Crate cdshealpix

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

Modules

compass_point

Module defining the commodity enums Cardinal (points) and MainWind (points).

external_edge

Module simply defining a the structure storing the external edges of an HEALPix cell and provided accesses according to the main wind directions. This strcut is common to nested and ring scheme.

nested

Module containing NESTED scheme methods

ring

Module containing RING scheme methods

special_points_finder

sph_geom

Module containing spherical geometry structures and methods like 3D vectors, polygon or cone on the unit sphere…

Constants

DEPTH_MAX

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).

F64_BUT_SIGN_BIT_MASK

Equals !F64_SIGN_BIT_MASK (the inverse of the f64 sign mask)

F64_SIGN_BIT_MASK

Mask to keep only the f64 sign

FOUR_OVER_PI

Constant = 4 / pi.

NSIDE_MAX

Constant = nside(29), i.e. the largest possible nside available when we store HEALPix hash on a u64.

ONE_OVER_TRANSITION_Z

Inverse of the limit on |z|=|sin(lat)| between the equatorial region and the polar caps. Equals 1/(2/3) = 1.5, see Eq. (1) in Gorsky2005.

PI_OVER_FOUR

Constant = pi / 4.

SQRT6

Constant = sqrt(6).

TRANSITION_LATITUDE

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.

TRANSITION_Z

Limit on |z|=|sin(lat)| between the equatorial region and the polar caps. Equals 2/3, see Eq. (1) in Gorsky2005.

TWICE_PI

Constant = 2 * pi.

Statics

LAT_OF_SQUARE_CELL

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).

Traits

Customf64

Simple trait used to implements pow2, twice and half on f64.

Functions

base_cell_from_proj_coo

Returns the hash of the base cell given the coordinates of a points in the Euclidean projection plane. The purpose so far is just to test and compare both speed and precision with the 45deg rotation solution.

best_starting_depth

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).

depth

Returns, for the given nside, the number of subdivision of a base-resolution cell (i.e. the depth). For the NESTED scheme only.

depth_from_n_hash_unsafe

Returns, for the given n_hash i.e. number of cells, the number of subdivision of a base-resolution cell (i.e. the depth). For the NESTED scheme only.

depth_unsafe

Same as depth except that this version does not check the argument, and thus does not panics if the argument is illegal.

direction_from_neighbour

Returns the direction of the given base cell from its neighbour base cell located in the given direction.

edge_cell_direction_from_neighbour

Returns the direction of a cell on the inner edge of the given base cell from its neighbour located at the given direction in a different base cell.

has_best_starting_depth

Returns true if the function best_starting_depth is valid for the given argument d_max_rad. So if d_max_rad < ~48 deg. d_max_rad is given in radians.

haversine_dist

is_depth

Returns true if the given argument is a valid depth, i.e. if it is <= DEPTH_MAX.

is_in_equatorial_region

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

is_in_projected_equatorial_region

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

is_nside

Returns true if the given argument is a valid nside for the NESTED scheme, i.e. if it is a power of 2, is != 0 and is <= NSIDE_MAX.

largest_center_to_vertex_distance

Returns an upper limit on the distance between a cell center around the given position and its furthest vertex.

largest_center_to_vertex_distance_with_radius

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

largest_center_to_vertex_distances_with_radius

Same as largest_center_to_vertex_distance_with_radius but making the computation for several depths at the same time.

n_hash

Returns the number of distinct hash value (the number of cells or pixel the unit sphere is devided in) at the given depth.

n_hash_unsafe

Same as n_hash except that this version does not panic if the given depth is out of range.

neighbour

Compute the base cell value which is the neighbour of the given base cell, in the given direction.
There is no neighbour:

nside

Returns, for the given depth, the number of cells along both axis of a base-resolution cell.

nside_square

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.

nside_square_unsafe

Same as nside_square except that this version does not check the argument, and thus does not panics if the argument is illegal.

nside_unsafe

Same as nside except that this version does not check the argument, and thus does not panics if the argument is illegal.

proj

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.

unproj

Unproject the given HEALPix projected points.
This unprojection is multi-purpose in the sense that: