Crate ellip

ELLIP

Ellip is an elliptic integral functions for Rust.

Why Ellip

Ellip is a pure-Rust implementation of elliptic integrals. This means there is no dependence on C++ libraries. Ellip also provides less common functions like Bulirsch’s cel and el. Some applications of elliptic integrals include computing the lengths of plane curves, magnetic field from magnets of various shapes (e.g., cylindrical), astrophysics, and string theory.

Example

Computing the circumference of an ellipse.

use ellip::*;

fn ellipse_length(a: f64, b: f64) -> Result<f64, StrErr> {
    Ok(8.0 * elliprg(0.0, a * a, b * b)?)
}

let ans = ellipse_length(5.0, 3.0).unwrap();
ellip::util::assert_close(ans, 25.526998863398124, 1e-15);

Features

Legendre’s complete integrals

• ellipk: Complete elliptic integral of the first kind (K).
• ellipe: Complete elliptic integral of the second kind (E).
• ellippi: Complete elliptic integral of the third kind (Π).
• ellipd: Complete elliptic integral of Legendre’s type (D).

Legendre’s incomplete integrals

• ellipf: Incomplete elliptic integral of the first kind (F).
• ellipeinc: Incomplete elliptic integral of the second kind (E).
• ellippiinc: Incomplete elliptic integral of the third kind (Π).
• ellipdinc: Incomplete elliptic integral of Legendre’s type (D).

Bulirsch’s integrals

• cel: General complete elliptic integral in Bulirsch’s form.
• cel1: Complete elliptic integral of the first kind in Bulirsch’s form.
• cel2: Complete elliptic integral of the second kind in Bulirsch’s form.
• el1: Incomplete elliptic integral of the first kind in Bulirsch’s form.
• el2: Incomplete elliptic integral of the second kind in Bulirsch’s form.
• el3: Incomplete elliptic integral of the third kind in Bulirsch’s form.

Carlson’s symmetric integrals

• elliprf: Symmetric elliptic integral of the first kind (RF).
• elliprg: Symmetric elliptic integral of the second kind (RG).
• elliprj: Symmetric elliptic integral of the third kind (RJ).
• elliprc: Degenerate elliptic integral of RF (RC).
• elliprd: Degenerate elliptic integral of the third kind (RD).

Testing

The function results are compared with Boost Math test data and Wolfram Engine test data. The accuracy report and the test data along with the test generation scripts can be found here.

Acknowledgment

Ellip is derived from multiple mathematic libraries. We thank the opensource contributors for making mathematic libraries free for all. Following are the main original works used in the development of Ellip. Detailed credits are available in the source code.

• SciPy
• Cephes Math Library
• Boost Math Library
• Russell Lab

References for original implementations are:

• NIST Digital Library, Chapter 19: Elliptic Integrals (Carlson, 2024).
• Numerical calculation of elliptic integrals and elliptic functions I (Bulirsch, 1965), II (Bulirsch, 1965), and III (Bulirsch, 1969).

Unicode-style mathematical notation are created using Diagon.

Re-exports

pub use legendre::ellipd;

pub use legendre::ellipe;

pub use legendre::ellipk;

pub use legendre::ellippi;

pub use legendre::ellipdinc;

pub use legendre::ellipeinc;

pub use legendre::ellipf;

pub use legendre::ellippiinc;

pub use bulirsch::cel;

pub use bulirsch::cel1;

pub use bulirsch::cel2;

pub use bulirsch::el1;

pub use bulirsch::el2;

pub use bulirsch::el3;

pub use carlson::elliprc;

pub use carlson::elliprd;

pub use carlson::elliprf;

pub use carlson::elliprg;

pub use carlson::elliprj;

Modules

bulirsch

Elliptic integral functions in Bulirsch’s form.

carlson

Elliptic integral functions in Carlson’s form.

legendre

Elliptic integral functions in Legendre’s form.

util

Utility functions like assert_close.

Type Aliases

StrErr

Static error str