py:54 · validation/unit/geometry/test_geometry_facade.py:73 · validation/unit/geometry/test_geometry_facade.py:75 · validation/unit/geometry/test_geometry_facade.py:9 · validation/unit/topology/test_filtration_builders.py:81

geometry.PointCloud.to_json

Serialize the finite coordinate rows.

PythonRust

(self) -> str

noether::geometry::PointCloud · Python projection

Python projection convenience over this Rust mathematical type

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, geom-core, convex-hull
GIL	attached / bounded
Semantic ID	noether/geometry/PointCloud@1

Executable usage: validation/unit/category/test_category.py:14 · validation/unit/category/test_category.py:15 · validation/unit/combin/test_acceptance_workload.py:30 · validation/unit/combin/test_acceptance_workload.py:65 · validation/unit/combin/test_acceptance_workload.py:86 · validation/unit/combin/test_graph_facade.py:106