## 1.4.3 Parameters for Normalization Transformation

The following are the parameters necessary for normalization transformation of U-coordinate into 2-D V-coordinate . Each parameter can be changed/inquired by the internal routines sgpget/sgpset of GRPH1, although utilities are available for setting similar parameters at once.

Viewport
(VXMIN, VXMAX, VYMIN, VYMAX)
This set of parameters that specifies the viewport is common for all transformations and is requisite. A viewport is usually a rectangular frame in which the axes are drawn. When clipping is specified, graphics outside the clipping rectangle is not drawn. The coordinates in VC for the lower left-hand and upper right-hand corners are used to specify the viewport.

These parameters can be changed/inquired with sgsvpt/sgqvpt.

Window
(UXMIN, UXMAX, UYMIN, UYMAX)
This set of parameters is necessary for setting the rectangular linear coordinate. This is specified using the coordinates in UC corresponding to the viewport.

In map projection coordinates, this information is not needed for GRPH1, but the GRPH2 package may inquire about this as "the lat-long range in scope." In this case, the viewport and the window does not usually coincide.

These parameters can be changed/inquired with sgswnd/sgqwnd.

Similar Transformation Parameters
(SIMFAC, VXOFF, VYOFF)
A set of parameters for similar transformation (scaling and origin shift) in a rectangular curvilinear coordinate and map projection transformations, defined by the function MATH1/CTRLIB for rectangular curvilinear coordinates, and MATH1/MAPLIB for map projection coordinates. In GRPH1, the value returned by these functions is multiplied by simfac to convert to a value in VC, and is set at the position with the origin shifted parallel from the center of the viewport by (VXOFF, VYOFF)

These parameters can be set/inquired with sgssim/sgqsim.

Map Coordinate Rotation Parameters
(PLX, PLY, PLROT)
Map projection transformations are always carried out within the longitudinal range of [-180º, 180º] and latitudinal range of [-90º, 90º] , so to project from an arbitrary viewpoint, the lat-long coordinates must be rotated before the projection. (PLX, PLY) specifies the latitude and longitude (UC) for positioning the pole of the projection coordinate (TC), and plrot specifies the angle of rotation of the projection coordinates around the pole.

The relationship between these parameters and the "Euler's angle( theta , phi , psi )" used generally to specify the angle of rotation in 3-D is theta = pi /2-ply, phi =plx, psi =plrot. (Consult math dictionaries for a definition of the Euler's angle.)

These parameters can be changed/inquired with sgsmpl/sgqmpl.

The actual procedures are as follows. In the initial state, UC coincides with TC (projection coordinate).

1.
TC is rotated around the north pole by phi .
(The meridian at plx longitude is set as the central meridian.)
2.
The north pole of TC is rotated in the direction of the central meridian by theta .
(The point at long. and lat.(PLX, PLY) is considered as the north pole of TC.)
3.
The projection is rotated around the north pole of the TC by psi .
(The angle between the meridian at longitude plx and the central meridian of TC is plrot.)

The transformation from TC to VC is determined by the similar transformation parameters above. The value of TC projected onto the origin of VC depends on the type of map projection.

 Projection TC projected onto origin Cylindrical Projection (0, 0) Azimuthal Projection (90, 0) Conic Projection Apex of Cone

The value of TC projected onto the origin corresponds to the case when the Euler's angle is (0,0,0), or when (PLX,PLY,PLROT)=( lambda ,90.,0).

To set the meridian at lambda as the central meridian in a normal cylindrical or a conic projection (on normal aspect), let (PLX,PLY)=( lambda ,90.,0).

To draw a map with long. and lat. of ( lambda , phi ) at the center in an azimuthal projection, let (PLX,PLY)=( lambda , phi ), and specify plrot as required. When plrot=0, the south pole is always below the origin.

In a cylindrical projection on the traverse aspect, specify a point on the Equator as (PLX,PLY). When plrot=0 , the south pole of UC is projected onto the central meridian. For example, if you wish to present the Arctic Ocean, the Atlantic Ocean, and the Southern Ocean as a continuous body with the traverse aspect, let (PLX, PLY, PLROT) = (60.,0.,-90.).

Standard Latitude
(STLAT1, STLAT2)
In the conic projection, the standard latitude is necessary besides the above parameters. Since the Lambert's conformal conical projection has two standard parallels, stlat2 also needs to be specified. In all other conic projections, only stlat1 is used.

These parameters can be inquired/changed with sgrget/sgrset.

(RSAT)
Originally, the orthographic projection views the Earth from line of infinity. But a projection that view the Earth from a satellite at a finite point is also available as option (Satellite View). To select this option in the orthographic projection, specify rsat to a value larger than 1. rsat is the "radius of the satellite orbit" when 1 is the radius of the Earth.

This parameter can be inquired/changed with sgrget/sgrset.