proj-core 0.5.0

Pure-Rust coordinate transformation library with no C dependencies
Documentation 
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use crate::ellipsoid::Ellipsoid;
use crate::error::Result;
use crate::projection::{
    ensure_finite_lon_lat, ensure_finite_xy, validate_angle, validate_latitude_param,
    validate_lon_lat, validate_offset, validate_projected, validate_scale,
};

/// Transverse Mercator projection.
///
/// The foundation for UTM zones and many national grid systems.
/// Uses series expansion for high accuracy.
pub(crate) struct TransverseMercator {
    /// Semi-major axis.
    a: f64,
    /// First eccentricity squared.
    e2: f64,
    /// Second eccentricity squared.
    ep2: f64,
    /// Semi-major axis times (1 - e^2).
    a_one_minus_e2: f64,
    /// Central meridian in radians.
    lon0: f64,
    /// Scale factor on central meridian.
    k0: f64,
    /// False easting in meters.
    false_easting: f64,
    /// False northing in meters.
    false_northing: f64,
    /// Meridional arc at the latitude of origin.
    m0: f64,
    /// Precomputed meridional arc coefficients.
    meridional_coeff0: f64,
    meridional_coeff2: f64,
    meridional_coeff4: f64,
    meridional_coeff6: f64,
    /// Denominator for the inverse meridional arc expansion.
    mu_denom: f64,
    /// Precomputed inverse series coefficients.
    e1_coeff2: f64,
    e1_coeff4: f64,
    e1_coeff6: f64,
}

impl TransverseMercator {
    pub(crate) fn new(
        ellipsoid: Ellipsoid,
        lon0: f64,
        lat0: f64,
        k0: f64,
        false_easting: f64,
        false_northing: f64,
    ) -> Result<Self> {
        validate_angle("central meridian", lon0)?;
        validate_latitude_param("latitude of origin", lat0)?;
        validate_scale("scale factor", k0)?;
        validate_offset("false easting", false_easting)?;
        validate_offset("false northing", false_northing)?;

        let a = ellipsoid.a;
        let e2 = ellipsoid.e2();
        let ep2 = ellipsoid.ep2();
        let e2_2 = e2 * e2;
        let e2_3 = e2_2 * e2;

        let meridional_coeff0 = a * (1.0 - e2 / 4.0 - 3.0 * e2_2 / 64.0 - 5.0 * e2_3 / 256.0);
        let meridional_coeff2 = a * (3.0 * e2 / 8.0 + 3.0 * e2_2 / 32.0 + 45.0 * e2_3 / 1024.0);
        let meridional_coeff4 = a * (15.0 * e2_2 / 256.0 + 45.0 * e2_3 / 1024.0);
        let meridional_coeff6 = a * (35.0 * e2_3 / 3072.0);
        let m0 = if lat0.abs() < 1e-12 {
            0.0
        } else {
            meridional_arc(
                lat0,
                meridional_coeff0,
                meridional_coeff2,
                meridional_coeff4,
                meridional_coeff6,
            )
        };

        let sqrt_one_minus_e2 = (1.0 - e2).sqrt();
        let e1 = (1.0 - sqrt_one_minus_e2) / (1.0 + sqrt_one_minus_e2);
        let e1_2 = e1 * e1;
        let e1_3 = e1_2 * e1;
        let e1_4 = e1_2 * e1_2;

        Ok(Self {
            a,
            e2,
            ep2,
            a_one_minus_e2: a * (1.0 - e2),
            lon0,
            k0,
            false_easting,
            false_northing,
            m0,
            meridional_coeff0,
            meridional_coeff2,
            meridional_coeff4,
            meridional_coeff6,
            mu_denom: meridional_coeff0,
            e1_coeff2: 3.0 * e1 / 2.0 - 27.0 * e1_3 / 32.0,
            e1_coeff4: 21.0 * e1_2 / 16.0 - 55.0 * e1_4 / 32.0,
            e1_coeff6: 151.0 * e1_3 / 96.0,
        })
    }

    /// Compute meridional arc length from equator to latitude phi.
    fn meridional_arc(&self, phi: f64) -> f64 {
        meridional_arc(
            phi,
            self.meridional_coeff0,
            self.meridional_coeff2,
            self.meridional_coeff4,
            self.meridional_coeff6,
        )
    }
}

fn meridional_arc(phi: f64, coeff0: f64, coeff2: f64, coeff4: f64, coeff6: f64) -> f64 {
    coeff0 * phi - coeff2 * (2.0 * phi).sin() + coeff4 * (4.0 * phi).sin()
        - coeff6 * (6.0 * phi).sin()
}

impl super::ProjectionImpl for TransverseMercator {
    fn forward(&self, lon: f64, lat: f64) -> Result<(f64, f64)> {
        validate_lon_lat(lon, lat)?;
        let k0 = self.k0;

        let phi = lat;
        let d_lon = lon - self.lon0;

