Struct LogCondition 

pub struct LogCondition {
    pub condition_number: f64,
    pub angle: f64,
    pub distance_to_cut_locus: f64,
    pub quality: LogQuality,
}

Conditioning information for logarithm computations.

The logarithm map on Lie groups becomes ill-conditioned near the cut locus (e.g., θ → π for SU(2), where the rotation axis becomes ambiguous). This structure provides quantitative information about the reliability of the computed logarithm.

Condition Number Interpretation

The condition number κ measures how sensitive the output is to input perturbations: |δ(log U)|/|log U| ≤ κ · |δU|/|U|

• κ ≈ 1: Well-conditioned, result is reliable
• κ ~ 10: Mildly ill-conditioned, expect ~1 digit loss of precision
• κ > 100: Severely ill-conditioned, result may be unreliable
• κ = ∞: At singularity (cut locus), axis is undefined

Example

let (log_u, cond) = g.log_with_condition()?;
if cond.is_well_conditioned() {
    // Safe to use log_u
} else if cond.is_usable() {
    // Use with caution, reduced precision
    eprintln!("Warning: log condition number = {:.1}", cond.condition_number);
} else {
    // Result unreliable
    return Err(...);
}

Fields

condition_number: f64

Condition number for the logarithm computation.

For SU(2), this is approximately 1/sin(θ/2) for rotation angle θ, which diverges as θ → π (approaching the cut locus at -I).

angle: f64

Rotation angle (for SO(3)/SU(2)) or equivalent metric.

Useful for understanding why conditioning is poor.

distance_to_cut_locus: f64

Distance from the cut locus (θ = 2π for SU(2), θ = π for SO(3)).

Values close to 0 indicate the element is near the cut locus where the logarithm is ill-defined or ill-conditioned.

quality: LogQuality

Quality assessment of the logarithm computation.

Implementations

impl LogCondition

pub fn from_angle(angle: f64) -> Self

Create a new LogCondition from the rotation angle.

For SU(2)/SO(3), the condition number is approximately 1/sin(θ/2) where θ is the rotation angle.

pub fn is_well_conditioned(&self) -> bool

Returns true if the result is well-conditioned (κ < 10).

pub fn is_usable(&self) -> bool

Returns true if the result is usable (κ < 100).

pub fn is_singular(&self) -> bool

Returns true if the result is at or near a singularity.

Trait Implementations

impl Clone for LogCondition

fn clone(&self) -> LogCondition

Returns a duplicate of the value.

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

Performs copy-assignment from source.

impl Debug for LogCondition

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

Formats the value using the given formatter.