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! Suppose we were given the job of figuring out the minimum amount of water
! pipe necessary to connect up the houses on a street to a water main. We
! have to determine where to dig a straight trench down the street for the
! water main pipe. Suppose that the houses are set back from the street by
! varying distances, and that each house should be about 5 meters from the
! water main. The problem is to find the trench which would result in the
! minimum usage of water pipe segments from the main to the houses. We also
! should determine the total length of the pipe we will need.
!
! Suppose the (x,y) coordinates of the connections to the houses are stored
! in a file, HOUSES.DAT. Remember from plane geometry that the distance from
! point (x,y) to a straight line with slope A and y-intercept B is given by
! D = |A*x+B-y|/sqrt(A^2+1), see the figure (waterpipe-fig.gif). We want
! to minimize the connecting pipe lengths by varying the free parameters A
! and B. The following small script does what we want.
!
! Since least squares is used for fitting, we take the square root of the
! expression. Note that we have more than one independent variable in the
! expression. Extension to multidimensional cases is straightforward.
! The problem of fitting to a scattered set of points with a straight line
! which is, on the average, a fixed number of units away will, in general,
! have two solutions and which one you get will depend on the starting values
! of the free parameters.
!
!
FILE = 'houses.dat'
LENGTH_GOAL = 5 ! try to achieve this goal
READ\-MESSAGES FILE X Y ! read in the house positions
SCALAR\FIT A B ! declare the free parameters
A = 1 ! initialize
B = 1 ! initialize
D[1:LEN(X)] = LENGTH_GOAL !
DISTANCE = 'ABS(A*X+B-Y)/SQRT(A*A+1)' !
FIT\-MESSAGES D=SQRT(DISTANCE) ! do the fit
FIT\UPDATE PIECES ! redo using calculated A and B
STATISTICS PIECES PSUM\SUM ! the total length of pipe needed
DISPLAY 'Total length of pipe = '//RCHAR(PSUM,'%4.2f')
waterpipesDone = 1
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