        let sin_phi = phi.sin();
        let cos_phi = phi.cos();
        let tan_phi = phi.tan();

        let n_val = self.a / (1.0 - self.e2 * sin_phi * sin_phi).sqrt();
        let t = tan_phi * tan_phi;
        let c = self.ep2 * cos_phi * cos_phi;
        let a_coeff = d_lon * cos_phi;

        let m = self.meridional_arc(phi);
        let a2 = a_coeff * a_coeff;

        let x = self.false_easting
            + k0 * n_val
                * (a_coeff
                    + (1.0 - t + c) * a2 * a_coeff / 6.0
                    + (5.0 - 18.0 * t + t * t + 72.0 * c - 58.0 * self.ep2) * a2 * a2 * a_coeff
                        / 120.0);

        let y = self.false_northing
            + k0 * (m - self.m0
                + n_val
                    * tan_phi
                    * (a2 / 2.0
                        + (5.0 - t + 9.0 * c + 4.0 * c * c) * a2 * a2 / 24.0
                        + (61.0 - 58.0 * t + t * t + 600.0 * c - 330.0 * self.ep2) * a2 * a2 * a2
                            / 720.0));

        ensure_finite_xy("Transverse Mercator", x, y)
    }

    fn inverse(&self, x: f64, y: f64) -> Result<(f64, f64)> {
        validate_projected(x, y)?;
        let k0 = self.k0;

        let m = self.m0 + (y - self.false_northing) / k0;
        let mu = m / self.mu_denom;

        let phi1 = mu
            + self.e1_coeff2 * (2.0 * mu).sin()
            + self.e1_coeff4 * (4.0 * mu).sin()
            + self.e1_coeff6 * (6.0 * mu).sin();

        let sin_phi1 = phi1.sin();
        let cos_phi1 = phi1.cos();
        let tan_phi1 = phi1.tan();
        let n1 = self.a / (1.0 - self.e2 * sin_phi1 * sin_phi1).sqrt();
        let t1 = tan_phi1 * tan_phi1;
        let c1 = self.ep2 * cos_phi1 * cos_phi1;
        let r1 = self.a_one_minus_e2 / (1.0 - self.e2 * sin_phi1 * sin_phi1).powf(1.5);
        let d = (x - self.false_easting) / (n1 * k0);
        let d2 = d * d;

        let lat = phi1
            - (n1 * tan_phi1 / r1)
                * (d2 / 2.0
                    - (5.0 + 3.0 * t1 + 10.0 * c1 - 4.0 * c1 * c1 - 9.0 * self.ep2) * d2 * d2
                        / 24.0
                    + (61.0 + 90.0 * t1 + 298.0 * c1 + 45.0 * t1 * t1
                        - 252.0 * self.ep2
                        - 3.0 * c1 * c1)
                        * d2
                        * d2
                        * d2
                        / 720.0);

        let lon = self.lon0
            + (d - (1.0 + 2.0 * t1 + c1) * d2 * d / 6.0
                + (5.0 - 2.0 * c1 + 28.0 * t1 - 3.0 * c1 * c1 + 8.0 * self.ep2 + 24.0 * t1 * t1)
                    * d2
                    * d2
                    * d
                    / 120.0)
                / cos_phi1;

        ensure_finite_lon_lat("Transverse Mercator", lon, lat)
    }
}

#[cfg(test)]
mod tests {
    use super::*;
    use crate::ellipsoid;
    use crate::projection::ProjectionImpl;

    const TOLERANCE: f64 = 0.01; // 1 cm

    fn utm(zone: u8, north: bool) -> TransverseMercator {
        let lon0 = ((zone as f64 - 1.0) * 6.0 - 180.0 + 3.0).to_radians();
        TransverseMercator::new(
            ellipsoid::WGS84,
            lon0,
            0.0,
            0.9996,
            500_000.0,
            if north { 0.0 } else { 10_000_000.0 },
        )
        .unwrap()
    }

    #[test]
    fn utm_zone_18n_new_york() {
        let proj = utm(18, true);
        let lon = (-74.006_f64).to_radians();
        let lat = 40.7128_f64.to_radians();
        let (x, y) = proj.forward(lon, lat).unwrap();

        assert!((x - 583960.0).abs() < 1.0);
        assert!(y > 4_500_000.0 && y < 4_510_000.0);

        let (lon_back, lat_back) = proj.inverse(x, y).unwrap();
        assert!((lon_back - lon).abs() < 1e-8);
        assert!((lat_back - lat).abs() < 1e-8);
    }

    #[test]
    fn utm_equator_central_meridian() {
        let proj = utm(31, true);
        let lon = 3.0_f64.to_radians();
        let lat = 0.0;
        let (x, y) = proj.forward(lon, lat).unwrap();
        assert!((x - 500000.0).abs() < TOLERANCE);
        assert!(y.abs() < TOLERANCE);
    }
}