Struct SphereBoostFlowBasis 

pub struct SphereBoostFlowBasis;

The three conformal-boost vector fields on the round S² in (lat, lon) coordinates — the non-isometric part of the conformal group, i.e. the gradient fields of the degree-1 spherical harmonics z, x, y.

Why these three modes, and why they are the right (and complete) pin

After the gauge-invariant fit the sphere chart’s residual freedom is the conformal group of (S², g_round), which is PSL(2, ℂ) (6-real-dim): the three rotation Killing fields (the isometries O(3)) plus the three boosts. The isometry defect ‖g_φ − g_ref‖² is null on the rotations (true isometries) and strictly positive on the boosts (a boost is a non-isometric conformal map: it scales the metric by a non-constant conformal factor μ(x) ≠ 1, compressing one hemisphere into a cap — the d = 2 analogue of the d = 1 circle-into-1.0-rad pathology). So minimizing the defect over the boost subspace breaks the conformal moduli down to the residual isometry group O(3) — exactly the finite-dim residual the certificate reports. The rotation modes are deliberately EXCLUDED: they are defect-null and would only insert null directions into the Gauss–Newton system (the same reason torus translations / patch rotations are excluded from their flow bases).

In (lat, lon) the boost fields are (with s = sin lat, c = cos lat):

K_z = ( c,                0                 )      (gradient of z = sin lat)
K_x = ( s·cos lon,       −sin lon / c       )      (gradient of x)
K_y = ( s·sin lon,        cos lon / c       )      (gradient of y)

K_z is globally smooth (pole-free); K_x, K_y carry the 1/c longitudinal pole the hairy-ball theorem guarantees somewhere. The flow is the same Euler convention as the torus / patch families, φ_θ(t) = t + Σ_k θ_k v_k(t), and the band guard (SPHERE_FLOW_POLE_MARGIN) keeps every evaluated row away from the 1/c singularity, so the three fields are smooth and bounded on the data and the pin is well-posed.

Implementations

impl SphereBoostFlowBasis

pub fn dim(&self) -> usize

The three boost modes are always present; the dimension is fixed at 3.

pub fn mode_layout(&self) -> [SphereBoostAxis; 3]

Mode identities in coefficient order: [Z, X, Y]. This IS the θ index layout — [Self::mode_samples] returns samples in the same order.

Trait Implementations

impl Clone for SphereBoostFlowBasis

fn clone(&self) -> SphereBoostFlowBasis

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Debug for SphereBoostFlowBasis

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

Formats the value using the given formatter.