unit_sphere/

lib.rs

// Copyright (c) 2024-2025 Ken Barker

// Permission is hereby granted, free of charge, to any person obtaining a copy
// of this software and associated documentation files (the "Software"),
// to deal in the Software without restriction, including without limitation the
// rights to use, copy, modify, merge, publish, distribute, sublicense, and/or
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// THE SOFTWARE.

//! # unit-sphere
//!
//! [![crates.io](https://img.shields.io/crates/v/unit-sphere.svg)](https://crates.io/crates/unit-sphere)
//! [![docs.io](https://docs.rs/unit-sphere/badge.svg)](https://docs.rs/unit-sphere/)
//! [![License](https://img.shields.io/badge/License-MIT-blue)](https://opensource.org/license/mit/)
//! [![Rust](https://github.com/kenba/unit-sphere-rs/actions/workflows/rust.yml/badge.svg)](https://github.com/kenba/unit-sphere-rs/actions)
//! [![codecov](https://codecov.io/gh/kenba/unit-sphere-rs/graph/badge.svg?token=G1H1XINERW)](https://codecov.io/gh/kenba/unit-sphere-rs)
//!
//! A library for performing geometric calculations on the surface of a sphere.
//!
//! The library uses a combination of spherical trigonometry and vector geometry
//! to perform [great-circle navigation](https://en.wikipedia.org/wiki/Great-circle_navigation)
//! on the surface of a unit sphere, see *Figure 1*.
//!
//! ![great circle arc](https://via-technology.aero/img/navigation/sphere/great_circle_arc.svg)
//!
//! *Figure 1 A Great Circle Arc*
//!
//! A [great circle](https://en.wikipedia.org/wiki/Great_circle) is the
//! shortest path between positions on the surface of a sphere.
//! It is the spherical equivalent of a straight line in planar geometry.
//!
//! ## Spherical trigonometry
//!
//! A great circle path between positions may be found using
//! [spherical trigonometry](https://en.wikipedia.org/wiki/Spherical_trigonometry).
//!
//! The [course](https://en.wikipedia.org/wiki/Great-circle_navigation#Course)
//! (initial azimuth) of a great circle can be calculated from the
//! latitudes and longitudes of the start and end points.
//! While great circle distance can also be calculated from the latitudes and
//! longitudes of the start and end points using the
//! [haversine formula](https://en.wikipedia.org/wiki/Haversine_formula).
//! The resulting distance in `Radians` can be converted to the required units by
//! multiplying the distance by the Earth radius measured in the required units.
//!
//! ## Vector geometry
//!
//! Points on the surface of a sphere and great circle poles may be represented
//! by 3D [vectors](https://www.movable-type.co.uk/scripts/latlong-vectors.html).
//! Many calculations are simpler using vectors than spherical trigonometry.
//!
//! ![Spherical Vector Coordinates](https://via-technology.aero/img/navigation/sphere/ecef_coordinates.svg)
//!
//! *Figure 2 Spherical Vector Coordinates*
//!
//! For example, the across track distance of a point from a great circle can
//! be calculated from the [dot product](https://en.wikipedia.org/wiki/Dot_product)
//! of the point and the great circle pole vectors.
//! While intersection points of great circles can simply be calculated from
//! the [cross product](https://en.wikipedia.org/wiki/Cross_product) of their
//! pole vectors.
//!
//! ## Design
//!
//! The `great_circle` module performs spherical trigonometric calculations
//! and the `vector` module performs vector geometry calculations.
//! See: [spherical vector geometry](https://via-technology.aero/navigation/spherical-vector-geometry/).
//!
//! The software uses types: `Angle`, `Degrees` and `Radians` from the
//! [angle-sc](https://crates.io/crates/angle-sc) crate.
//!
//! The library is declared [no_std](https://docs.rust-embedded.org/book/intro/no-std.html)
//! so it can be used in embedded applications.
//!
//! ## Example
//!
//! The following example calculates the intersection between two Great Circle `Arc`s
//! it is taken from Charles Karney's original solution to
//! [Intersection between two geodesic lines](https://sourceforge.net/p/geographiclib/discussion/1026621/thread/21aaff9f/#fe0a).
//!
//! ```rust
//! use unit_sphere::{Arc, Degrees, LatLong, calculate_intersection_point};
//! use angle_sc::is_within_tolerance;
//!
//! let istanbul = LatLong::new(Degrees(42.0), Degrees(29.0));
//! let washington = LatLong::new(Degrees(39.0), Degrees(-77.0));
//! let reyjavik = LatLong::new(Degrees(64.0), Degrees(-22.0));
//! let accra = LatLong::new(Degrees(6.0), Degrees(0.0));
//!
//! let arc1 = Arc::try_from((&istanbul, &washington)).unwrap();
//! let arc2 = Arc::try_from((&reyjavik, &accra)).unwrap();
//!
//! let intersection_point = calculate_intersection_point(&arc1, &arc2).unwrap();
//! let lat_long = LatLong::from(&intersection_point);
//! // Geodesic intersection latitude is 54.7170296089477
//! assert!(is_within_tolerance(54.72, lat_long.lat().0, 0.05));
//! // Geodesic intersection longitude is -14.56385574430775
//! assert!(is_within_tolerance(-14.56, lat_long.lon().0, 0.02));
//! ```

