Struct So3Cover 

pub struct So3Cover(/* private fields */);

A finite geodesic-ball cover of So3 centred on icosahedral rotations.

The cover supplies Bounded domains and nodes for the global simplicial and geodesic algorithms.

Trait Implementations

impl AsRef<So3<Coords<R64, 3>>> for So3Cover

fn as_ref(&self) -> &So3<Coords<R64, 3>>

Converts this type into a shared reference of the (usually inferred) input type.

impl Bounded<So3<Coords<R64, 3>>, So3<Coords<R64, 3>>, Coords<R64, 3>> for So3Cover

Radius of the geodesic-ball domains of So3Cover.

The 60 nodes are the icosahedral rotation group I ≅ A₅ ⊂ SO(3) — the image of the 120 icosian unit quaternions (the vertices of the 600-cell) under the double cover S³ → SO(3). In the bi-invariant metric d = |identity_log| (half the rotation angle; diameter π/2), the pairwise distances realised between nodes are exactly

  π/5 ≈ 0.628,   π/3 ≈ 1.047,   2π/5 ≈ 1.257,   π/2 ≈ 1.571

and the covering radius of the node set is ≈ 0.3857 (the circumradius of a cell of the 600-cell). The radius ρ = 0.42 is chosen so that:

• covering: ρ > 0.3857, so the 60 open balls cover SO(3);
• goodness: ρ < π/4, the convexity radius of SO(3) ≅ RP³, so every ball is geodesically convex and all intersections of balls are convex, hence contractible or empty — an open good cover;
• faithful 1-skeleton: two equal balls overlap iff their centres are closer than 2ρ = 0.84, which separates π/5 from π/3 with a wide margin on both sides — the nerve’s edges are exactly the 600-cell’s edges (mod ±1), and the computation is robust to floating-point error;
• faithful 2-skeleton: every triangle of the overlap graph is an equilateral triangle of side π/5 with spherical circumradius ≈ 0.365 < ρ, so all three balls genuinely share a point — mutual pairwise overlap coincides with triple intersection, and the triangles of the nerve are exactly the 600-cell’s 2-faces (mod ±1).

The nerve of this cover is therefore the hemi-600-cell: the classical vertex-transitive 60-vertex triangulation of RP³ with f-vector (60, 360, 600, 300), obtained from the boundary complex of the 600-cell by identifying antipodes. By the nerve theorem the nerve is homotopy equivalent to SO(3), and π₁ computed from its 2-skeleton is ⟨x | x²⟩ ≅ Z/2Z.

fn sdf(&self, v: &Coords<R64, 3>) -> R64

impl BuildNodes<So3Cover> for So3Cover

fn build_nodes() -> Vec<Self>

impl Chart<So3<Coords<R64, 3>>, Coords<R64, 3>> for So3Cover

type Global = So3<Coords<R64, 3>>

The result of mapping local coordinates back onto the manifold.

fn to_local(&self, point: &So3<Coords<R64, 3>>) -> Option<Coords<R64, 3>>

fn to_global(&self, coord: Coords<R64, 3>) -> So3<Coords<R64, 3>>

fn chart_at(p: &So3<Coords<R64, 3>>) -> Self

fn local_distance(&self, other: &P) -> Option<<V::F as Field>::Fixed>

where V: Euclidean,

Calculates the distance between self and other in local coordinates, based at &self.

fn check_local_inverse(p: &P) -> bool

where P: PartialEq,

impl Clone for So3Cover

fn clone(&self) -> So3Cover

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for So3Cover

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl ExpMap<So3<Coords<R64, 3>>, Coords<R64, 3>> for So3Cover

fn base_point(&self) -> P

fn check_base_point_is_origin(&self) -> bool

where V: Form,

fn check_preservation_of_origin(&self) -> bool

where V: Form,

fn check_chart_at_base_point(&self) -> bool

where V: Form,

If a chart centred at p exists, chart_at(p) returns it. Formally: chart_at(p).base_point() == p whenever p is the base point of some valid chart in this atlas.

impl From<So3<Coords<R64, 3>>> for So3Cover

fn from(value: So3<Coords<R64, 3>>) -> Self

Converts to this type from the input type.

impl NerveComplexParameters<So3<Coords<R64, 3>>, Coords<R64, 3>, So3<Coords<R64, 3>>, So3Cover> for So3Cover

fn overestimation_bound() -> Option<(V::F, V::F)>

The one irreducible assumption: graph distance on the 1-skeleton overestimates true geodesic distance by at most a factor 1 + e.

fn max_candidate_paths() -> usize

Hard cap on how many candidate edge-paths are straightened. Raising it strengthens the guarantee; lowering it trades certification for speed.

fn max_frontier() -> usize

Cap on heap entries. Prefixes vastly outnumber completions, and this is the quantity that threatens memory. Independent of the above: no ratio between prefixes-in-flight and completions exists.

fn max_rescues() -> usize

Cap on local insertions before declaring the charts unusable. when flowing a polyline, sometimes a point might go out of the injectivity radius of its neighbors, in that case, we insert a point between them to try to rescue the polyline.

fn max_straightening_iterations(n: usize) -> usize

Iteration cap for the flow, as a function of vertex count.

fn max_samples() -> usize

Cap on samples per same_basin comparison. Exceeding it means the polyline is longer than max_samples · δ_s, and no comparison at that spacing could prove two prefixes share a basin — so the prune is declined rather than performed on insufficient evidence.

fn max_canonical_generators() -> usize

Generator count above which fundamental_group returns a correct but non-canonical presentation.

fn prefix_smoothing_sweeps() -> usize

Sweeps of the discrete geodesic flow applied to a prefix before it is compared for basin membership.

fn max_basins_per_class() -> usize

fn get_neighbors(i: usize) -> impl Iterator<Item = usize>

Returns the indices of nodes whose bounded domains overlap the bounded domain of this node — the 1-skeleton of the nerve.

impl PartialEq for So3Cover

fn eq(&self, other: &So3Cover) -> bool

Equality operator ==.

fn ne(&self, other: &Rhs) -> bool

Inequality operator !=.

impl StructuralPartialEq for So3Cover

impl TangentBundle<So3<Coords<R64, 3>>, Coords<R64, 3>> for So3Cover

fn check_universal_centring(p: P) -> bool

where V: Form,