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//! Implementation of the HEALPix framework. //! See papers: //! * Gorsky2005: "HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data //! Distributed on the Sphere", Górski, K. M. et al., 2005; 2005ApJ...622..759G. //! * Calabretta2004: "Mapping on the HEALPix grid", Calabretta, M. R., 2004; 2004astro.ph.12607C //! * Calabretta2007: "Mapping on the HEALPix grid", Calabretta, M. R. et Roukema, B. F., 2007; 2007MNRAS.381..865C //! * Reinecke2015: "Efficient data structures for masks on 2D grids", Reinecke, M. et Hivon, E., 2015; 2015A&A...580A.132R #![cfg_attr(test, feature(test))] #[cfg(test)] extern crate test; use std::sync::Once; use std::f64::consts::{PI}; const SQRT2: f64 = 1.41421356237309504880_f64; const SQRT6: f64 = 2.44948974278317809819_f64; const ONE_OVER_SQRT6: f64 = 0.40824829046386301636_f64; const HALF: f64 = 0.5_f64; /// Upper limit on sqrt(3(1-|z|)) to consider that we are not near from the poles const EPS_POLE: f64 = 1e-13_f64; /// Constant = pi/2. /// /// ```rust /// use cdshealpix::{HALF_PI}; /// use std::f64::consts::PI; /// assert_eq!(PI / 2f64, HALF_PI); /// ``` pub const HALF_PI: f64 = HALF * PI; /// Constant = 4/pi. /// /// ```rust /// use cdshealpix::{FOUR_OVER_PI}; /// use std::f64::consts::PI; /// assert_eq!(4f64 / PI, FOUR_OVER_PI); /// ``` pub const FOUR_OVER_PI: f64 = 4_f64 / PI; /// Constant = pi/4. /// /// ```rust /// use cdshealpix::{PI_OVER_FOUR}; /// use std::f64::consts::PI; /// assert_eq!(PI / 4f64, PI_OVER_FOUR); /// ``` pub const PI_OVER_FOUR: f64 = 0.25_f64 * PI; /// Constant = 29, i.e. the largest possible depth we can store on a signed positive long /// (4 bits for base cells + 2 bits per depth + 2 remaining bits (1 use in the unique notation). /// /// ```rust /// use cdshealpix::{DEPTH_MAX}; /// assert_eq!(29, DEPTH_MAX); /// ``` pub const DEPTH_MAX: u8 = 29; /// Constant = nside(29), i.e. the largest possible nside available when we store HEALPix hash /// on a u64. /// /// ```rust /// use cdshealpix::{DEPTH_MAX, NSIDE_MAX, nside}; /// assert_eq!(nside(DEPTH_MAX), NSIDE_MAX); /// ``` pub const NSIDE_MAX: u32 = 536870912; /// Limit on the latitude (in radians) between the equatorial region and the polar caps. /// Equals asin(2/3) = 0.7297276562269663 radians ~= 41,81 degrees. /// Written $\theta_X$ in Calabretta2007. /// /// ```rust /// use cdshealpix::{TRANSITION_LATITUDE}; /// assert_eq!(f64::asin(2f64 / 3f64), TRANSITION_LATITUDE); /// ``` pub const TRANSITION_LATITUDE: f64 = 0.72972765622696636344_f64; // asin(2/3) /// Limit on |z|=|sin(lat)| between the equatorial region and the polar caps. /// Equals 2/3, see Eq. (1) in Gorsky2005. pub const TRANSITION_Z: f64 = 2_f64 / 3_f64; const ONE_OVER_TRANSITION_Z: f64 = 1.5_f64; const F64_SIGN_BIT_MASK: u64 = 0x8000000000000000; const F64_BUT_SIGN_BIT_MASK: u64 = 0x7FFFFFFFFFFFFFFF; /// For each HEALPix depth, stores the smallest distance from an edge of a cell to the opposite /// edge of the same cell. If the radius of a cone is smaller than this distance, we know that /// it will overlap maximum 9 pixels (the pixel containing the center of the cone plus /// the 8 neighbours). /// In practice, this distance if the distance between the point of coordinate /// (0, TRANSITION_LATITUDE) and it nearest point on the Northeast edge of the /// cell of base hash 0 and coordinates in the base hash (x=0, y=nside-1). /// IMPORTANT REMARK: /// - this value is larger than the smallest center to vertex distance /// - this value x2 is larger than the smallest diagonal (NS or EW) /// - this value x2 is larger than the smallest edge /// - BUT there is no case in which the value is larger than the four center-to-vertex distance /// - BUT there is no case in which the value x2 is larger than both diagonals /// => this radius is smaller than the smaller circumcircle radius (=> no cone having the smaller /// -edge-to-opposite-edge-radius radius can contains the 4 vertices of a cell (but 3 is ok) /// vertices static SMALLER_EDGE2OPEDGE_DIST: [f64; 30] = [ 0.8410686705685088, // depth = 0 0.37723631722170053, // depth = 1 0.18256386461918295, // depth = 2 0.09000432499034523, // depth = 3 0.04470553761855741, // depth = 4 0.02228115704023076, // depth = 5 0.011122977211214961, // depth = 6 0.005557125022105058, // depth = 7 0.0027774761500209185, // depth = 8 0.0013884670480328143, // depth = 9 6.941658374603201E-4, // depth = 10 3.4706600585087755E-4, // depth = 11 1.7352877579970442E-4, // depth = 12 8.676333125510362E-5, // depth = 13 4.338140148342286E-5, // depth = 14 2.1690634707822447E-5, // depth = 15 1.084530084565172E-5, // depth = 16 5.422646295795749E-6, // depth = 17 2.711322116099695E-6, // depth = 18 1.3556608000873442E-6, // depth = 19 6.778303355805395E-7, // depth = 20 3.389151516386149E-7, // depth = 21 1.69457571754776E-7, // depth = 22 8.472878485272006E-8, // depth = 23 4.236439215502565E-8, // depth = 24 2.1182195982014308E-8, // depth = 25 1.0591097960375205E-8, // depth = 26 5.295548939447981E-9, // depth = 27 2.647774429917369E-9, // depth = 28 1.3238871881399636E-9 // depth = 29 ]; // Idee pour cone: surface du cone => ncells !! (facteur variant 8, 4, 2, ... 1.2 /// Latitude, in the equatorial region, for which the distance from the cell center to its four /// vertices is almost equal on the sky (i.e. the shape of the cell on the sky is close to a square). /// The larger the depth, the better the approximation (based on differential calculus). /// > dX = dY = 1 / nside (center to vertex distance) /// > X = 4/pi * lon => dX = 4/pi dlon /// > Y = 3/2 * sin(lat) => dY = 3/2 * cos(lat) dlat /// > dlon * cos(lat) = dlat (same distance on the sky) /// > => cos^2(lat) = 2/3 * 4/pi /// > => lat = arccos(sqrt(2/3 * 4/pi)) ~= 22.88 deg ~= 0.39934 rad /// /// ```rust /// use cdshealpix::{TRANSITION_Z, FOUR_OVER_PI, LAT_OF_SQUARE_CELL}; /// assert!(f64::abs(f64::acos(f64::sqrt(TRANSITION_Z * FOUR_OVER_PI)) - LAT_OF_SQUARE_CELL) < 1e-15_f64); /// ``` pub static LAT_OF_SQUARE_CELL: f64 = 0.39934019947897773410_f64; /// Simply the consine of LAT_OF_SQUARE_CELL static COS_LAT_OF_SQUARE_CELL: f64 = 0.92131773192356127804_f64; /// Array storing pre-computed values for each of the 30 possible depth (from 0 to 29) /// Info: I would have prefered to compute those quantities at compilation time, and thus have /// a `static CSTS_C2V: [ConstantsC2V; 30]`. Unfortunately: /// - macro do not seems work with static arrays /// - const fn is not stable and can only use const fn (so no min/max/sin/ ...) :o/ static mut CSTS_C2V: [Option<ConstantsC2V>; 30] = [ None, None, None, None, None, None, None, None, None, None, None, None, None, None, None, None, None, None, None, None, None, None, None, None, None, None, None, None, None, None ]; // Found here: https://stackoverflow.com/questions/28656387/initialize-a-large-fixed-size-array-with-non-copy-types // I wanted to use it to set a static array. So far it is not possible. // I hope in the future to be able to compute such an array at compilation time. /*macro_rules! make_array { ($n: expr, $constructor: expr) => { { let mut items: [_; $n] = mem::uninitialized(); for (i, place) in items.iter_mut().enumerate() { ptr::write(place, $constructor(i as u8)); } items } } } static CSTS_C2V: [ConstantsC2V; 30] = unsafe { make_array!