#![cfg_attr(not(test), no_std)]

extern crate angle_sc;
extern crate nalgebra as na;

pub mod great_circle;
pub mod vector;

use angle_sc::min;
pub use angle_sc::{Angle, Degrees, Radians, Validate};
use thiserror::Error;

/// Test whether a latitude in degrees is a valid latitude.
///
/// I.e. whether it lies in the range: -90.0 <= degrees <= 90.0
#[must_use]
pub fn is_valid_latitude(degrees: f64) -> bool {
    (-90.0..=90.0).contains(&degrees)
}

/// Test whether a longitude in degrees is a valid longitude.
///
/// I.e. whether it lies in the range: -180.0 <= degrees <= 180.0
#[must_use]
pub fn is_valid_longitude(degrees: f64) -> bool {
    (-180.0..=180.0).contains(&degrees)
}

/// A position as a latitude and longitude pair of `Degrees`.
#[derive(Clone, Copy, Debug, PartialEq)]
pub struct LatLong {
    lat: Degrees,
    lon: Degrees,
}

impl Validate for LatLong {
    /// Test whether a `LatLong` is valid.
    ///
    /// I.e. whether the latitude lies in the range: -90.0 <= lat <= 90.0
    /// and the longitude lies in the range: -180.0 <= lon <= 180.0
    fn is_valid(&self) -> bool {
        is_valid_latitude(self.lat.0) && is_valid_longitude(self.lon.0)
    }
}

impl LatLong {
    #[must_use]
    pub const fn new(lat: Degrees, lon: Degrees) -> Self {
        Self { lat, lon }
    }

    #[must_use]
    pub const fn lat(&self) -> Degrees {
        self.lat
    }

    #[must_use]
    pub const fn lon(&self) -> Degrees {
        self.lon
    }

    /// Determine whether the `LatLong` is South of  a.
    ///
    /// It compares the latitude of the two points.
    /// * `a` - the other `LatLong`.
    ///
    /// returns true if South of a, false otherwise.
    #[must_use]
    pub fn is_south_of(&self, a: &Self) -> bool {
        self.lat.0 < a.lat.0
    }

    /// Determine whether the `LatLong` is West of `LatLong` a.
    ///
    /// It compares the longitude difference of the two points.
    /// * `a`, `b` - the points.
    ///
    /// returns true if a is West of b, false otherwise.
    #[must_use]
    pub fn is_west_of(&self, a: &Self) -> bool {
        (a.lon() - self.lon).0 < 0.0
    }
}

/// A Error type for an invalid `LatLong`.
#[derive(Error, Debug, PartialEq)]
pub enum LatLongError {
    #[error("invalid latitude value: `{0}`")]
    Latitude(f64),
    #[error("invalid longitude value: `{0}`")]
    Longitude(f64),
}

impl TryFrom<(f64, f64)> for LatLong {
    type Error = LatLongError;

    /// Attempt to convert a pair of f64 values in latitude, longitude order.
    ///
    /// return a valid `LatLong` or a `LatLongError`.
    fn try_from(lat_long: (f64, f64)) -> Result<Self, Self::Error> {
        if !is_valid_latitude(lat_long.0) {
            Err(LatLongError::Latitude(lat_long.0))
        } else if !is_valid_longitude(lat_long.1) {
            Err(LatLongError::Longitude(lat_long.1))
        } else {
            Ok(Self::new(Degrees(lat_long.0), Degrees(lat_long.1)))
        }
    }
}

/// Calculate the azimuth and distance along the great circle of point b from
/// point a.
/// * `a`, `b` - the start and end positions
///
/// returns the great-circle azimuth relative to North and distance of point b
/// from point a.
#[must_use]
pub fn calculate_azimuth_and_distance(a: &LatLong, b: &LatLong) -> (Angle, Radians) {
    let a_lat = Angle::from(a.lat);
    let b_lat = Angle::from(b.lat);
    let delta_long = Angle::from((b.lon, a.lon));
    (
        great_circle::calculate_gc_azimuth(a_lat, b_lat, delta_long),
        great_circle::calculate_gc_distance(a_lat, b_lat, delta_long),
    )
}

