Problem A: The global positioning system GPS uses the fact that a receiver can get the
difference of distances to two satellites. Each GPS satellite sends periodically signals
which are triggered by an atomic clock. While the distance to each satellite is not
known, the difference from the distances to two satellites can be determined from the
time delay of the two signals. This clever trick has the consequence that
the receiver does not need to contain an atomic clock itself. a) (7 points) Given two satellites P=(2,0,0), Q=(0,0,0) in space. Identify the quadric of all points X, such that the distance d(X,P) to P is by 1 larger than the distance d(X,Q) to Q. b) (1 point) Assume we have three satellites P,Q and R in space and that the receiver at X knows the distances d(X,P) - d(X,Q) and d(X,P) - d(X,R). Why do we know the distance d(X,Q) - d(X,R) also? Conclude that 3 satellites are not enough to determine the location of the receiver. c) (2 points) Assume we have 4 satellites P,Q,R,S in space and that the receiver knows all the distance differences from X to any pair of satellites from the 4. What is in general the set of points for which these distances match? Conclude that with some additional rough location information we can determine the GPS receiver position when 4 satellites are visiable. |
Problem B: Find the surface whose points have the property that the distance to the x-axes is half the distance to the parametrized line r(t) = (1,0,0) + t (0,0,1)." |