Help for AUXILIARY

PURPOSE:
To compute the conformal or authalic to planetocentric auxiliary coordinate
mapping which optimizes map projections of irregularly shaped objects.
Note: AREAISO is the alternative method.

EXECUTION:
auxiliary inp=(E1,F1,G1,E2,F2,G2,..e,f,g) out=(lat,lon,residual,dlat,dlon)

where:

E,F,G are the output of EFGISO for the iso.
E1,F1,G1 are smoothed more than En,Fn,Gn.
e,f,g are the output of EFGISO for the sphere.

lat and lon are maps in rectangular projections the same size and format as
the input images. These images contain the planetocentric lat and lon
respectively which map to the conformal or authalic lat and lon of their pixel
coordinates. ie: the contents are the centric coordinates of the auxiliary
arguments.

residual is the Tissot indicatrix angle in degrees for the conformal case:
 sin(residual/2)=(a-b)/(a+b) 
or the area ratio for the authalic case:
 residual=sqrt(eg-f*f)/sqrt(EG-F*F)
 
residual is in the same format as lat & lon.

dlat and dlon are images of the displacement of the computed centric 
coordinates from their initial auxiliary values.


METHOD:

The following steps are executed for every output pixel:

1. Compute the conformat or authalic auxiliary coordinate from the output pixel 
   location.

2. Convert this to input image coordinates.

3. Read the values for e, f, g from input images 4-6.

4. Guess that the centric lat & lon is the same as the auxiliary lat & lon.

5. Convert the centric lat & lon to input image coordinates.

6. This gives values for E, F, G from input images 1-3.

7. From efg & EFG compute the constraint violation.

8. Estimate a new centric location by moving along the gradient downhill
   by a bit.

9. Iterate steps 5-8 to find the smallest constraint violation.


These steps are repeated for each set of three input images.


 the Tissot indicatrix angle omega
where sin(omega/2)=(a-b)/(a+b)
and a and b are the axes of an ellipse on the projection resulting from 
the incorrect projection of a circle on the ISO.

where E,F,and G are measured on the iso (irregularly shaped object).
and e,f,and g are measured from the sphere.

e = (dx/dp)**2+(dy/dp)**2+(dz/dp)**2
f = (dx/dp)*(dx/dl)+(dy/dp)*(dy/dl)+(dz/dp)*(dz/dl)
g = (dx/dl)**2+(dy/dl)**2+(dz/dl)**2

where:
x and y are cartesian coordinates on the map projection
( sample is x, and line is -y).
p is latitude
l is longitude

HISTORY:
9-1-98  J Lorre. 
COGNIZANT PROGRAMMER:  Jean Lorre