/// Calculate the distance along the great circle of point b from point a.
///
/// See: [Haversine formula](https://en.wikipedia.org/wiki/Haversine_formula).
/// This function is less accurate than `calculate_azimuth_and_distance`.
/// * `a`, `b` - the start and end positions
///
/// returns the great-circle distance of point b from point a in `Radians`.
#[must_use]
pub fn haversine_distance(a: &LatLong, b: &LatLong) -> Radians {
    let a_lat = Angle::from(a.lat);
    let b_lat = Angle::from(b.lat);
    let delta_lat = Angle::from((b.lat, a.lat));
    let delta_long = Angle::from(b.lon - a.lon);
    great_circle::calculate_haversine_distance(a_lat, b_lat, delta_long, delta_lat)
}

/// A `Vector3d` is a [nalgebra](https://crates.io/crates/nalgebra) `Vector3<f64>`.
#[allow(clippy::module_name_repetitions)]
pub type Vector3d = na::Vector3<f64>;

impl From<&LatLong> for Vector3d {
    /// Convert a `LatLong` to a point on the unit sphere.
    ///
    /// @pre |lat| <= 90.0 degrees.
    /// * `lat` - the latitude.
    /// * `lon` - the longitude.
    ///
    /// returns a `Vector3d` of the point on the unit sphere.
    fn from(a: &LatLong) -> Self {
        vector::to_point(Angle::from(a.lat), Angle::from(a.lon))
    }
}

impl From<&Vector3d> for LatLong {
    /// Convert a point to a `LatLong`
    fn from(value: &Vector3d) -> Self {
        Self::new(
            Degrees::from(vector::latitude(value)),
            Degrees::from(vector::longitude(value)),
        )
    }
}

/// An `Arc` of a Great Circle on a unit sphere.
#[derive(Clone, Copy, Debug, PartialEq)]
pub struct Arc {
    /// The start point of the `Arc`.
    a: Vector3d,
    /// The right hand pole of the Great Circle of the `Arc`.
    pole: Vector3d,
    /// The length of the `Arc`.
    length: Radians,
    /// The half width of the `Arc`.
    half_width: Radians,
}

impl Validate for Arc {
    /// Test whether an `Arc` is valid.
    ///
    /// I.e. both a and pole are on the unit sphere and are orthogonal and
    /// both length and `half_width` are not negative.
    fn is_valid(&self) -> bool {
        vector::is_unit(&self.a)
            && vector::is_unit(&self.pole)
            && vector::are_orthogonal(&self.a, &self.pole)
            && !self.length.0.is_sign_negative()
            && !self.half_width.0.is_sign_negative()
    }
}

impl Arc {
    /// Construct an `Arc`
    ///
    /// * `a` - the start point of the `Arc`.
    /// * `pole` - the right hand pole of the Great Circle of the `Arc`.
    /// * `length` - the length of the `Arc`.
    /// * `half_width` - the half width of the `Arc`.
    #[must_use]
    pub const fn new(a: Vector3d, pole: Vector3d, length: Radians, half_width: Radians) -> Self {
        Self {
            a,
            pole,
            length,
            half_width,
        }
    }

    /// Construct an `Arc`
    ///
    /// * `a` - the start position
    /// * `azimuth` - the azimuth at a.
    /// * `length` - the length of the `Arc`.
    #[must_use]
    pub fn from_lat_lon_azi_length(a: &LatLong, azimuth: Angle, length: Radians) -> Self {
        Self::new(
            Vector3d::from(a),
            vector::calculate_pole(Angle::from(a.lat()), Angle::from(a.lon()), azimuth),
            length,
            Radians(0.0),
        )
    }

    /// Construct an `Arc` from the start and end positions.
    ///
    /// Note: if the points are the same or antipodal, the pole will be invalid.
    /// * `a`, `b` - the start and end positions
    #[must_use]
    pub fn between_positions(a: &LatLong, b: &LatLong) -> Self {
        let (azimuth, length) = calculate_azimuth_and_distance(a, b);
        let a_lat = Angle::from(a.lat());
        // if a is at the North or South pole
        if a_lat.cos().0 < great_circle::MIN_VALUE {
            // use b's longitude
            Self::from_lat_lon_azi_length(&LatLong::new(a.lat(), b.lon()), azimuth, length)
        } else {
            Self::from_lat_lon_azi_length(a, azimuth, length)
        }
    }

    /// Set the `half_width` of an `Arc`.
    ///
    /// * `half_width` - the half width of the `Arc`.
    #[must_use]
    pub const fn set_half_width(&mut self, half_width: Radians) -> &mut Self {
        self.half_width = half_width;
        self
    }

