Routines to provide the index of the template pixel associated with a given HEALPix pixel, for a resolution parameter ${N_{\rm side}}$ .
Any pixel can be matched in shape to a single of these templates by a combination of a rotation around the polar axis with reflexion(s) around a meridian and/or the equator.
The template pixels are all located in the Northern Hemisphere, or on the Equator. They are chosen to have their center located at

$\displaystyle z=\cos(\theta)\ge 2/3,$ $\textstyle \$ $\displaystyle 0< \phi \leq \pi/2,$

$\displaystyle %[Nside*(Nside+2)/4] 2/3 > z \geq 0,$ $\textstyle \$ $\displaystyle \phi=0, \quad{\rm or}\quad \phi=\frac{\pi}{4{N_{\rm side}}}. %[Nside]$

They are numbered continuously from 0, starting at the North Pole, with the index increasing in $\phi$ , and then increasing for decreasing z.

Location in HEALPix directory tree: src/f90/mod/pix_tools.f90

FORMAT

call template_pixel_nest, template_pixel_ring( nside, pixel_nest, template, reflexion )

FORMAT

call template_pixel_nest, template_pixel_ring( nside, pixel_ring, template, reflexion )

ARGUMENTS

name & dimensionality	kind	in/out	description

nside	I4B	IN	the HEALPix ${N_{\rm side}}$ parameter.
pixel_nest	I4B	IN	NESTED scheme pixel identification number over the range {0, $12{N_{\rm side}}^2-1$ }.
pixel_ring	I4B	IN	RING scheme pixel identification number over the range {0, $12{N_{\rm side}}^2-1$ }.
template	I4B	OUT	identification number(s) of the template matching in shape the pixel(s) provided (the numbering scheme of the pixel templates is the same for both routines).
reflexion	I4B	OUT	in {0, 3} encodes the transformation(s) to apply to each pixel provided to match exactly in shape and position its respective template. 0: rotation around the polar axis only, 1: rotation + East-West swap (ie, reflexion around meridian), 2: rotation + North-South swap (ie, reflexion around Equator), 3: rotation + East-West and North-South swaps

EXAMPLE:

call template_pixel_ring(256, 500000, template, reflexion)

Returns in template the index of the template pixel (16663) whose shape matches that of the pixel #500000 for ${N_{\rm side}}=256$ . Upon return reflexion will contain 2, meaning that the template must be reflected around a meridian and around the equator (and then rotated around the polar axis) in order to match the pixel.

RELATED ROUTINES

This section lists the routines related to template_pixel_nest, template_pixel_ring

nside2templates: returns the number of template pixel shapes available for a given ${N_{\rm side}}$ .
same_shape_pixels_ring
same_shape_pixels_nest: return the ordered list of pixels having the same shape as a given pixel template

Eric Hivon 2010-06-18

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