Crate arithmetica 

Exact geometric predicates for f64 input.

orient2d, incircle, orient3d and insphere return the sign of their determinant exactly, for every finite input: any magnitudes, subnormals and exactly degenerate configurations included. A NaN or infinite coordinate returns NonFinite rather than a plausible-looking Sign.

use arithmetica::{Sign, orient2d};

assert_eq!(orient2d([0.0, 0.0], [1.0, 0.0], [0.0, 1.0]), Ok(Sign::Positive));
assert_eq!(orient2d([0.0, 0.0], [1.0, 1.0], [2.0, 2.0]), Ok(Sign::Zero));
assert!(orient2d([f64::NAN, 0.0], [1.0, 0.0], [0.0, 1.0]).is_err());

Contents

• Predicates: orient2d, incircle, orient3d, insphere.
• Exact 2D constructions: RationalCoord2d, built from f64 input or as the exact intersection of two lines, with the exact predicates rational_coords_equal, rational_compare_along and orient2d_rational. RationalCoord2d::f64_bracket reports the two f64 neighbors of an exact coordinate and its exact position between them; no function in the crate returns an approximation of a value.
• Expansion arithmetic: the error-free primitives in expansion, exact under the conditions their documentation states.
• Stage counters: counters, behind the stage-counters feature.

How the predicates stay exact

Each predicate is a ladder of stages, and each stage either certifies a sign or passes the input on:

1. a filter in native f64 with the underflow-protected error bound of Bartels, Fisikopoulos and Weiser (2023), valid for every finite input;
2. Shewchuk’s adaptive expansion stages, run only inside an exactness window of magnitudes where binary64 provably behaves as his unbounded-exponent model, with input moved there by an exact power-of-two rescale when possible;
3. exact fixed-width integer arithmetic for everything else.

The README and docs/exactness_window.md in the repository give the full argument and the references.

Re-exports

pub use rational::RationalCoord2d;

pub use rational::orient2d_rational;

pub use rational::rational_compare_along;

pub use rational::rational_coords_equal;

Modules

counters

Optional counters recording how often each predicate reaches its exact fallback.

expansion

Low-level floating-point expansion arithmetic primitives.

rational

Exact predicates over constructed points.

Structs

F64Bracket

The two representable f64 values that bracket an exact value, and the value’s exact position between them.

NonFinite

Error returned by a geometric predicate when an input coordinate is not finite (NaN, +∞, or −∞).

Enums

BracketPosition

Where an exact value sits relative to its two nearest representable f64 neighbors.

Sign

All predicates in this crate return Sign rather than f64 or bool. The three variants correspond to the three mathematically distinct outcomes of a predicate determinant.

Functions

incircle

Tests whether point pd lies inside the circumcircle of triangle pa, pb, pc.

insphere

Tests whether pe lies inside the circumsphere of pa, pb, pc, pd.

orient2d

Tests the orientation of the triangle formed by pa, pb, and pc.

orient3d

Tests the orientation of the tetrahedron formed by four 3D points.