    /// The start point of the `Arc`.
    #[must_use]
    pub const fn a(&self) -> Vector3d {
        self.a
    }

    /// The right hand pole of the Great Circle at the start point of the `Arc`.
    #[must_use]
    pub const fn pole(&self) -> Vector3d {
        self.pole
    }

    /// The length of the `Arc`.
    #[must_use]
    pub const fn length(&self) -> Radians {
        self.length
    }

    /// The half width of the `Arc`.
    #[must_use]
    pub const fn half_width(&self) -> Radians {
        self.half_width
    }

    /// The azimuth at the start point.
    #[must_use]
    pub fn azimuth(&self) -> Angle {
        vector::calculate_azimuth(&self.a, &self.pole)
    }

    /// The direction vector of the `Arc` at the start point.
    #[must_use]
    pub fn direction(&self) -> Vector3d {
        vector::direction(&self.a, &self.pole)
    }

    /// A position vector at distance along the `Arc`.
    #[must_use]
    pub fn position(&self, distance: Radians) -> Vector3d {
        vector::position(&self.a, &self.direction(), Angle::from(distance))
    }

    /// The end point of the `Arc`.
    #[must_use]
    pub fn b(&self) -> Vector3d {
        self.position(self.length)
    }

    /// The mid point of the `Arc`.
    #[must_use]
    pub fn mid_point(&self) -> Vector3d {
        self.position(Radians(0.5 * self.length.0))
    }

    /// The position of a perpendicular point at distance from the `Arc`.
    ///
    /// * `point` a point on the `Arc`'s great circle.
    /// * `distance` the perpendicular distance from the `Arc`'s great circle.
    ///
    /// returns the point at perpendicular distance from point.
    #[must_use]
    pub fn perp_position(&self, point: &Vector3d, distance: Radians) -> Vector3d {
        vector::position(point, &self.pole, Angle::from(distance))
    }

    /// The position of a point at angle from the `Arc` start, at `Arc` length.
    ///
    /// * `angle` the angle from the `Arc` start.
    ///
    /// returns the point at angle from the `Arc` start, at `Arc` length.
    #[must_use]
    pub fn angle_position(&self, angle: Angle) -> Vector3d {
        vector::rotate_position(&self.a, &self.pole, angle, Angle::from(self.length))
    }

    /// The `Arc` at the end of an `Arc`, just the point if `half_width` is zero.
    ///
    /// @param `at_b` if true the `Arc` at b, else the `Arc` at a.
    ///
    /// @return the end `Arc` at a or b.
    #[must_use]
    pub fn end_arc(&self, at_b: bool) -> Self {
        let p = if at_b { self.b() } else { self.a };
        let pole = vector::direction(&p, &self.pole);
        if self.half_width.0 < great_circle::MIN_VALUE {
            Self::new(p, pole, Radians(0.0), Radians(0.0))
        } else {
            let a = self.perp_position(&p, self.half_width);
            Self::new(a, pole, self.half_width + self.half_width, Radians(0.0))
        }
    }

    /// Calculate great-circle along and across track distances of point from
    /// the `Arc`.
    ///
    /// * `point` - the point.
    ///
    /// returns the along and across track distances of the point in Radians.
    #[must_use]
    pub fn calculate_atd_and_xtd(&self, point: &Vector3d) -> (Radians, Radians) {
        vector::calculate_atd_and_xtd(&self.a, &self.pole(), point)
    }

    /// Calculate the shortest great-circle distance of a point from the `Arc`.
    ///
    /// * `point` - the point.
    ///
    /// returns the shortest distance of a point from the `Arc` in Radians.
    #[must_use]
    pub fn shortest_distance(&self, point: &Vector3d) -> Radians {
        let (atd, xtd) = self.calculate_atd_and_xtd(point);
        if vector::intersection::is_alongside(atd, self.length, Radians(4.0 * f64::EPSILON)) {
            xtd.abs()
        } else {
            let sq_a = vector::sq_distance(&self.a, point);
            let sq_b = vector::sq_distance(&self.b(), point);
            let sq_d = min(sq_a, sq_b);
            great_circle::e2gc_distance(libm::sqrt(sq_d))
        }
    }
}

/// A Error type for an invalid `Arc`.
#[derive(Error, Debug, PartialEq)]
pub enum ArcError {
    #[error("positions are too close: `{0}`")]
    PositionsTooClose(f64),
    #[error("positions are too far apart: `{0}`")]
    PositionsTooFar(f64),
}

impl TryFrom<(&LatLong, &LatLong)> for Arc {
    type Error = ArcError;

