% GPS problems

These problems all work toward applications in **GPS navigation**.

**Geographical note:** for our purposes, the Earth is a sphere with
a radius of 6,378,137 meters.



Numeric computation with some conditionals
----------------------------------------

Functions `north`, `south`, `east`, `west`.  Each takes three
arguments $dd$ $mm$ $ss.sss$ and returns an angle in radians.  
(No conditionals)

Function `radians->degrees`  
(No conditionals)

Structure `gps-posn` for GPS coordinates

Convert GPS coordinates to human-readable string  
(**conditionals**)

Function `gps->degrees-posn` (no conditionals)

Functions `flat-world-distance`, `great-circle-distance`
(probably no conditionals; need to check).  Return distance in meters

  - Flat-world distance should be self-explanatory.

  - The easiest way to compute great-circle distance is to draw two
    vectors from the center of the earth and to compute the angle
    between them.  The [dot product and law of
    cosines](http://www.math.washington.edu/~king/coursedir/m445w04/notes/vector/dotproduct.html)
    will be helpful here.


Functions `feet->meters`, `miles->meters`, `meters->feet`,
`meters->miles`.  
(no conditionals)

Function `meters->english`, converts meters into a string of English
units that uses feet for distances under 1000 feet and miles (to the
nearest tenth) for distances of 1000 feet or more:

   31 ft
   789 ft
   0.1 mi
   10.7 mi

Structure GPS-step.  Headings are in *degrees* always:

    (define-struct gps-step meters heading)

Function `take-step : gps-posn gps-step -> gps-posn`

Function `flat-world-heading`.  Analog of the law $A + (B - A) = B$ is

    (almost-equal?
       (gps-step A (make-gps-step (flat-world-distance B A) (flat-world-heading B A)))
       B)

Unspecified data type `gps-path`.
Teachpack function `follow-path` to compute destination from starting
point or total distance traveled.

Given intervals of latitude and longitude, find a way to convert GPS
coordinates to $(x,y)$ coordinates for display in a map.