(30, |depth| new_cst_c2v(depth)) }; */ /* /// Defines a simple range `from` a given index (inclusive) `to` a given index (exclusive) pub struct Range { pub from: u64, pub to: u64, } impl Range { pub fn new(from: u64, to: u64) -> Range { if from >= to { panic!("Illegal argument: from '{}' must be lower than to '{}'.", from, to); } new_unsafe(from, to) } fn new_unsafe(from: u64, to: u64) -> Range { Range{from, to} } } */ /// See the get_or_create function, each object is used for the lazy instantiation of the /// layer of the corresponding depht. /// Info: Unfortunatly Default::default(); do no work with static arrays :o/ static CSTS_C2V_INIT: [Once; 30] = [ Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new(), Once::new() ]; /// Lazy factory method: instantiate a new Layer at the first call for a given depth; after the /// first call, returns an already instantiated Layer. /// # Info /// This method resort to a double-checked lock, ensuring thread-safety. fn get_or_create(depth: u8) -> &'static ConstantsC2V { unsafe { // Inspired from the Option get_or_insert_with method, modified to ensure thread safety with // https://doc.rust-lang.org/std/sync/struct.Once.html // This implements a double-checked lock match CSTS_C2V[depth as usize] { Some(ref v) => return v, None => { CSTS_C2V_INIT[depth as usize].call_once(|| { CSTS_C2V[depth as usize] = Some(ConstantsC2V::new(depth)); }); }, } match CSTS_C2V[depth as usize] { Some(ref v) => v, _ => unreachable!(), } } } struct ConstantsC2V { slope_npc: f64, intercept_npc: f64, slope_eqr: f64, intercept_eqr: f64, coeff_x2_eqr: f64, coeff_cst_eqr: f64, } impl ConstantsC2V { fn new(depth: u8) -> ConstantsC2V { let nside = nside_unsafe(depth); let dist_cw = 1.0_f64 / (nside as f64); // Center to West (or East) vertex distance on the transition latitude let one_min_dist_cw = 1.0_f64 - dist_cw; // NPC, see comment of function largest_c2v_dist_in_npc() let lat_north = f64::asin(1_f64 - (pow2(one_min_dist_cw) / 3_f64)); let mut d_min = lat_north - TRANSITION_LATITUDE; let mut d_max: f64 = sphe_dist( squared_half_segment( PI_OVER_FOUR * dist_cw, d_min, lat_north.cos(), TRANSITION_LATITUDE.cos())); // - linear approx let slope_npc: f64 = (d_max - d_min) / (PI_OVER_FOUR * one_min_dist_cw); // a = (yB - yA) / (xB - xA); with xA = 0 let intercept_npc: f64 = d_min; // b = yA - a * xA ; with xA = 0 // EQR TOP, see comment of function largest_c2v_dist_in_eqr_top() d_min = FOUR_OVER_PI * dist_cw * COS_LAT_OF_SQUARE_CELL; d_max = TRANSITION_LATITUDE - f64::asin(one_min_dist_cw * TRANSITION_Z); let slope_eqr: f64 = (d_max - d_min) / (TRANSITION_LATITUDE - LAT_OF_SQUARE_CELL); // a = (yB - yA) / (xB - xA) let intercept_eqr: f64 = d_min - slope_eqr * LAT_OF_SQUARE_CELL; // b = yA - a * xA // EQR BOTTOM, see comment of function largest_c2v_dist_in_eqr_bottom() d_max = FOUR_OVER_PI * dist_cw; let coeff_cst_eqr: f64 = d_max; let coeff_x2_eqr: f64 = (d_min - d_max) / pow2(LAT_OF_SQUARE_CELL); // Struct creation ConstantsC2V { slope_npc, intercept_npc, slope_eqr, intercept_eqr, coeff_x2_eqr, coeff_cst_eqr } } } #[inline] fn haversine_dist(p1_lon: f64, p1_lat: f64, p2_lon: f64, p2_lat: f64) -> f64 { let shs = squared_half_segment( p2_lon - p1_lon, p2_lat - p1_lat, p1_lat.cos(), p2_lat.cos()); sphe_dist(shs) } /// Returns the angular distance corresponding to the given squared half great-circle arc segment #[inline] fn sphe_dist(squared_half_segment: f64) -> f64 { squared_half_segment.sqrt().asin().twice() } /// Returns `(s/2)^2` with `s` the segment (i.e. the Euclidean distance) between /// the two given points `P1` and `P2` on the unit-sphere. /// We recall that `s = 2 sin(ad/2)` with `ad` the angular distance between the two points. /// # Input /// - `dlon` the longitude difference, i.e. (P2.lon - P1.lon), in radians /// - `dlat` the latitude difference, i.e. (P2.lat - P1.lat), in radians /// - `cos_lat1` cosine of the latitude of the first point /// - `cos_lat2` cosine of the latitude of the second point #[inline] fn squared_half_segment(dlon: f64, dlat: f64, cos_lat1: f64, cos_lat2: f64) -> f64 { dlat.half().sin().pow2() + cos_lat1 * cos_lat2 * dlon.half().sin().pow2() } #[inline] fn to_squared_half_segment(spherical_distance: f64) -> f64 { spherical_distance.half().sin().pow2() } /*#[inline] fn to_squared_half_segments(spherical_distances: &mut [f64]) { spherical_distances.iter_mut().for_each(|d| *d = to_squared_half_segment(*d)); }*/ #[inline] fn pow2(x: f64) -> f64 { x * x } /*impl f64 { #[inline] pub fn pow2(self) -> f64 { self * self } }*/ /// Simple trait used to implements `pow2`, `twice` and `half` on f64. pub trait Customf64 { #[inline] fn pow2(self) -> f64; #[inline] fn twice(self) -> f64; #[inline] fn half(self) -> f64; #[inline] fn div_eucl(self, rhs: f64) -> f64; } impl Customf64 for f64 { /// Returns x^2 #[inline] fn pow2(self) -> f64 { self * self // or powi ? } /// Returns 2 * x #[inline] fn twice(self) -> f64 { 2.0 * self // self + self (I hope the compiler know the simple shift bit to be used for x2) } /// Returns x / 2 #[inline] fn half(self) -> f64 { 0.5 * self } /// [Duplicated code](https://doc.rust-lang.org/std/primitive.f64.html#method.div_euc), because /// it is unstable so far. #[inline] fn div_eucl(self, rhs: f64) -> f64 { let q = (self / rhs).trunc(); if self % rhs < 0.0 { return if rhs > 0.0 { q - 1.0 } else { q + 1.0 } } q } } /// All types that implement `Write` get methods defined in `WriteBytesExt` /// for free. // impl<F: f64> Customf64 for F {}*/ /// Returns an upper limit on the distance between a cell center around the given position /// and its furthest vertex. /// # Params /// - `depth` the depth of the cell /// - `lon` the longitude of the point on the unit-sphere, in radians /// - `lat` the latitude of the point on the unit-sphere, in radians /// /// # Result /// The following plot shows, for the depth 8, the real largest distances (in red) and the result /// of this method (in blue). /// WARNING: the units of `(lon, lat)` on the plot are *degrees*, while the distance is in *mas* /// Credit: plot made using [TOPCAT](http://www.star.bris.ac.uk/~mbt/topcat/) /// ![CenterToVertexDist](https://raw.githubusercontent.com/cds-astro/cds-healpix-rust/master/resources/4doc/d_center_vertex.png) /// pub fn largest_center_to_vertex_distance(depth: u8, lon: f64, lat: f64) -> f64 { // Specific case for depth 0 if depth == 0 { return HALF_PI - TRANSITION_LATITUDE; } // Regular case let lat_abs = lat.abs(); if lat_abs >= TRANSITION_LATITUDE { largest_c2v_dist_in_npc(lon, get_or_create(depth)) } else if lat_abs >= LAT_OF_SQUARE_CELL { largest_c2v_dist_in_eqr_top(lat_abs, get_or_create(depth)) } else { largest_c2v_dist_in_eqr_bottom(lat_abs, get_or_create(depth)) } } /// Returns an upper limit on the distance between a cell center and it furthest vertex, for /// all the cells in the region covered by a cone of given center and radius. /// It is an extension of [largest_center_to_vertex_distance](#fn.largest_center_to_vertex_distance) /// # Params /// - `depth` the depth of the cell /// - `lon` the longitude of the point on the unit-sphere, in radians /// - `lat` the latitude of the point on the unit-sphere, in radians /// - `radius` the radius of the cone, in radians /// pub fn largest_center_to_vertex_distance_with_radius(depth: u8, lon: f64, lat: f64, radius: f64) -> f64 { // Specific case for depth 0 if depth == 0 { return HALF_PI - TRANSITION_LATITUDE; } // Regular case let lat_abs = lat.abs(); let lat_max = lat_abs + radius; let lat_min = lat_abs - radius; if lat_max >= TRANSITION_LATITUDE { largest_c2v_dist_in_npc_with_radius(lon, radius, get_or_create(depth)) } else if lat_min >= LAT_OF_SQUARE_CELL { largest_c2v_dist_in_eqr_top_with_radius(lat_abs, radius, get_or_create(depth)) } else if lat_max <= LAT_OF_SQUARE_CELL { largest_c2v_dist_in_eqr_bottom_with_radius(lat_abs, radius, get_or_create(depth)) } else { let csts = get_or_create(depth); f64::max( largest_c2v_dist_in_eqr_top_with_radius(lat_abs, radius, csts), largest_c2v_dist_in_eqr_bottom_with_radius(lat_abs, radius, csts) ) } } /// Same as [largest_center_to_vertex_distance_with_radius](#fn.largest_center_to_vertex_distance_with_radius) /// but making the computation for several depths at the same time. pub fn largest_center_to_vertex_distances_with_radius(mut from_depth: u8, to_depth: u8, lon: f64, lat: f64, radius: f64) -> Box<[f64]> { let mut vec: Vec<f64> = Vec::with_capacity((to_depth - from_depth) as usize); // Specific case for depth 0 if from_depth == 0 { vec.push(HALF_PI - TRANSITION_LATITUDE); from_depth = 1_u8; } // Regular case let lat_abs = lat.abs(); let lat_max = lat_abs + radius; let lat_min = lat_abs - radius; if lat_max >= TRANSITION_LATITUDE { let mut lon = (PI_OVER_FOUR - (lon % HALF_PI)).abs(); lon = f64::min(lon + radius, PI_OVER_FOUR); for depth in from_depth..to_depth { let csts = get_or_create(depth); vec.push(linear_approx(lon, csts.slope_npc, csts.intercept_npc)); } } else if lat_min >= LAT_OF_SQUARE_CELL { for depth in from_depth..to_depth { vec.push(largest_c2v_dist_in_eqr_top(lat_max, get_or_create(depth))); } } else if lat_max <= LAT_OF_SQUARE_CELL { let val_min = f64::max(lat_min, 0_f64); for depth in from_depth..to_depth { vec.push(largest_c2v_dist_in_eqr_bottom(val_min, get_or_create(depth))); } } else { let val_max = f64::min(lat_max, TRANSITION_LATITUDE); let val_min = f64::max(lat_min, 0_f64); for depth in from_depth..to_depth { let csts = get_or_create(depth); vec.push( f64::max( largest_c2v_dist_in_eqr_top(val_max, csts), largest_c2v_dist_in_eqr_bottom(val_min, csts) ) ); } } vec.into_boxed_slice() } /// Returns an upper limit on distance between the center of a cell and its furthest vertex. /// We assumes that the cell center is located in the North polar cap region (OR IS ON THE /// TRANSITION LATITUDE). /// We use a linear upper limit based on the longitude. /// - At the transition latitude, we note CN the distance at a base cell border between: /// - Cell center (C): (x_c = 1/nside, y_c = 1) => (lon_c = pi/4 * 1/nside, lat_c = TRANSITION_LATITUDE) /// - North vertex (N): (x_n = 0, y_c = 2 - (1 + 1/nside)) => (lon_n = 0, lat_n = asin(1 - ((1 - 1/nside)^2 / 3)) /// - using the Haversine formula and SC defined below: /// > dMax = CN = 2 * asin(sqrt( sin^2(SC/2) + sin^2(pi/8) * cos(pi/4 * 1/nside))) /// - At the transition latitude, we not SC the distance at a base cell center between: /// - South vertex (S): (lon_s = pi/4, lat_s = TRANSITION_LATITUDE) /// - Cell center: (lon_c = pi/4, lat_c = asin(1 - ((1 - 1/nside)^2 / 3)) /// > dMin = SC = asin(1 - ((1 - 1/nside)^2 / 3)) - TRANSITION_LATITUDE /// - finally, linear approx: /// > d = ((lon % pi/2) - pi/4) * (dMax - dMin)/(pi/4 * (1 - 1/nside)) + dMin #[inline] fn largest_c2v_dist_in_npc(lon: f64, csts: &ConstantsC2V) -> f64 { let lon = (PI_OVER_FOUR - (lon % HALF_PI)).abs(); debug_assert!(0_f64 <= lon && lon <= PI_OVER_FOUR); linear_approx(lon, csts.slope_npc, csts.intercept_npc) } /// Same as the above method, but taking into account an additional radius #[inline] fn largest_c2v_dist_in_npc_with_radius(lon: f64, radius: f64, csts: &ConstantsC2V) -> f64 { debug_assert!(0_f64 < radius); let mut lon = (PI_OVER_FOUR - (lon % HALF_PI)).abs(); debug_assert!(0_f64 <= lon && lon <= PI_OVER_FOUR); lon = f64::min(lon + radius, PI_OVER_FOUR); linear_approx(lon, csts.slope_npc, csts.intercept_npc) } /// Returns an upper limit on distance between the center of a cell and its furthest vertex. /// We assumes that the cell center is located in the equatorial region, /// above the latitude at which cells are squares. /// We use a linear upper limit based on the latitude. /// - At the latitude in which cells are squares (lat = LAT_OF_SQUARE_CELL), we note CE the /// Center-to-East distance: CE (=CW) = 4/pi * 1/nside * cos(LAT_OF_SQUARE_CELL) /// > dMin = 4/pi * 1/nside * cos(LAT_OF_SQUARE_CELL) /// - At the latitude under the transition latitude, we note CN the distance with N on the /// transition latitude and C at latitude: y_c = 1 - 1/nside => lat_c = arcsin(y_c * 2/3) /// > dMax = TRANSITION_LATITUDE - arcsin(y_c * 2/3) /// - finally, linear approx: /// > d = (lat - LAT_OF_SQUARE_CELL) * (dMax - dMin)/(TRANSITION_LATITUDE - LAT_OF_SQUARE_CELL) + dMin #[inline] fn largest_c2v_dist_in_eqr_top(lat_abs: f64, csts: &ConstantsC2V) -> f64 { debug_assert!(LAT_OF_SQUARE_CELL <= lat_abs && lat_abs < TRANSITION_LATITUDE); linear_approx(lat_abs, csts.slope_eqr, csts.intercept_eqr) } /// Same as the above method, but taking into account an additional radius #[inline] fn largest_c2v_dist_in_eqr_top_with_radius(lat_abs: f64, radius: f64, csts: &ConstantsC2V) -> f64 { debug_assert!(0_f64 < radius); debug_assert!(LAT_OF_SQUARE_CELL <= lat_abs && lat_abs < TRANSITION_LATITUDE); largest_c2v_dist_in_eqr_top(f64::min(lat_abs + radius, TRANSITION_LATITUDE), csts) } /// Returns an upper limit on distance between the center of a cell and its furthest vertex. /// We assumes that the cell center is located in the equatorial region, /// bellow the latitude at which cells are squares. /// We use a parabola approximation upper limit based on the latitude. /// - At lat = 0, d_max = pi/4 * 1/nside /// - At lat = LAT_OF_SQUARE_CELL, d_min = 4/pi * 1/nside * cos(LAT_OF_SQUARE_CELL) /// - Parabola approx: a * lat^2 + b = dist /// - At lat = 0, b = d_max /// - At lat = LAT_OF_SQUARE_CELL, a * LAT_OF_SQUARE_CELL^2 + d_max = d_min /// - => a = (d_min - d_max) / LAT_OF_SQUARE_CELL^2 /// - => a = d_max (cos(LAT_OF_SQUARE_CELL) - 1) / LAT_OF_SQUARE_CELL^2 /// - => dist = d_max * (1 + (cos(LAT_OF_SQUARE_CELL) - 1) / LAT_OF_SQUARE_CELL^2 * lat^2) /// - => dist = d_max * (1 - (1 - cos(LAT_OF_SQUARE_CELL)) / LAT_OF_SQUARE_CELL^2 * lat^2) #[inline] fn largest_c2v_dist_in_eqr_bottom(lat_abs: f64, csts: &ConstantsC2V) -> f64 { debug_assert!