    /// Construct an `Arc` from a pair of positions.
    ///
    /// * `params` - the start and end positions
    fn try_from(params: (&LatLong, &LatLong)) -> Result<Self, Self::Error> {
        // Convert positions to vectors
        let a = Vector3d::from(params.0);
        let b = Vector3d::from(params.1);
        // Calculate the great circle pole
        vector::normalise(&a.cross(&b), vector::MIN_SQ_NORM).map_or_else(
            || {
                let sq_d = vector::sq_distance(&a, &b);
                if sq_d < 1.0 {
                    Err(ArcError::PositionsTooClose(sq_d))
                } else {
                    Err(ArcError::PositionsTooFar(sq_d))
                }
            },
            |pole| {
                Ok(Self::new(
                    a,
                    pole,
                    great_circle::e2gc_distance(vector::distance(&a, &b)),
                    Radians(0.0),
                ))
            },
        )
    }
}

/// Calculate the great-circle distances along a pair of `Arc`s to their
/// closest intersection point or their coincident arc distances if the
/// `Arc`s are on coincident Great Circles.
///
/// * `arc1`, `arc2` the `Arc`s.
///
/// returns the distances along the first `Arc` and second `Arc` to the intersection
/// point or to their coincident arc distances if the `Arc`s do not intersect.
#[must_use]
pub fn calculate_intersection_distances(arc1: &Arc, arc2: &Arc) -> (Radians, Radians) {
    vector::intersection::calculate_intersection_point_distances(
        &arc1.a,
        &arc1.pole,
        arc1.length(),
        &arc2.a,
        &arc2.pole,
        arc2.length(),
        &(0.5 * (arc1.mid_point() + arc2.mid_point())),
    )
}

/// Calculate whether a pair of `Arc`s intersect and (if so) where.
///
/// * `arc1`, `arc2` the `Arc`s.
///
/// returns the distance along the first `Arc` to the second `Arc` or None if they
/// don't intersect.
///
/// # Examples
/// ```
/// use unit_sphere::{Arc, Degrees, LatLong, calculate_intersection_point};
/// use angle_sc::is_within_tolerance;
///
/// let istanbul = LatLong::new(Degrees(42.0), Degrees(29.0));
/// let washington = LatLong::new(Degrees(39.0), Degrees(-77.0));
/// let reyjavik = LatLong::new(Degrees(64.0), Degrees(-22.0));
/// let accra = LatLong::new(Degrees(6.0), Degrees(0.0));
///
/// let arc1 = Arc::try_from((&istanbul, &washington)).unwrap();
/// let arc2 = Arc::try_from((&reyjavik, &accra)).unwrap();
///
/// // Calculate the intersection point position
/// let intersection_point = calculate_intersection_point(&arc1, &arc2).unwrap();
/// let lat_long = LatLong::from(&intersection_point);
///
/// // The expected latitude and longitude are from:
/// // <https://sourceforge.net/p/geographiclib/discussion/1026621/thread/21aaff9f/#fe0a>
///
/// // Geodesic intersection latitude is 54.7170296089477
/// assert!(is_within_tolerance(54.72, lat_long.lat().0, 0.05));
/// // Geodesic intersection longitude is -14.56385574430775
/// assert!(is_within_tolerance(-14.56, lat_long.lon().0, 0.02));
/// ```
#[must_use]
pub fn calculate_intersection_point(arc1: &Arc, arc2: &Arc) -> Option<Vector3d> {
    let (distance1, distance2) = calculate_intersection_distances(arc1, arc2);

    // Determine whether both distances are within both arcs
    if vector::intersection::is_alongside(distance1, arc1.length(), Radians(4.0 * f64::EPSILON))
        && vector::intersection::is_alongside(distance2, arc2.length(), Radians(4.0 * f64::EPSILON))
    {
        Some(arc1.position(distance1.clamp(arc1.length())))
    } else {
        None
    }
}

#[cfg(test)]
mod tests {
    use super::*;
    use angle_sc::{is_within_tolerance, Degrees};

    #[test]
    fn test_is_valid_latitude() {
        // value < -90
        assert!(!is_valid_latitude(-90.0001));
        // value = -90
        assert!(is_valid_latitude(-90.0));
        // value = 90
        assert!(is_valid_latitude(90.0));
        // value > 90
        assert!(!is_valid_latitude(90.0001));
    }

    #[test]
    fn test_is_valid_longitude() {
        // value < -180
        assert!(!is_valid_longitude(-180.0001));
        // value = -180
        assert!(is_valid_longitude(-180.0));
        // value = 180
        assert!(is_valid_longitude(180.0));
        // value > 180
        assert!(!is_valid_longitude(180.0001));
    }

