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Alternatively, if a grid of height points has been measured or has been 
computed, discreet points of the contour line can be computed by interpolating for 
the contour height along the grid lines. Usually, the grid is made sufficiently 
dense to allow linear interpolation between the grid points. An exception is a 
program described by Kusch and Naser [51], where the linear interpolation is 
preceded by a densification of the grid points with the help of eq. (12a). 
If local polynomial surfaces have been computed, it is natural to derive 
the position of points on the grid lines and in the interior of the grid elements 
by solving for the contour line in each local surface. Only Jancaitis and 
Junkins [38] state explicitly that they do this. 
10.2 Contour threading 
If the discreet points on a contour line have been determined by inter- 
polating along the grid lines, these points must now be sorted in the correct 
sequence. This has been called contour threading. A difficulty can occur if a 
rectangular grid is used but local polynomial surfaces are not computed. In that 
case, linear interpolation along the grid lines may show that certain grid elements 
are crossed by two contour lines of the same height and thus that four points on 
the sides of the grid element must be properly connected in pairs. 
Stanger [68] determines the proper connections by a combination of 
distance and direction arguments. Connelly [25], having found one of the four 
points as the point of entry of a contour line into an element, chooses the nearest 
of the three remaining points as exit point. This may lead to different results 
depending on which of the four points is used as the first point. A simple and 
unambiguous solution would be to determine the proper connections from a local 
bilinear polynomial. 
In several programs, interpolation between the discreet points is per- 
formed by straight-line connections. Because of the density of the points, this 
will often produce acceptably smooth-looking contour lines. More sophisticated 
interpolations make use of one of the two procedures described in the next two 
sections of this report. The appearance of the contour lines can then be further 
improved by discarding discreet points that are too close together [41] or by 
choosing discreet points only on the boundaries of greatest slope of each grid 
element [45]. 
10.3 Interpolation by functions f(x,y) * o 
A smooth curvilinear connection between the discreet points of a contour 
line can be made, independently of any local polynomial surfaces, by an analytical 
formulation of the connecting curve. Such a formulation can be provided by a 
separate subroutine and, in some cases, it is a feature of the built-in program of 
an automatic drafting table. On the curve, points are then computed at such short 
distances that straight-line connections produce an apparently smooth curve. 
The formulation of one of the x,y coordinates as an explicit function of 
the other one poses the problem that for a sharply curved section of a contour line 
such a function is not necessarily one-valued. If it is not, a polynomial cannot 
be used. Laurikanen has shown with examples that, even if the function is one- 
valued, the explicit representation of one coordinate by a polynomial in the other 
one can lead to widely different results for different orientations of the 
coordinate system. Therefore, this explicit representation is not advisable. 
Marckwardt [59] solves this problem by using an auxiliary coordinate 
system for each interpolation between two discreet points. The origin of this 
system is in one of the points and the x-axis is along the chord. The y-coordinate 
is written as a third-degree polynomial in x. For the computation of the 
coefficients, the tangents at the end points are used but it is not specified how 
those are determined. Even in this auxiliary coordinate system, y can be a