(0_f64 <= lat_abs && lat_abs <= LAT_OF_SQUARE_CELL); csts.coeff_x2_eqr * pow2(lat_abs) + csts.coeff_cst_eqr } /// Same as the above method, but taking into account an additional radius. #[inline] fn largest_c2v_dist_in_eqr_bottom_with_radius(lat_abs: f64, radius: f64, csts: &ConstantsC2V) -> f64 { debug_assert!(0_f64 < radius); debug_assert!(0_f64 <= lat_abs && lat_abs <= LAT_OF_SQUARE_CELL); largest_c2v_dist_in_eqr_bottom(f64::max(lat_abs - radius, 0_f64), csts) } #[inline] fn linear_approx(x: f64, slope: f64, intercept: f64) -> f64 { slope * x + intercept } /// Returns, for the given depth, the number of cells along both axis of a base-resolution cell. /// /// # Input /// - `depth` must be in `[0, 29]` /// /// # Output /// - `nside` = 2^`depth` /// /// # Panics /// If `depth` is not valid (see [is_depth](fn.is_depth.html)), this method panics. /// /// # Examples /// /// ```rust /// use cdshealpix::{nside}; /// /// assert_eq!(1, nside(0)); /// assert_eq!(2, nside(1)); /// assert_eq!(4, nside(2)); /// assert_eq!(8, nside(3)); /// assert_eq!(16, nside(4)); /// assert_eq!(32, nside(5)); /// assert_eq!(64, nside(6)); /// assert_eq!(128, nside(7)); /// assert_eq!(256, nside(8)); /// assert_eq!(512, nside(9)); /// assert_eq!(1024, nside(10)); /// assert_eq!(2048, nside(11)); /// assert_eq!(4096, nside(12)); /// assert_eq!(8192, nside(13)); /// assert_eq!(16384, nside(14)); /// assert_eq!(32768, nside(15)); /// assert_eq!(65536, nside(16)); /// assert_eq!(131072, nside(17)); /// assert_eq!(262144, nside(18)); /// assert_eq!(524288, nside(19)); /// assert_eq!(1048576, nside(20)); /// assert_eq!(2097152, nside(21)); /// assert_eq!(4194304, nside(22)); /// assert_eq!(8388608, nside(23)); /// assert_eq!(16777216, nside(24)); /// assert_eq!(33554432, nside(25)); /// assert_eq!(67108864, nside(26)); /// assert_eq!(134217728, nside(27)); /// assert_eq!(268435456, nside(28)); /// assert_eq!(536870912, nside(29)); /// // Using a for loop... /// for depth in 0..29 { /// assert_eq!(2u32.pow(depth), nside(depth as u8)); /// } /// ``` #[inline] pub fn nside(depth: u8) -> u32 { check_depth(depth); nside_unsafe(depth) } /// Same as [nside](fn.nside.html) except that this version does not check the argument, and thus /// does not panics if the argument is illegal. #[inline] pub fn nside_unsafe(depth: u8) -> u32 { 1_u32 << depth } /// Returns, for the given difference of depth, the number of cells small cells the large cell /// contains. If the small cell level is 0, the result is the sqaured nside. /// /// # Input /// - `delta_depth` must be in `[0, 29]` /// /// # Output /// - `nside^2` = 2^2*`delta_depth` /// /// # Panics /// If `delta_depth` is not in `[0, 29]`. /// /// # Examples /// /// ```rust /// use cdshealpix::{nside_square}; /// /// for delta_depth in 0..29_u8 { /// assert_eq!(2u64.pow(2 * delta_depth as u32), nside_square(delta_depth)); /// } /// ``` #[inline] pub fn nside_square(delta_depth: u8) -> u64 { check_depth(delta_depth); nside_square_unsafe(delta_depth) } /// Same as [nside_square](fn.nside_square.html) except that this version does not check the argument, /// and thus does not panics if the argument is illegal. #[inline] pub fn nside_square_unsafe(delta_depth: u8) -> u64 { 1_u64 << (delta_depth << 1) } #[inline] fn check_depth(depth: u8) { assert!(is_depth(depth), "Expected depth in [0, 29]"); } /// Returns `true` if the given argument is a valid depth, i.e. if it is <= [DEPTH_MAX](constant.DEPTH_MAX.html). #[inline] pub fn is_depth(depth: u8) -> bool { depth <= DEPTH_MAX } /// Returns, for the given `nside`, the number of subdivision of a base-resolution cell (i.e. the depth). /// /// # Input /// - `nside` must be a power of 2 in `[0, 2^29]` /// /// # Output /// - `depth` = `log2(nside)` /// /// # Panics /// If `nside` is not valid (see [is_nside](fn.is_nside.html)), this method panics. /// /// # Examples /// /// ```rust /// use cdshealpix::{nside, depth}; /// /// for d in 0..29 { /// assert_eq!(d, depth(nside(d as u8))); /// } /// ``` #[inline] pub fn depth(nside: u32) -> u8 { check_nside(nside); depth_unsafe(nside) } /// Same as [depth](fn.depth.html) except that this version does not check the argument, and thus /// does not panics if the argument is illegal. #[inline] pub fn depth_unsafe(nside: u32) -> u8 { nside.trailing_zeros() as u8 } #[inline] fn check_nside(nside: u32) { assert!(is_nside(nside), "Nside must be a power of 2 in [1-2^29]"); } /// Returns `true` if the given argument is a valid `nside`, i.e. /// if it is a power of 2, is != 0 and is <= [NSIDE_MAX](constant.NSIDE_MAX.html). #[inline] pub fn is_nside(nside: u32) -> bool { is_pow_of_2(nside) && nside > 0 && nside <= NSIDE_MAX } /// Determines if an integer is a power of two, including 0. /// Taken from the "Bit Twiddling Hacks" web page of Sean Eron Anderson. #[inline] fn is_pow_of_2(x: u32) -> bool { (x & (x - 1)) == 0 } /// Returns the number of distinct hash value (the number of cells or pixel the unit sphere is /// devided in) at the given `depth`. /// /// # Input /// - `depth` must be in `[0, 29]` /// /// # Output /// - `n_hash` = `12 * nside^2` /// /// # Panics /// If `depth` is not valid (see [is_depth](fn.is_depth.html)), this method panics. /// /// # Examples /// /// ```rust /// use cdshealpix::{n_hash}; /// /// assert_eq!(12u64, n_hash(0u8)); /// assert_eq!(48u64, n_hash(1u8)); /// assert_eq!(192u64, n_hash(2u8)); /// assert_eq!(768u64, n_hash(3u8)); /// assert_eq!(3072u64, n_hash(4u8)); /// assert_eq!(12288u64, n_hash(5u8)); /// assert_eq!(49152u64, n_hash(6u8)); /// assert_eq!(196608u64, n_hash(7u8)); /// assert_eq!(786432u64, n_hash(8u8)); /// assert_eq!(3145728u64, n_hash(9u8)); /// assert_eq!(12582912u64, n_hash(10u8)); /// assert_eq!(50331648u64, n_hash(11u8)); /// assert_eq!(201326592u64, n_hash(12u8)); /// assert_eq!(805306368u64, n_hash(13u8)); /// assert_eq!(3221225472u64, n_hash(14u8)); /// assert_eq!(12884901888u64, n_hash(15u8)); /// assert_eq!(51539607552u64, n_hash(16u8)); /// assert_eq!(206158430208u64, n_hash(17u8)); /// assert_eq!(824633720832u64, n_hash(18u8)); /// assert_eq!(3298534883328u64, n_hash(19u8)); /// assert_eq!(13194139533312u64, n_hash(20u8)); /// assert_eq!(52776558133248u64, n_hash(21u8)); /// assert_eq!(211106232532992u64, n_hash(22u8)); /// assert_eq!(844424930131968u64, n_hash(23u8)); /// assert_eq!