   ### Aitoff Projection

(Used for Low-Resolution All-Sky Maps)

The Aitoff equal area projection was used to provide photometrically correct maps of the entire celestial sphere for the Low-Resolution All-Sky Maps. Galactic coordinates were chosen as a convenient and natural coordinate system.

The transformation equations for conversion between line and sample number (i.e., y and x pixels) in a map with Aitoff projection and Galactic coordinates on the sky are shown below.

Forward:
Define
```      scale:  2 pixels per degree in the map
l, b:  Galactic coordinates of a given position
l0:  Galactic longitude of the map center
RHO = arccos[cos(b) x cos({l - l0}/2)]
THETA = arcsin[cos(b) x sin({l - l0}/2)/sin(RHO)]```
then the pixel coordinates in the image are
```
SAMPLE = -4 x scale x 180/pi x sin(RHO/2) x sin(THETA)
LINE = (±)2 x scale x 180/pi x sin(RHO/2) x cos(THETA),

where the '+' applies to b < 0 and the '-' to b >= 0.
```
Reverse:
Define
```      Y = -LINE/(2 x scale x 180/pi)
X = -SAMPLE/(2 x scale x 180/pi)
A = (4 - X^2 - 4 x Y^2)^0.5```
then the Galactic coordinates are
```
b = 180/pi x arcsin(A x Y)
l = l0 + 2 x 180/pi x arcsin[A x X/(2 x cos{b})]
```

### Equivalent Cylindrical Projection

(Used for Galactic Plane Maps)

The Lambert normal equivalent cylindrical projection was used to provide an equal area projection of the sky within 10° of the Galactic plan for the Galactic Plane Maps. The projection cylinder is tangent to the celestial sphere at the Galactic equator and the projection proceeds by projecting radially outward from each point on the polar axis of the Galactic coordinate system in a plane parallel to the equatorial plane. The maximum angular distortion (deviation of bearing) is 0.9°. The equal area property of the transformation preseves photometric accuracy when integrating fluxes for an extended source.

The transformation equations for conversion between line and sample number (i.e., y and x pixels) in a map with cylindrical projection and Galactic coordinates on the sky are shown below.

Forward:
Define
```      scale:  30 pixels per degree
l, b:  Galactic coordinates of a given position
l0:  Galactic longitude of the map center```
then the pixel coordinates in the image are
```
LINE = -scale x 180/pi x sin(b)
SAMPLE = -scale x (l - l0)
```
Reverse:
```      l = l0 - SAMPLE/scale
b = -arcsin[LINE/(scale x 180/pi)]
```

### Gnomonic (Tangent Plane) Projection

(Used for ISSA, ISSA Galactic Plane Mosaics, and Sky Flux Maps)

The gnomonic projection produces a geometric projection of the celestial sphere onto a tangent plane from a center of projection at the center of the sphere. Each individual field has its own tangent projection plane with the tangent point at the center of the field. This projection is neither conformal (angle preserving) nor equivalent (equal area) but does have the property that all straight lines in the projection are great circles on the sphere. All projections were done so that the sky coordinate associated with a pixel refers to the position at the center of the pixel. In the Sky Flux images, which have 16.5° extents, the maximum distortion of angles is 0.6° and the maximum distortion of area is 6%. The area distortion is approximately proportional to the inverse cube of the cosine of the angular displacement from the center of the field. The distortion is in the sense to make extended areas cover more square arcminute pixels than their true solid angles would require. This results in over-estimating fluxes when integrating sources within fixed intensity contours.

The transformation equations for conversion between line and sample number (i.e., y and x pixels) in a map with gnomonic projection and right ascension and declination on the sky are shown below. (Note that Galactic coordinates are used instead for the ISSA Galactic Plane Mosaics; the transformation equations are otherwise the same.)

Forward:
Define
```      scale:  number of pixels per degree in the map
alpha, delta:  Equatorial coordinates of a given position
alpha0, delta0:  Equatorial coordinates of the map center
A = cos(delta) x cos(alpha - alpha0)
F = scale x (180/pi)/[sin(delta0) x sin(delta) + A x cos(delta0)]```
then the pixel coordinates in the image are
```
LINE = -F x [cos(delta0) x sin(delta) - A x sin(delta0)]
SAMPLE = -F x cos(delta) x sin(alpha - alpha0)
```
Reverse:
Define
```      X = SAMPLE/(scale x 180/pi)
Y = LINE/(scale x 180/pi)
D = arctan[(X^2 + Y^2)^0.5]
B = arctan(-X/Y)
XX = sin(delta0) x sin(D) x cos(B) + cos(delta0) x cos(D)
YY = sin(D) x sin(B)```
then the right ascension and declination are
```
alpha = alpha0 + arctan(YY/XX)
delta = arcsin[sin(delta0) x cos(D) - cos(delta0) x sin(D) x cos(B)]
```
NOTE: The arctangent functions for B and alpha must be four-quadrant arctangents.

### Orthographic Projection

(Used for Faint Source Survey Plates)

The orthographic projection, like the gnomonic projection, is a projection of the celestial sphere onto a tangent plane, but the center of projection is infinitely distant from the tangent plane. Each individual field has its own tangent projection plane with the tangent point at the center of the field. All projections were done so that the sky coordinate associated with a pixel refers to the position at the center of the pixel.

The transformation equations for conversion between line and sample number in a map with orthographic projection and right ascension and declination on the sky are shown below.

Forward:
Define
```      scale:  number of pixels per degree in the map (for the FSS
plates, 240 at 12 and 25 microns, 120 at 60 and 100 microns)
alpha, delta:  Equatorial coordinates of a given position
alpha0, delta0:  Equatorial coordinates of the map center
A = cos(delta) x cos(alpha - alpha0)```
then the pixel coordinates in the image are
```
LINE = scale x (180/pi) x [A x cos(delta0) - sin(delta) x cos(delta0)]
SAMPLE = -scale x (180/pi) x cos(delta) x sin(alpha - alpha0)
```
Reverse:
Define
```      X = SAMPLE/(scale x 180/pi)
Y = LINE/(scale x 180/pi)
D = arccos[(X^2 + Y^2)^0.5]
B = arctan(-X/Y)
XX = sin(delta0) x sin(D) x cos(B) + cos(delta0) x cos(D)
YY = sin(D) x sin(B)```
then the right ascension and declination are
```
alpha = alpha0 + arctan(YY/XX)
delta = arcsin[sin(delta0) x cos(D) - cos(delta0) x sin(D) x cos(B)]
```
NOTE: The arctangent functions for B and alpha must be four-quadrant arctangents.

Reference:
Infrared Astronomical Satellite (IRAS) Catalogs and Atlases, vol. 1, Explanatory Supplement, 1988, ed. C. Beichman, et al., NASA RP-1190 (Washington, DC: GPO)