    #[test]
    fn test_latlong_traits() {
        let a = LatLong::try_from((0.0, 90.0)).unwrap();

        assert!(a.is_valid());

        let a_clone = a.clone();
        assert!(a_clone == a);

        assert_eq!(Degrees(0.0), a.lat());
        assert_eq!(Degrees(90.0), a.lon());

        assert!(!a.is_south_of(&a));
        assert!(!a.is_west_of(&a));

        let b = LatLong::try_from((-10.0, -91.0)).unwrap();
        assert!(b.is_south_of(&a));
        assert!(b.is_west_of(&a));

        println!("LatLong: {:?}", a);

        let invalid_lat = LatLong::try_from((91.0, 0.0));
        assert_eq!(Err(LatLongError::Latitude(91.0)), invalid_lat);
        println!("invalid_lat: {:?}", invalid_lat);

        let invalid_lon = LatLong::try_from((0.0, 181.0));
        assert_eq!(Err(LatLongError::Longitude(181.0)), invalid_lon);
        println!("invalid_lon: {:?}", invalid_lon);
    }

    #[test]
    fn test_vector3d_traits() {
        let a = LatLong::try_from((0.0, 90.0)).unwrap();
        let point = Vector3d::from(&a);

        assert_eq!(0.0, point.x);
        assert_eq!(1.0, point.y);
        assert_eq!(0.0, point.z);

        assert_eq!(Degrees(0.0), Degrees::from(vector::latitude(&point)));
        assert_eq!(Degrees(90.0), Degrees::from(vector::longitude(&point)));

        let result = LatLong::from(&point);
        assert_eq!(a, result);
    }

    #[test]
    fn test_great_circle_90n_0n_0e() {
        let a = LatLong::new(Degrees(90.0), Degrees(0.0));
        let b = LatLong::new(Degrees(0.0), Degrees(0.0));
        let (azimuth, dist) = calculate_azimuth_and_distance(&a, &b);

        assert!(is_within_tolerance(
            core::f64::consts::FRAC_PI_2,
            dist.0,
            f64::EPSILON
        ));
        assert_eq!(180.0, Degrees::from(azimuth).0);

        let dist = haversine_distance(&a, &b);
        assert!(is_within_tolerance(
            core::f64::consts::FRAC_PI_2,
            dist.0,
            f64::EPSILON
        ));
    }

    #[test]
    fn test_great_circle_90s_0n_50e() {
        let a = LatLong::new(Degrees(-90.0), Degrees(0.0));
        let b = LatLong::new(Degrees(0.0), Degrees(50.0));
        let (azimuth, dist) = calculate_azimuth_and_distance(&a, &b);

        assert!(is_within_tolerance(
            core::f64::consts::FRAC_PI_2,
            dist.0,
            f64::EPSILON
        ));
        assert_eq!(0.0, Degrees::from(azimuth).0);

        let dist = haversine_distance(&a, &b);
        assert!(is_within_tolerance(
            core::f64::consts::FRAC_PI_2,
            dist.0,
            f64::EPSILON
        ));
    }

    #[test]
    fn test_great_circle_0n_60e_0n_60w() {
        let a = LatLong::new(Degrees(0.0), Degrees(60.0));
        let b = LatLong::new(Degrees(0.0), Degrees(-60.0));
        let (azimuth, dist) = calculate_azimuth_and_distance(&a, &b);

        assert!(is_within_tolerance(
            2.0 * core::f64::consts::FRAC_PI_3,
            dist.0,
            2.0 * f64::EPSILON
        ));
        assert_eq!(-90.0, Degrees::from(azimuth).0);

        let dist = haversine_distance(&a, &b);
        assert!(is_within_tolerance(
            2.0 * core::f64::consts::FRAC_PI_3,
            dist.0,
            2.0 * f64::EPSILON
        ));
    }

    #[test]
    fn test_arc() {
        // Greenwich equator
        let g_eq = LatLong::new(Degrees(0.0), Degrees(0.0));

        // 90 degrees East on the equator
        let e_eq = LatLong::new(Degrees(0.0), Degrees(90.0));

        let mut arc = Arc::between_positions(&g_eq, &e_eq);
        let arc = arc.set_half_width(Radians(0.01));
        assert!(arc.is_valid());
        assert_eq!(Radians(0.01), arc.half_width());

        assert_eq!(Vector3d::from(&g_eq), arc.a());
        assert_eq!(Vector3d::new(0.0, 0.0, 1.0), arc.pole());
        assert!(is_within_tolerance(
            core::f64::consts::FRAC_PI_2,
            arc.length().0,
            f64::EPSILON
        ));
        assert_eq!(Angle::from(Degrees(90.0)), arc.azimuth());
        let b = Vector3d::from(&e_eq);
        assert!(is_within_tolerance(
            0.0,
            vector::distance(&b, &arc.b()),
            f64::EPSILON
        ));

        let mid_point = arc.mid_point();
        assert_eq!(0.0, mid_point.z);
        assert!(is_within_tolerance(
            45.0,
            Degrees::from(vector::longitude(&mid_point)).0,
            32.0 * f64::EPSILON
        ));

        let start_arc = arc.end_arc(false);
        assert_eq!(0.02, start_arc.length().0);

        let start_arc_a = start_arc.a();
        assert_eq!(start_arc_a, arc.perp_position(&arc.a(), Radians(0.01)));

        let angle_90 = Angle::from(Degrees(90.0));
        let pole_0 = Vector3d::new(0.0, 0.0, 1.0);
        assert!(vector::distance(&pole_0, &arc.angle_position(angle_90)) <= f64::EPSILON);

        let end_arc = arc.end_arc(true);
        assert_eq!(0.02, end_arc.length().0);

        let end_arc_a = end_arc.a();
        assert_eq!(end_arc_a, arc.perp_position(&arc.b(), Radians(0.01)));
    }

    #[test]
    fn test_north_and_south_poles() {
        let north_pole = LatLong::new(Degrees(90.0), Degrees(0.0));
        let south_pole = LatLong::new(Degrees(-90.0), Degrees(0.0));

        let (azimuth, distance) = calculate_azimuth_and_distance(&south_pole, &north_pole);
        assert_eq!(0.0, Degrees::from(azimuth).0);
        assert_eq!(core::f64::consts::PI, distance.0);

        let (azimuth, distance) = calculate_azimuth_and_distance(&north_pole, &south_pole);
        assert_eq!(180.0, Degrees::from(azimuth).0);
        assert_eq!(core::f64::consts::PI, distance.0);

        // 90 degrees East on the equator
        let e_eq = LatLong::new(Degrees(0.0), Degrees(50.0));

        let arc = Arc::between_positions(&north_pole, &e_eq);
        assert!(is_within_tolerance(
            e_eq.lat().0,
            LatLong::from(&arc.b()).lat().abs().0,
            1e-13
        ));
        assert!(is_within_tolerance(
            e_eq.lon().0,
            LatLong::from(&arc.b()).lon().0,
            50.0 * f64::EPSILON
        ));

        let arc = Arc::between_positions(&south_pole, &e_eq);
        assert!(is_within_tolerance(
            e_eq.lat().0,
            LatLong::from(&arc.b()).lat().abs().0,
            1e-13
        ));
        assert!(is_within_tolerance(
            e_eq.lon().0,
            LatLong::from(&arc.b()).lon().0,
            50.0 * f64::EPSILON
        ));

        let w_eq = LatLong::new(Degrees(0.0), Degrees(-140.0));

        let arc = Arc::between_positions(&north_pole, &w_eq);
        assert!(is_within_tolerance(
            w_eq.lat().0,
            LatLong::from(&arc.b()).lat().abs().0,
            1e-13
        ));
        assert!(is_within_tolerance(
            w_eq.lon().0,
            LatLong::from(&arc.b()).lon().0,
            256.0 * f64::EPSILON
        ));

        let arc = Arc::between_positions(&south_pole, &w_eq);
        assert!(is_within_tolerance(
            w_eq.lat().0,
            LatLong::from(&arc.b()).lat().abs().0,
            1e-13
        ));
        assert!(is_within_tolerance(
            w_eq.lon().0,
            LatLong::from(&arc.b()).lon().0,
            256.0 * f64::EPSILON
        ));

        let invalid_arc = Arc::try_from((&north_pole, &north_pole));
        assert_eq!(Err(ArcError::PositionsTooClose(0.0)), invalid_arc);
        println!("invalid_arc: {:?}", invalid_arc);

        let arc = Arc::between_positions(&north_pole, &north_pole);
        assert_eq!(north_pole, LatLong::from(&arc.b()));

        let invalid_arc = Arc::try_from((&north_pole, &south_pole));
        assert_eq!(Err(ArcError::PositionsTooFar(4.0)), invalid_arc);
        println!("invalid_arc: {:?}", invalid_arc);

        let arc = Arc::between_positions(&north_pole, &south_pole);
        assert_eq!(south_pole, LatLong::from(&arc.b()));

        let arc = Arc::between_positions(&south_pole, &north_pole);
        assert_eq!(north_pole, LatLong::from(&arc.b()));

        let arc = Arc::between_positions(&south_pole, &south_pole);
        assert_eq!(south_pole, LatLong::from(&arc.b()));
    }

    #[test]
    fn test_arc_atd_and_xtd() {
        // Greenwich equator
        let g_eq = LatLong::new(Degrees(0.0), Degrees(0.0));