(3377699720527872u64, n_hash(24u8)); /// assert_eq!(13510798882111488u64, n_hash(25u8)); /// assert_eq!(54043195528445952u64, n_hash(26u8)); /// assert_eq!(216172782113783808u64, n_hash(27u8)); /// assert_eq!(864691128455135232u64, n_hash(28u8)); /// assert_eq!(3458764513820540928u64, n_hash(29u8)); /// ``` /// #[inline] pub fn n_hash(depth: u8) -> u64 { check_depth(depth); n_hash_unsafe(depth) } /// Same as [n_hash](fn.n_hash.html) except that this version does not panic if the given `depth` is /// out of range. #[inline] pub fn n_hash_unsafe(depth: u8) -> u64 { 12u64 << (depth << 1u8) } /// Returns `true` if the function [best_starting_depth](fn.best_starting_depth.html) is valid /// for the given argument `d_max_rad`. So if `d_max_rad < ~48 deg`. `d_max_rad` is given in radians. /// /// ```rust /// use cdshealpix::{has_best_starting_depth}; /// use std::f64::consts::PI; /// /// assert!(!has_best_starting_depth(PI / 3f64)); /// assert!(has_best_starting_depth(PI / 4f64)); /// ``` #[inline] pub fn has_best_starting_depth(d_max_rad: f64) -> bool { d_max_rad < SMALLER_EDGE2OPEDGE_DIST[0] } /// Returns the the smallest depth (in `[0, 29]`) at which a shape having the given largest distance /// from its center to a border overlaps a maximum of 9 cells (the cell containing the center of /// the shape plus the 8 neighbouring cells). /// Info: internally, unrolled binary search loop on 30 pre-computed values (one by depth). /// @return -1 if the given distance is very large (> ~48deg), else returns the smallest depth /// (in [0, 29]) at which a shape having the given largest distance from its center to a border /// overlaps a maximum of 9 cells (the cell containing the center of the shape plus the 8 /// neighbouring cells). /// /// # Input /// - `d_max_rad` largest possible distance, in radians, between the center and the border of a shape /// /// # Output /// - `depth` = the smallest depth (in `[0, 29]`) at which a shape having the given largest distance /// from its center to a border overlaps a maximum of 9 cells (the cell containing the center of the /// shape plus the 8 neighbouring cells). /// /// # Panics /// If the given distance is very large (> ~48deg), this function is not valid since the 12 base /// cells could be overlaped by the shape /// (see [has_best_starting_depth](fn.has_best_starting_depth.html)). Thus it panics. /// /// # Examples /// /// ```rust /// use cdshealpix::{best_starting_depth}; /// use std::f64::consts::PI; /// /// assert_eq!(0, best_starting_depth(PI / 4f64)); // 45 deg /// assert_eq!(5, best_starting_depth(0.0174533)); // 1 deg /// assert_eq!(7, best_starting_depth(0.0043632)); // 15 arcmin /// assert_eq!(9, best_starting_depth(0.0013)); // 4.469 arcmin /// assert_eq!(15, best_starting_depth(1.454E-5)); // 3 arcsec /// assert_eq!(20, best_starting_depth(6.5E-7)); // 0.134 arcsec /// assert_eq!(22, best_starting_depth(9.537E-8)); // 20 mas /// ``` #[inline] pub fn best_starting_depth(d_max_rad: f64) -> u8 { // Could have used an Option assert!(d_max_rad < SMALLER_EDGE2OPEDGE_DIST[0], "Too large value, use first function has_best_starting_depth"); // Unrolled binary search loop if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[29] { 29 } else if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[15] { if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[22] { if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[25] { if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[27] { if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[28] { 28 } else { 27 } } else if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[26] { 26 } else { 25 } } else if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[24] { 24 } else if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[23] { 23 } else { 22 } } else if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[18] { if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[20] { if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[21] { 21 } else { 20 } } else if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[19] { 19 } else { 18 } } else if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[17] { 17 } else if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[16] { 16 } else { 15 } } else if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[7] { if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[11] { if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[13] { if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[14] { 14 } else { 13 } } else if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[12] { 12 } else { 11 } } else if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[9] { if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[10] { 10 } else { 9 } } else if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[8] { 8 } else { 7 } } else if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[3] { if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[5] { if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[6] { 6 } else { 5 } } else if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[4] { 4 } else { 3 } } else if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[2] { 2 } else if d_max_rad < SMALLER_EDGE2OPEDGE_DIST[1] { 1 } else { 0 } } /// Performs the HEALPix projection: `(x, y) = proj(lon, lat)`. /// The chosen scale is such that: base cell vertices and center coordinates are integers; /// the distance from a cell center to its vertices equals one. /// This projection is multi-purpose in the sense that if `lon` is in `[-pi, pi]`, then /// `x` is in `[-4, 4]` and if `lon` is in `[0, 2pi]`, then `x` is in `[0, 8]`. /// It means that a same position on the sphere can lead to different positions in the projected /// Euclidean plane. /// /// Simplified projection formulae are: /// - Equatorial region /// ```math /// \boxed{ /// \left\{ /// \begin{array}{lcl} /// X & = & \alpha \times \frac{4}{\pi} \\ /// Y & = & \sin(\delta) \times \frac{3}{2} /// \end{array} /// \right. /// } /// \Rightarrow /// \left\{ /// \begin{array}{lcl} /// \alpha \in [0, 2\pi] & \leadsto & X \in [0, 8] \\ /// \sin\delta \in [-\frac{2}{3}, \frac{2}{3}] & \leadsto & Y \in [-1, 1] /// \end{array} /// \right. /// ``` /// - Polar caps: /// ```math /// \boxed{ /// \left\{ /// \begin{array}{lcl} /// t & = & \sqrt{3(1-\sin\delta)} \\ /// X & = & (\alpha\frac{4}{\pi} - 1)t+1 \\ /// Y & = & 2 - t /// \end{array} /// \right. /// } /// \Rightarrow /// \left\{ /// \begin{array}{l} /// \alpha \in [0, \frac{\pi}{2}] \\ /// \sin\delta \in ]\frac{2}{3}, 1] /// \end{array} /// \right. /// \leadsto /// \begin{array}{l} /// t \in [0, 1[ \\ /// X \in ]0, 2[ \\ /// Y \in ]1, 2] /// \end{array} /// ``` /// /// It is the responsibility of the caller to homogenize the result according to its needs. /// ![Proj](https://raw.githubusercontent.com/cds-astro/cds-healpix-rust/master/resources/4doc/hpx_proj.png) /// /// # Inputs /// - `lon` longitude in radians, support positive and negative reasonably large values /// (naive approach, no Cody-Waite nor Payne Hanek range reduction). /// - `lat` latitude in radians, must be in `[-pi/2, pi/2]` /// /// # Output /// - `(x, y)` the projected planar Euclidean coordinates of the point /// of given coordinates `(lon, lat)` on the unit sphere /// - `lon` ≤ `0` => `x in [-8, 0]` /// - `lon` ≥ `0` => `x in [0, 8]` /// - `y in [-2, 2]` /// /// # Panics /// If `lat` **not in** `[-pi/2, pi/2]`, this method panics. /// /// # Examples /// To obtain the WCS projection (see Calabretta2007), you can write: /// ```rust /// use cdshealpix::{HALF_PI, PI_OVER_FOUR, proj}; /// use std::f64::consts::{PI}; /// /// let lon = 25.1f64; /// let lat = 46.7f64; /// /// let (mut x, mut y) = proj(lon.to_radians(), lat.to_radians()); /// if x > 4f64 { /// x -= 8f64; /// } /// x *= PI_OVER_FOUR; /// y *= PI_OVER_FOUR; /// /// assert!