Tessellation

ROTIR discretizes the stellar surface into quadrilateral pixels using one of two tessellation schemes. The choice affects pixel uniformity, neighbor structure, and multi-resolution capabilities.

HEALPix scheme (nested ordering)

The HEALPix (Hierarchical Equal Area isoLatitude Pixelization) scheme divides the sphere into equal-area pixels arranged in a nested hierarchy. ROTIR uses the nested HEALPix ordering, which arranges pixels in a hierarchical quad-tree structure. This enables efficient multi-resolution operations (each pixel at level n splits into 4 children at level n+1). This is the recommended scheme for most applications.

n = 3                                   # resolution parameter
tessels = tessellation_healpix(n)       # nside = 2^n = 8, npix = 768

The resolution parameter n determines the number of pixels:

Conversion functions:

• nside2npix(nside) – returns 12 * nside^2
• npix2n(npix) – returns the resolution parameter n

Pixel structure

Each pixel has 5 associated points: 4 corner vertices and 1 center. These are stored in the tessellation as:

• tessels.unit_xyz – Cartesian coordinates (npix, 5, 3) on the unit sphere
• tessels.unit_spherical – spherical coordinates (r, theta, phi) in (npix, 5, 3)

where theta is the colatitude (0 at north pole, pi at south pole) and phi is the longitude.

Visual comparison

HEALPix wireframe at n=3 (768 pixels) compared with a longitude/latitude grid (20x40 = 800 pixels):

HEALPix (nested, n=3)	Lon/Lat (20x40)

Mesh overlay showing pixel edges on a limb-darkened sphere:

HEALPix (nested, n=3)	Lon/Lat (20x40)

Neighbors and total variation

HEALPix pixels have 8 neighbors (6 or 7 at special locations). The neighbor structure is used for total variation regularization:

tv_info = tv_neighbors_healpix(n)

This returns a tuple containing the sparse difference matrix and Hessian used by the TV regularization terms. For reconstructions where some pixels are never visible, use:

tv_info = tv_neighbors_healpix_visible(n, stars)

This excludes invisible pixels from the regularization, preventing artifacts at the limb.

Longitude/latitude scheme

The longitude/latitude grid is a regular grid in colatitude and longitude:

ntheta = 50                                     # latitude rings
nphi = 50                                       # pixels per ring
tessels = tessellation_latlong(ntheta, nphi)     # npix = 2500

All rings have the same number of pixels (nphi), giving a total of ntheta * nphi pixels. The grid ranges over colatitude [0, pi) and longitude [0, 2*pi).

Neighbor information for total variation:

tv_info = tv_neighbors_longlat(ntheta, nphi)

Spots

ROTIR provides two spot-creation interfaces.

Any tessellation (HEALPix or lon/lat) — works with stellar_geometry:

tmap = make_circ_spot(tmap, star_geometry, spot_radius, lat, long;
                       bright_frac=0.8)

• spot_radius in degrees, lat in [-90, 90], long in [-180, 180] (or [0, 360])
• Uses Euclidean chord distance in 3D, so spots remain circular on non-spherical surfaces (ellipsoids, rapid rotators, Roche lobes)
• A fill-fraction variant is also available: make_circ_spot_spotfill(star_geometry, spot_radius, lat, long; profile="flat") where profile can be "flat" or "linear" (linearly decreasing toward the edge)

Lon/lat grid only — pixel-index-based (legacy):

tmap = make_circ_spot(tmap, ntheta, nphi, spot_radius, latitude, longitude;
                       bright_frac=0.8)
tmap = make_spot_move(tmap, ntheta, nphi, period, B_rot, tepochs)

A cool spot created with make_circ_spot on a lon/lat sphere:

Resolution levels

HEALPix resolution progression on a rapid rotator (pixel edges shown):

n=2 (192 px)	n=3 (768 px)	n=4 (3072 px)

Choosing a scheme

For image reconstruction from interferometry, HEALPix is recommended due to equal-area pixels and multi-resolution support.