        // 90 degrees East on the equator
        let e_eq = LatLong::new(Degrees(0.0), Degrees(90.0));

        let arc = Arc::try_from((&g_eq, &e_eq)).unwrap();
        assert!(arc.is_valid());

        let start_arc = arc.end_arc(false);
        assert_eq!(0.0, start_arc.length().0);

        let start_arc_a = start_arc.a();
        assert_eq!(arc.a(), start_arc_a);

        let longitude = Degrees(1.0);

        // Test across track distance
        // Accuracy drops off outside of this range
        for lat in -83..84 {
            let latitude = Degrees(lat as f64);
            let latlong = LatLong::new(latitude, longitude);
            let point = Vector3d::from(&latlong);

            let expected = (lat as f64).to_radians();
            let (atd, xtd) = arc.calculate_atd_and_xtd(&point);
            assert!(is_within_tolerance(1_f64.to_radians(), atd.0, f64::EPSILON));
            assert!(is_within_tolerance(expected, xtd.0, 2.0 * f64::EPSILON));

            let d = arc.shortest_distance(&point);
            assert!(is_within_tolerance(
                libm::fabs(expected),
                d.0,
                2.0 * f64::EPSILON
            ));
        }

        let point = Vector3d::from(&g_eq);
        let d = arc.shortest_distance(&point);
        assert_eq!(0.0, d.0);

        let point = Vector3d::from(&e_eq);
        let d = arc.shortest_distance(&point);
        assert_eq!(0.0, d.0);

        let latlong = LatLong::new(Degrees(0.0), Degrees(-1.0));
        let point = Vector3d::from(&latlong);
        let d = arc.shortest_distance(&point);
        assert!(is_within_tolerance(1_f64.to_radians(), d.0, f64::EPSILON));

        let point = -point;
        let d = arc.shortest_distance(&point);
        assert!(is_within_tolerance(89_f64.to_radians(), d.0, f64::EPSILON));
    }

    #[test]
    fn test_arc_intersection_point() {
        // Karney's example:
        // Istanbul, Washington, Reyjavik and Accra
        // from: <https://sourceforge.net/p/geographiclib/discussion/1026621/thread/21aaff9f/#fe0a>
        let istanbul = LatLong::new(Degrees(42.0), Degrees(29.0));
        let washington = LatLong::new(Degrees(39.0), Degrees(-77.0));
        let reyjavik = LatLong::new(Degrees(64.0), Degrees(-22.0));
        let accra = LatLong::new(Degrees(6.0), Degrees(0.0));

        let arc1 = Arc::try_from((&istanbul, &washington)).unwrap();
        let arc2 = Arc::try_from((&reyjavik, &accra)).unwrap();

        let intersection_point = calculate_intersection_point(&arc1, &arc2).unwrap();
        let lat_long = LatLong::from(&intersection_point);
        // Geodesic intersection latitude is 54.7170296089477
        assert!(is_within_tolerance(54.72, lat_long.lat().0, 0.05));
        // Geodesic intersection longitude is -14.56385574430775
        assert!(is_within_tolerance(-14.56, lat_long.lon().0, 0.02));

        // Switch arcs
        let intersection_point = calculate_intersection_point(&arc2, &arc1).unwrap();
        let lat_long = LatLong::from(&intersection_point);
        // Geodesic intersection latitude is 54.7170296089477
        assert!(is_within_tolerance(54.72, lat_long.lat().0, 0.05));
        // Geodesic intersection longitude is -14.56385574430775
        assert!(is_within_tolerance(-14.56, lat_long.lon().0, 0.02));
    }

    #[test]
    fn test_arc_intersection_same_great_circles() {
        let south_pole_1 = LatLong::new(Degrees(-88.0), Degrees(-180.0));
        let south_pole_2 = LatLong::new(Degrees(-87.0), Degrees(0.0));

        let arc1 = Arc::try_from((&south_pole_1, &south_pole_2)).unwrap();

        let intersection_lengths = calculate_intersection_distances(&arc1, &arc1);
        assert_eq!(Radians(0.0), intersection_lengths.0);
        assert_eq!(Radians(0.0), intersection_lengths.1);

        let intersection_point = calculate_intersection_point(&arc1, &arc1).unwrap();
        assert_eq!(0.0, vector::sq_distance(&arc1.a(), &intersection_point));

        let south_pole_3 = LatLong::new(Degrees(-85.0), Degrees(0.0));
        let south_pole_4 = LatLong::new(Degrees(-86.0), Degrees(0.0));
        let arc2 = Arc::try_from((&south_pole_3, &south_pole_4)).unwrap();
        let intersection_point = calculate_intersection_point(&arc1, &arc2);
        assert!(intersection_point.is_none());
    }
}