(-PI <= x && x <= PI); /// assert!(-HALF_PI <= y && y <= HALF_PI); /// ``` /// /// Other test example: /// ```rust /// use cdshealpix::{TRANSITION_LATITUDE, HALF_PI, PI_OVER_FOUR, proj}; /// /// let (x, y) = proj(0.0, 0.0); /// assert_eq!(0f64, x); /// assert_eq!(0f64, y); /// /// assert_eq!((0.0, 1.0), proj(0.0 * HALF_PI, TRANSITION_LATITUDE)); /// /// fn dist(p1: (f64, f64), p2: (f64, f64)) -> f64 { /// f64::sqrt((p2.0 - p1.0) * (p2.0 - p1.0) + (p2.1 - p1.1) * (p2.1 - p1.1)) /// } /// assert!(dist((0.0, 0.0), proj(0.0 * HALF_PI, 0.0)) < 1e-15); /// assert!(dist((0.0, 1.0), proj(0.0 * HALF_PI, TRANSITION_LATITUDE)) < 1e-15); /// assert!(dist((1.0, 2.0), proj(0.0 * HALF_PI + PI_OVER_FOUR, HALF_PI)) < 1e-15); /// assert!(dist((2.0, 1.0), proj(1.0 * HALF_PI, TRANSITION_LATITUDE)) < 1e-15); /// assert!(dist((3.0, 2.0), proj(1.0 * HALF_PI + PI_OVER_FOUR, HALF_PI)) < 1e-15); /// assert!(dist((4.0, 1.0), proj(2.0 * HALF_PI , TRANSITION_LATITUDE)) < 1e-15); /// assert!(dist((5.0, 2.0), proj(2.0 * HALF_PI + PI_OVER_FOUR, HALF_PI)) < 1e-15); /// assert!(dist((6.0, 1.0), proj(3.0 * HALF_PI, TRANSITION_LATITUDE)) < 1e-15); /// assert!(dist((7.0, 2.0), proj(3.0 * HALF_PI + PI_OVER_FOUR, HALF_PI)) < 1e-15); /// assert!(dist((0.0, 1.0), proj(4.0 * HALF_PI, TRANSITION_LATITUDE)) < 1e-15); /// assert!(dist((0.0, 0.0), proj(4.0 * HALF_PI, 0.0)) < 1e-15); /// assert!(dist((0.0, -1.0), proj(0.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); /// assert!(dist((1.0, -2.0), proj(0.0 * HALF_PI + PI_OVER_FOUR, -HALF_PI)) < 1e-15); /// assert!(dist((2.0, -1.0), proj(1.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); /// assert!(dist((3.0, -2.0), proj(1.0 * HALF_PI + PI_OVER_FOUR, -HALF_PI)) < 1e-15); /// assert!(dist((4.0, -1.0), proj(2.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); /// assert!(dist((5.0, -2.0), proj(2.0 * HALF_PI + PI_OVER_FOUR, -HALF_PI)) < 1e-15); /// assert!(dist((6.0, -1.0), proj(3.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); /// assert!(dist((7.0, -2.0), proj(3.0 * HALF_PI + PI_OVER_FOUR, -HALF_PI)) < 1e-15); /// assert!(dist((0.0, -1.0), proj(4.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); /// /// assert!(dist((-0.0, 0.0), proj(-0.0 * HALF_PI, 0.0)) < 1e-15); /// assert!(dist((-0.0, 1.0), proj(-0.0 * HALF_PI, TRANSITION_LATITUDE)) < 1e-15); /// assert!(dist((-1.0, 2.0), proj(-0.0 * HALF_PI - PI_OVER_FOUR, HALF_PI)) < 1e-15); /// assert!(dist((-2.0, 1.0), proj(-1.0 * HALF_PI, TRANSITION_LATITUDE)) < 1e-15); /// assert!(dist((-3.0, 2.0), proj(-1.0 * HALF_PI - PI_OVER_FOUR, HALF_PI)) < 1e-15); /// assert!(dist((-4.0, 1.0), proj(-2.0 * HALF_PI , TRANSITION_LATITUDE)) < 1e-15); /// assert!(dist((-5.0, 2.0), proj(-2.0 * HALF_PI - PI_OVER_FOUR, HALF_PI)) < 1e-15); /// assert!(dist((-6.0, 1.0), proj(-3.0 * HALF_PI, TRANSITION_LATITUDE)) < 1e-15); /// assert!(dist((-7.0, 2.0), proj(-3.0 * HALF_PI - PI_OVER_FOUR, HALF_PI)) < 1e-15); /// assert!(dist((-0.0, 1.0), proj(-4.0 * HALF_PI, TRANSITION_LATITUDE)) < 1e-15); /// assert!(dist((-0.0, 0.0), proj(-4.0 * HALF_PI, 0.0)) < 1e-15); /// assert!(dist((-0.0, -1.0), proj(-0.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); /// assert!(dist((-1.0, -2.0), proj(-0.0 * HALF_PI - PI_OVER_FOUR, -HALF_PI)) < 1e-15); /// assert!(dist((-2.0, -1.0), proj(-1.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); /// assert!(dist((-3.0, -2.0), proj(-1.0 * HALF_PI - PI_OVER_FOUR, -HALF_PI)) < 1e-15); /// assert!(dist((-4.0, -1.0), proj(-2.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); /// assert!(dist((-5.0, -2.0), proj(-2.0 * HALF_PI - PI_OVER_FOUR, -HALF_PI)) < 1e-15); /// assert!(dist((-6.0, -1.0), proj(-3.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); /// assert!(dist((-7.0, -2.0), proj(-3.0 * HALF_PI - PI_OVER_FOUR, -HALF_PI)) < 1e-15); /// assert!(dist((-0.0, -1.0), proj(-4.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); /// ``` #[inline] pub fn proj(lon: f64, lat: f64) -> (f64, f64) { check_lat(lat); let lon = abs_sign_decompose(lon); let lat = abs_sign_decompose(lat); let x = pm1_offset_decompose(lon.abs * FOUR_OVER_PI); let mut xy = (x.pm1, lat.abs); if is_in_equatorial_region(lat.abs) { proj_cea(&mut xy); } else { proj_collignon(&mut xy); } apply_offset_and_signs(&mut xy, x.offset, lon.sign, lat.sign); xy } /// Unproject the given HEALPix projected points. /// This unprojection is multi-purpose in the sense that: /// - if input `x` in `[-8, 0[`, then output `lon` in `[-2pi, 0]` /// - if input `x` in `[ 0, 8]`, then output `lon` in `[0, 2pi]` /// - output `lat` always in `[-pi/2, pi/2]` /// /// # Inputs /// - `x` the projected coordinate along the x-axis, supports positive and negative reasonably /// large values with a naive approach (no Cody-Waite nor Payne Hanek range reduction). /// - `y` the projected coordinate along te x-axis, must be in `[-2, 2]` /// /// # Output /// - `(lon, lat)` in radians, the position on the unit sphere whose projected coordinates are /// the input coordinates `(x, y)`. /// - if `x <= 0`, then `lon` in `[-2pi, 0]`; /// - else if `x >= 0`, the `lon` in `[0, 2pi]` /// - `lat` always in `[-pi/2, pi/2]`. /// /// # Panics /// If `y` **not in** `[-2, 2]`, this method panics. /// /// # Examples /// To obtain the WCS un-projection (see Calabretta2007), you can write: /// ```rust /// use cdshealpix::{HALF_PI, FOUR_OVER_PI, unproj}; /// use std::f64::consts::{PI}; /// /// let x = 2.1f64; /// let y = 0.36f64; /// /// let (mut lon, mut lat) = unproj(x * FOUR_OVER_PI, y * FOUR_OVER_PI); /// if lon < 0f64 { /// lon += 2f64 * PI; /// } /// /// assert!(0f64 <= lon && lon <= 2f64 * PI); /// assert!