The above coordinates are for the parking when locating the cache. PREAMBLE
To appreciate the philosophy behind the MP series one does need to have a fundamental understanding of Geocaching. This is certainly true when you walk backwards and forwards and scrutinize the changing latitude and longitudes on your GPS receiver. At this point you realize that there is a one to one mapping between your readout and the x and y positions on a Cartesian Grid. Thus, somewhat unconsciously, you are using the methods of Analytic
Geometry.
For this reason we have constructed this series as a tutorial to gain an understanding of some of the fundamental concepts of Analytic Geometry. In
MP#1, we explored the properties of a straight line, y =mx + b. (Two nonparallel lines will eventually cross). In this geocache, we will examine the properties of a line that is not perfectly straight.
Line curvature requires the addition of a quadratic term in the linear expression viz. y = a*x^2 + b*x+c
Another way of viewing this expression is to regard the expansion as a power series to the second order. This is the generalized equation for a geometrical parabola.
THE PROBLEM
The problem is to determine the most northerly point (coordinates) of a smooth arc that pass through the points I (1000, 1207), II (2000, 2334), and III (3000, 527).
Analysis
The shape of this function (curve) is established by the a, b and c parameters. These constants can be readily be found using a regression procedure. Many of are listed under a google search (regression java) Note: a true regression requires four points. In this case, use a package that uses three data points to produce an exact fit. To locate the most northerly point, you need to regress back to your high school days. Does dy/dx = 0 strike a chord.
Divide the resulting x-coordinate by 10000 and add it to: N43 08.395
Divide the resulting y-coordinate by 10000 and add it to: W079 27.409
Bring your own pencil and the cache is small.