Positioning satellites

1. A method comprising:

receiving, in a mobile terminal, first information for defining a first algebraic function that relates a time variable to a first mapped variable;

using, in the mobile terminal, the function to convert a time variable value to a first value of the first mapped variable;

converting, in the mobile terminal, the first value of the first mapped variable to a second value of a second mapped variable; and

positioning, in the mobile terminal, a satellite by converting the second value of the second mapped variable value to a position value that positions a satellite;

wherein a second algebraic function that relates the first mapped variable to the second mapped variable is used to convert the first value of the first mapped variable to a second value of a second mapped variable, the second algebraic function is an algebraic solution to a differential equation of motion of the satellite, and the first and second algebraic functions are expressed using complex exponential functions.

2. The method as claimed in claim 1 , wherein the first algebraic function is based upon an algebraic solution to a geometric algebraic differential equation of motion of the satellite.

3. The method as claimed in claim 1 , wherein the first information includes coefficients for defining the first algebraic function.

4. The method as claimed in claim 1 , wherein the differential equation of motion is

2

⁢

ⅆ

2

ⅆ

s

2

⁢

U

-

EU

=

U

⁢

r

_

⁢

f

_

where

E is constant

U is defined by r =U † σ 1 U

σ 1 has a constant unit length

s is defined by

ⅆ

ⅆ

s

=

r

⁢

ⅆ

ⅆ

t

and

f is expressed using a geometric algebra in terms of U.

5. The method as claimed in claim 1 , further comprising receiving second information for defining the second algebraic function.

6. The method as claimed in claim 5 , wherein the second information includes coefficients for defining the second algebraic function.

7. The method as claimed in claim 1 , wherein the first and second algebraic functions are expressed as weighted summations of complex exponential functions.

8. The method as claimed in claim 1 , further comprising storing the position of the satellite in memory.

9. An apparatus comprising:

processing circuitry; a memory storing instructions which when loaded into the processor enable the apparatus to:

use a function that relates a time variable to a first mapped variable, and is defined by received information, to convert a time variable value to a first value of the first mapped variable;

convert the first value of the first mapped variable to a second value of a second mapped variable; and

position a satellite by converting the second value of the second mapped variable value to a position value that positions a satellite;

wherein a second algebraic function that relates the first mapped variable to the second mapped variable is used to convert the first value of the first mapped variable to a second value of a second mapped variable, the second algebraic function is an algebraic solution to a differential equation of motion of the satellite, and the first and second algebraic functions are expressed using complex exponential functions.

10. An apparatus comprising:

means for using a function that relates a time variable to a first mapped variable, and is defined by received information, to convert a time variable value to a first value of the first mapped variable;

means for converting the first value of the first mapped variable to a second value of a second mapped variable; and

means for positioning a satellite by converting the second value of the second mapped variable value to a position value that positions a satellite;

wherein a second algebraic function that relates the first mapped variable to the second mapped variable is used to convert the first value of the first mapped variable to a second value of a second mapped variable, the second algebraic function is an algebraic solution to a differential equation of motion of the satellite, and the first and second algebraic functions are expressed using complex exponential functions.

11. A non-transitory computer-readable medium containing computer instructions stored therein for causing a computer processor to:

use a function that relates a time variable to a first mapped variable, and is defined by received information, to convert a time variable value to a first value of the first mapped variable;

convert the first value of the first mapped variable to a second value of a second mapped variable; and

position a satellite by converting the second value of the second mapped variable value to a position value that positions a satellite;

wherein a second algebraic function that relates the first mapped variable to the second mapped variable is used to convert the first value of the first mapped variable to a second value of a second mapped variable, the second algebraic function is an algebraic solution to a differential equation of motion of the satellite, and the first and second algebraic functions are expressed using complex exponential functions.

12. A method comprising:

positioning, in a mobile terminal, a satellite using an algebraic solution to the equation

2

⁢

ⅆ

2

ⅆ

s

2

⁢

U

-

EU

=

U

⁢

r

_

⁢

f

_

where

E is a constant

U is defined by r =U † σ 1 U

σ 1 has a constant unit length

s is defined by

ⅆ

ⅆ

s

=

r

⁢

ⅆ

ⅆ

t

 and

f is expressed using geometric algebra in terms of U; and

wherein a second algebraic function that relates a first mapped variable to a second mapped variable is used to convert a first value of the first mapped variable to a second value of a second mapped variable, the second algebraic function is an algebraic solution to a differential equation of motion of the satellite, and the first and second algebraic functions are expressed using complex exponential functions.

13. The method as claimed in claim 12 , wherein f is at least one of the following:

a) a gravitational field created by the uneven distribution of the earth's mass expressed as location dependent values

or

b) a geometric algebraic expression of a multi-pole function

or

c) f is expressed using an empirically determined, location independent constant or

d) f is expressed using an empirically determined, initial-conditions dependent constant

f

or

e) f is expressed using perturbation

theory

or

f) is expressed as a constant function of U for a particular co-ordinate system and particular initial conditions.

14. An apparatus comprising:

means for determining an algebraic solution to the equation

2

⁢

ⅆ

2

ⅆ

s

2

⁢

U

-

EU

=

U

⁢

r

_

⁢

f

_

where

E is a constant

U is defined by r =U † σ 1 U

σ 1 has a constant unit length

s is defined by

ⅆ

ⅆ

s

=

r

⁢

ⅆ

ⅆ

t

 and

f is expressed using geometric algebra in terms of U,

and

means for positioning the satellite using the algebraic solution; and

wherein a second algebraic function that relates the first mapped variable to the second mapped variable is used to convert the first value of the first mapped variable to a second value of a second mapped variable, the second algebraic function is an algebraic solution to a differential equation of motion of the satellite, and the first and second algebraic functions are expressed using complex exponential functions.

15. A non-transitory computer-readable medium containing computer instructions stored therein for causing a computer processor to:

determine an algebraic solution to the equation

2

⁢

ⅆ

2

ⅆ

s

2

⁢

U

-

EU

=

U

⁢

r

_

⁢

f

_

where

E is a constant

U is defined by r =U † σ 1 U

σ 1 has a constant unit length

s is defined by

ⅆ

ⅆ

s

=

r

⁢

ⅆ

ⅆ

t

 and

f is expressed using geometric algebra in terms of U,

and

position at least one satellites using the algebraic solution; and

wherein a second algebraic function that relates the first mapped variable to the second mapped variable is used to convert the first value of the first mapped variable to a second value of a second mapped variable, the second algebraic function is an algebraic solution to a differential equation of motion of the satellite, and the first and second algebraic functions are expressed using complex exponential functions.