(-HALF_PI <= lat && lat <= HALF_PI); /// ``` /// /// Other test example: /// ```rust /// use cdshealpix::{TRANSITION_LATITUDE, HALF_PI, PI_OVER_FOUR, proj, unproj}; /// /// fn dist(p1: (f64, f64), p2: (f64, f64)) -> f64 { /// let sindlon = f64::sin(0.5 * (p2.0 - p1.0)); /// let sindlat = f64::sin(0.5 * (p2.1 - p1.1)); /// 2f64 * f64::asin(f64::sqrt(sindlat * sindlat + p1.1.cos() * p2.1.cos() * sindlon * sindlon)) /// } /// /// let points: [(f64, f64); 40] = [ /// (0.0 * HALF_PI, 0.0), /// (1.0 * HALF_PI, TRANSITION_LATITUDE), /// (2.0 * HALF_PI, TRANSITION_LATITUDE), /// (3.0 * HALF_PI, TRANSITION_LATITUDE), /// (4.0 * HALF_PI, TRANSITION_LATITUDE), /// (0.0 * HALF_PI + PI_OVER_FOUR, HALF_PI), /// (1.0 * HALF_PI + PI_OVER_FOUR, HALF_PI), /// (2.0 * HALF_PI + PI_OVER_FOUR, HALF_PI), /// (3.0 * HALF_PI + PI_OVER_FOUR, HALF_PI), /// (4.0 * HALF_PI + PI_OVER_FOUR, HALF_PI), /// (0.0 * HALF_PI, 0.0), /// (1.0 * HALF_PI, -TRANSITION_LATITUDE), /// (2.0 * HALF_PI, -TRANSITION_LATITUDE), /// (3.0 * HALF_PI, -TRANSITION_LATITUDE), /// (4.0 * HALF_PI, -TRANSITION_LATITUDE), /// (0.0 * HALF_PI + PI_OVER_FOUR, -HALF_PI), /// (1.0 * HALF_PI + PI_OVER_FOUR, -HALF_PI), /// (2.0 * HALF_PI + PI_OVER_FOUR, -HALF_PI), /// (3.0 * HALF_PI + PI_OVER_FOUR, -HALF_PI), /// (4.0 * HALF_PI + PI_OVER_FOUR, -HALF_PI), /// (-0.0 * HALF_PI, 0.0), /// (-1.0 * HALF_PI, TRANSITION_LATITUDE), /// (-2.0 * HALF_PI, TRANSITION_LATITUDE), /// (-3.0 * HALF_PI, TRANSITION_LATITUDE), /// (-4.0 * HALF_PI, TRANSITION_LATITUDE), /// (-0.0 * HALF_PI + PI_OVER_FOUR, HALF_PI), /// (-1.0 * HALF_PI + PI_OVER_FOUR, HALF_PI), /// (-2.0 * HALF_PI + PI_OVER_FOUR, HALF_PI), /// (-3.0 * HALF_PI + PI_OVER_FOUR, HALF_PI), /// (-4.0 * HALF_PI + PI_OVER_FOUR, HALF_PI), /// (-0.0 * HALF_PI, 0.0), /// (-1.0 * HALF_PI, -TRANSITION_LATITUDE), /// (-2.0 * HALF_PI, -TRANSITION_LATITUDE), /// (-3.0 * HALF_PI, -TRANSITION_LATITUDE), /// (-4.0 * HALF_PI, -TRANSITION_LATITUDE), /// (-0.0 * HALF_PI + PI_OVER_FOUR, -HALF_PI), /// (-1.0 * HALF_PI + PI_OVER_FOUR, -HALF_PI), /// (-2.0 * HALF_PI + PI_OVER_FOUR, -HALF_PI), /// (-3.0 * HALF_PI + PI_OVER_FOUR, -HALF_PI), /// (-4.0 * HALF_PI + PI_OVER_FOUR, -HALF_PI) /// ]; /// /// for (lon, lat) in points.iter() { /// let (x, y): (f64, f64) = proj(*lon, *lat); /// assert!(dist((*lon, *lat), unproj(x, y)) < 1e-15); /// } /// ``` #[inline] pub fn unproj(x: f64, y: f64) -> (f64, f64) { check_y(y); let x = abs_sign_decompose(x); let y = abs_sign_decompose(y); let lon = pm1_offset_decompose(x.abs); let mut lonlat= (lon.pm1, y.abs); if is_in_projected_equatorial_region(y.abs) { deproj_cea(&mut lonlat); } else { deproj_collignon(&mut lonlat); } apply_offset_and_signs(&mut lonlat, lon.offset, x.sign, y.sign); lonlat.0 *= PI_OVER_FOUR; lonlat } /// Verify that the latitude is in [-PI/2, PI/2], panics if not. #[inline] fn check_lat(lat: f64) { assert!(-HALF_PI <= lat && lat <= HALF_PI); } /// Verify that the projected y coordinate is in [-2, 2], panics if not. #[inline] fn check_y(y: f64) { assert!(-2f64 <= y && y <= 2f64); } /// Returns `true` if the point of given (absolute value of) latitude is in the equatorial region, /// and `false` if it is located in one of the two polar caps #[inline] pub fn is_in_equatorial_region(abs_lat: f64) -> bool { abs_lat <= TRANSITION_LATITUDE } /// Returns `true` if the point of given (absolute value of) y coordinate in the projected plane /// is in the equatorial region, and `false` if it is located in one of the two polar caps #[inline] pub fn is_in_projected_equatorial_region(abs_y: f64) -> bool { abs_y <= 1f64 } // Returns the absolute value of the given double together with its bit of sign struct AbsAndSign { abs: f64, sign: u64, } #[inline] fn abs_sign_decompose(x: f64) -> AbsAndSign { let bits = f64::to_bits(x); AbsAndSign { abs: f64::from_bits(bits & F64_BUT_SIGN_BIT_MASK), sign: bits & F64_SIGN_BIT_MASK, } } // Decompose the given positive real value in // --* an integer offset in [1, 3, 5, 7] (*PI/4) and // --* a real value in [-1.0, 1.0] (*PI/4) struct OffsetAndPM1 { offset: u8, // = 1, 3, 5 or 7 pm1: f64, // in [-1.0, 1.0] } #[inline] fn pm1_offset_decompose(x: f64) -> OffsetAndPM1 { let floor: u8 = x as u8; let odd_floor: u8 = floor | 1u8; OffsetAndPM1 { offset: odd_floor & 7u8, // value modulo 7 pm1: x - (odd_floor as f64), } } // Cylindrical Equal Area projection #[inline] fn proj_cea(xy: &mut (f64, f64)) { let (_, ref mut y) = *xy; *y = f64::sin(*y) * ONE_OVER_TRANSITION_Z; } #[inline] fn deproj_cea(lonlat: &mut (f64, f64)) { let (_, ref mut lat) = *lonlat; *lat = f64::asin((*lat) * TRANSITION_Z); } // Collignon projection #[inline] fn proj_collignon(xy: &mut (f64, f64)) { let (ref mut x, ref mut y) = *xy; *y = SQRT6 * f64::cos(HALF * *y + PI_OVER_FOUR); *x *= *y; *y = 2f64 - *y; } #[inline] fn deproj_collignon(lonlat: &mut (f64, f64)) { let (ref mut lon, ref mut lat) = *lonlat; *lat = 2f64 - *lat; if is_not_near_from_pole(*lat) { // Rare, so few risks of branch miss-prediction *lon /= *lat; deal_with_numerical_approx_in_edges(lon); } // in case of pole, lon = lat = 0 (we avoid NaN due to division by lat=0) *lat *= ONE_OVER_SQRT6; *lat = 2f64 * f64::acos(*lat) - HALF_PI; } #[inline] fn is_not_near_from_pole(sqrt_of_three_time_one_minus_sin_of: f64) -> bool { // In case of pole: x = y = 0 sqrt_of_three_time_one_minus_sin_of > EPS_POLE } #[inline] fn deal_with_numerical_approx_in_edges(lon: &mut f64) { if *lon > 1f64 { *lon = 1f64; } else if *lon < -1f64 { *lon = -1f64; } } // Shift x by the given offset and apply lon and lat signs to x and y respectively #[inline] fn apply_offset_and_signs(ab: &mut (f64, f64), off: u8, a_sign: u64, b_sign: u64) { let (ref mut a, ref mut b) = *ab; *a += off as f64; *a = f64::from_bits(f64::to_bits(*a) | a_sign); *b = f64::from_bits(f64::to_bits(*b) | b_sign); } // Import module compass point pub mod compass_point; use crate::compass_point::{MainWind}; use crate::compass_point::MainWind::*; /// Compute the base cell value which is the neighbour of the given base cell, in the given direction. /// There is no neighbour: /// - in the North and South directions for the equatorial region cells (i.e. cells 4, 5, 6 and 7) /// - in the East and West directions for: /// - the north polar cap cells (i.e. cells 0, 1, 2 and 3) /// - the south polar cap cells (i.e. cells 8, 9, 10 and 11) pub fn neighbour(base_cell: u8, direction: MainWind) -> Option<u8> { if direction == MainWind::C { Some(base_cell) } else { let d0h_mod_4 = base_cell & 3_u8; // <=> base_cell modulo 4 match base_cell >> 2 { // <=> basce_cell / 4 0 => ncp_neighbour(d0h_mod_4, direction), 1 => eqr_neighbour(d0h_mod_4, direction), 2 => spc_neighbour(d0h_mod_4, direction), _ => panic!("Base cell must be in [0, 12["), } } } fn ncp_neighbour(d0h_mod_4: u8, direction: MainWind) -> Option<u8> { match direction { S => base_cell_opt(iden(d0h_mod_4), 2), SE => base_cell_opt(next(d0h_mod_4), 1), SW => base_cell_opt(iden(d0h_mod_4), 1), NE => base_cell_opt(next(d0h_mod_4), 0), NW => base_cell_opt(prev(d0h_mod_4), 0), N => base_cell_opt(oppo(d0h_mod_4), 0), _ => None, } } fn eqr_neighbour(d0h_mod_4: u8, direction: MainWind) -> Option<u8> { match direction { SE => base_cell_opt(iden(d0h_mod_4), 2), E => base_cell_opt(next(d0h_mod_4), 1), SW => base_cell_opt(prev(d0h_mod_4), 2), NE => base_cell_opt(iden(d0h_mod_4), 0), W => base_cell_opt(prev(d0h_mod_4), 1), NW => base_cell_opt(prev(d0h_mod_4), 0), _ => None, } } fn spc_neighbour(d0h_mod_4: u8, direction: MainWind) -> Option<u8> { match direction { S => base_cell_opt(oppo(d0h_mod_4), 2), SE => base_cell_opt(next(d0h_mod_4), 2), SW => base_cell_opt(prev(d0h_mod_4), 2), NE => base_cell_opt(next(d0h_mod_4), 1), NW => base_cell_opt(iden(d0h_mod_4), 1), N => base_cell_opt(iden(d0h_mod_4), 0), _ => None, } } /// Returns (mod4 - 1) in [0, 2], and 3 if mod4 == 0 (i.e. the previous value in [0, 3] range) #[inline] fn prev(mod4: u8) -> u8 { debug_assert!(mod4 < 4); (((mod4 as i8) - 1) & 3) as u8 } /// Returns (mod4 + 1) in [1, 3], and 0 if mod4 == 3 (i.e. the next value in [0, 3] range) #[inline] fn next(mod4: u8) -> u8 { debug_assert!(mod4 < 4); (mod4 + 1) & 3 } /// Returns (mod4 + 2) in [2, 3], and 0 if mod4 == 2 and 1 if mod4 == 3 (i.e. the opposite value in [0, 3] range) #[inline] fn oppo(mod4: u8) -> u8 { debug_assert!(mod4 < 4); (mod4 + 2) & 3 } /// Returns the input value: useless, just used to improve code legibility #[inline] fn iden(mod4: u8) -> u8 { debug_assert!(mod4 < 4); mod4 } #[inline] fn base_cell_opt(i: u8, j: u8) -> Option<u8> { Some(base_cell(i, j)) } /// Compute the base cell from its (i, j) coordinates: /// - i: index along the longitude axis ( = base_cell modulo 4) /// - j: index along the latitude axis ( = base_cell / 4) /// - = 0 for the cells covering the north polar cap /// - = 1 for the cells with are only in the equatorial region /// - = 2 for the cells covering the south polar cap #[inline] fn base_cell(i: u8, j: u8) -> u8 { debug_assert!(i < 4 && j < 3); (j << 2) + i } /// Module containing NESTED scheme methods pub mod nested; /// No need to make those public! mod sph_geom; mod special_points_finder; #[cfg(test)] mod tests { use super::*; #[test] fn testok_nside() { assert_eq!(1, nside(0)); assert_eq!(2, nside(1)); assert_eq!(4, nside(2)); assert_eq!(8, nside(3)); assert_eq!(16, nside(4)); assert_eq!(32, nside(5)); assert_eq!(64, nside(6)); assert_eq!(128, nside(7)); assert_eq!(256, nside(8)); assert_eq!(512, nside(9)); assert_eq!(1024, nside(10)); assert_eq!(2048, nside(11)); assert_eq!(4096, nside(12)); assert_eq!(8192, nside(13)); assert_eq!(16384, nside(14)); assert_eq!(32768, nside(15)); assert_eq!(65536, nside(16)); assert_eq!(131072, nside(17)); assert_eq!(262144, nside(18)); assert_eq!(524288, nside(19)); assert_eq!(1048576, nside(20)); assert_eq!(2097152, nside(21)); assert_eq!(4194304, nside(22)); assert_eq!(8388608, nside(23)); assert_eq!(16777216, nside(24)); assert_eq!(33554432, nside(25)); assert_eq!(67108864, nside(26)); assert_eq!(134217728, nside(27)); assert_eq!(268435456, nside(28)); assert_eq!(536870912, nside(29)); } #[test] #[should_panic] fn testpanic_nside() { nside(30); } #[test] fn testok_proj() { fn dist(p1: (f64, f64), p2: (f64, f64)) -> f64 { f64::sqrt((p2.0 - p1.0) * (p2.0 - p1.0) + (p2.1 - p1.1) * (p2.1 - p1.1)) } // println!("{:?}", proj(0.0, TRANSITION_LATITUDE)); // println!("{}", dist((0.0, 1.0), proj(0.0, TRANSITION_LATITUDE))); assert!(dist((0.0, 0.0), proj(0.0 * HALF_PI, 0.0)) < 1e-15); assert!(dist((0.0, 1.0), proj(0.0 * HALF_PI, TRANSITION_LATITUDE)) < 1e-15); assert!(dist((1.0, 2.0), proj(0.0 * HALF_PI + PI_OVER_FOUR, HALF_PI)) < 1e-15); assert!(dist((2.0, 1.0), proj(1.0 * HALF_PI, TRANSITION_LATITUDE)) < 1e-15); assert!(dist((3.0, 2.0), proj(1.0 * HALF_PI + PI_OVER_FOUR, HALF_PI)) < 1e-15); assert!(dist((4.0, 1.0), proj(2.0 * HALF_PI, TRANSITION_LATITUDE)) < 1e-15); assert!(dist((5.0, 2.0), proj(2.0 * HALF_PI + PI_OVER_FOUR, HALF_PI)) < 1e-15); assert!(dist((6.0, 1.0), proj(3.0 * HALF_PI, TRANSITION_LATITUDE)) < 1e-15); assert!(dist((7.0, 2.0), proj(3.0 * HALF_PI + PI_OVER_FOUR, HALF_PI)) < 1e-15); assert!(dist((0.0, 1.0), proj(4.0 * HALF_PI, TRANSITION_LATITUDE)) < 1e-15); assert!(dist((0.0, 0.0), proj(4.0 * HALF_PI, 0.0)) < 1e-15); assert!(dist((0.0, -1.0), proj(0.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); assert!(dist((1.0, -2.0), proj(0.0 * HALF_PI + PI_OVER_FOUR, -HALF_PI)) < 1e-15); assert!(dist((2.0, -1.0), proj(1.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); assert!(dist((3.0, -2.0), proj(1.0 * HALF_PI + PI_OVER_FOUR, -HALF_PI)) < 1e-15); assert!(dist((4.0, -1.0), proj(2.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); assert!(dist((5.0, -2.0), proj(2.0 * HALF_PI + PI_OVER_FOUR, -HALF_PI)) < 1e-15); assert!(dist((6.0, -1.0), proj(3.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); assert!(dist((7.0, -2.0), proj(3.0 * HALF_PI + PI_OVER_FOUR, -HALF_PI)) < 1e-15); assert!(dist((0.0, -1.0), proj(4.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); assert!(dist((-0.0, 0.0), proj(-0.0 * HALF_PI, 0.0)) < 1e-15); assert!(dist((-0.0, 1.0), proj(-0.0 * HALF_PI, TRANSITION_LATITUDE)) < 1e-15); assert!(dist((-1.0, 2.0), proj(-0.0 * HALF_PI - PI_OVER_FOUR, HALF_PI)) < 1e-15); assert!(dist((-2.0, 1.0), proj(-1.0 * HALF_PI, TRANSITION_LATITUDE)) < 1e-15); assert!(dist((-3.0, 2.0), proj(-1.0 * HALF_PI - PI_OVER_FOUR, HALF_PI)) < 1e-15); assert!(dist((-4.0, 1.0), proj(-2.0 * HALF_PI, TRANSITION_LATITUDE)) < 1e-15); assert!(dist((-5.0, 2.0), proj(-2.0 * HALF_PI - PI_OVER_FOUR, HALF_PI)) < 1e-15); assert!(dist((-6.0, 1.0), proj(-3.0 * HALF_PI, TRANSITION_LATITUDE)) < 1e-15); assert!(dist((-7.0, 2.0), proj(-3.0 * HALF_PI - PI_OVER_FOUR, HALF_PI)) < 1e-15); assert!(dist((-0.0, 1.0), proj(-4.0 * HALF_PI, TRANSITION_LATITUDE)) < 1e-15); assert!(dist((-0.0, 0.0), proj(-4.0 * HALF_PI, 0.0)) < 1e-15); assert!(dist((-0.0, -1.0), proj(-0.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); assert!(dist((-1.0, -2.0), proj(-0.0 * HALF_PI - PI_OVER_FOUR, -HALF_PI)) < 1e-15); assert!(dist((-2.0, -1.0), proj(-1.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); assert!(dist((-3.0, -2.0), proj(-1.0 * HALF_PI - PI_OVER_FOUR, -HALF_PI)) < 1e-15); assert!(dist((-4.0, -1.0), proj(-2.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); assert!(dist((-5.0, -2.0), proj(-2.0 * HALF_PI - PI_OVER_FOUR, -HALF_PI)) < 1e-15); assert!(dist((-6.0, -1.0), proj(-3.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); assert!(dist((-7.0, -2.0), proj(-3.0 * HALF_PI - PI_OVER_FOUR, -HALF_PI)) < 1e-15); assert!(dist((-0.0, -1.0), proj(-4.0 * HALF_PI, -TRANSITION_LATITUDE)) < 1e-15); } #[test] #[should_panic] fn testpanic_proj_1() { proj(0.0, -1.58); } #[test] #[should_panic] fn testpanic_proj_2() { proj(3.14159, -1.58); } #[test] fn testok_shs() { let ang_dist = 0.24; let shs = to_squared_half_segment(ang_dist); let ang_dist_2 = sphe_dist(shs); assert!((ang_dist - ang_dist_2).abs() < 1e-4); } }