DESIGNATE TORUS

Syntax

Fixed

   DESIGNATE @int(1001:10099)
      CENTER @body
      REFERENCE TORUS
      (3:3){ LATITUDE  @number
           | LONGITUDE @number
           | RADIUS    @number }
      (0:5){ ORIGIN LONGITUDE @number
           | ORIGIN LATITUDE  @number
           | ORIGIN RADIUS    @number
           | POLAR  LATITUDE  @number
           | POLAR  LONGITUDE @number }
      (0:3){ FROM  @calendar
           | TO    @calendar
           | EPOCH @calendar }

Moving

   DESIGNATE @int(1001:10099)
      CENTER @body
      REFERENCE TORUS
      (3:3){ LATITUDE  (1:2)@number
           | LONGITUDE (1:2)@number
           | RADIUS    (1:2)@number }
      (0:5){ ORIGIN LONGITUDE @number
           | ORIGIN LATITUDE  @number
           | ORIGIN RADIUS    @number
           | POLAR  LATITUDE  @number
           | POLAR  LONGITUDE @number }
      EPOCH @calendar
      (0:2){ FROM  @calendar
           | TO    @calendar }

Description

When creating a designated object in this system, you must specify the latitude, longitude, and radius of the object within the torus reference frame. You may also define the frame itself by specifying a tilt and an offset.

Informal definition

Start with the cartographic frame attached to the reference body. Tilt the frame so that the north pole (the positive z-axis) runs through some latitude and longitude. Now move the frame to center it at some point away from the center of the reference body. Now rotate it again so that the x-z plane contains the sub-Earth point. The new frame is fixed to the reference body and swings around as the body rotates.

Formal Definition

The torus frame is an apparent Earth meridian frame.

The polar latitude and longitude, if specified, define a direction vector, in the cartographic frame attached to the reference body. The z-axis of the torus frame is parallel to this direction vector.

The origin latitude, longitude, and radius, if specified, define a point within the cartographic frame. The torus frame is centered at this point.

Let E be the vector from Earth to the apparent center of the torus frame. Then the y-axis of the torus frame is the cross-product

   z x -E

and the x-axis of the frame is the cross-product

   y x z

Let P be a point in space, and let V be the position vector from the origin of the torus frame to P.

The latitude of P is the angle between the equator and V. North latitudes are positive, south latitudes are negative.

The longitude of P is the angle between the x-axis and the projection of V onto the x-y plane. Longitude is positive from x toward y. (As viewed from the surface of the earth, objects at 90 degrees longitude and 0 degrees latitude will rise before the center of the TRS. Objects at -90 degrees longitude and zero degrees latitude will rise after the center of the TRS.)

The radius of P is the distance from the origin of the torus frame to P (the magnitude of V).

Default values

If the tilt and offset terms that define the torus frame are not supplied, they default to the nominal values:

   ORIGIN LONGITUDE   0
   ORIGIN LATITUDE    0
   ORIGIN RADIUS      0
   POLAR  LATITUDE    0
   POLAR  LONGITUDE   0

Examples

Example 1

Place a designated object at the ansa (end) of a torus of particles at five Jupiter radii from the center of Jupiter.

   DEFINE ANSA 1001;
 
   DESIGNATE ANSA
        CENTER        JUPITER
        REFERENCE       TORUS
        LATITUDE    0 DEGREES
        LONGITUDE  90 DEGREES
        RADIUS      357000 KM
        POLAR LATITUDE   83 DEGREES
        POLAR LONGITUDE 202 DEGREES;