- 17 > 
two-valued function of x. If it is, the polynomial will instead produce a one- 
valued function which produces a sharp point in the contour line at one of the 
ends of the segment. Rhindt [63] states that the ECU system fits cubic polynomials 
between the discreet points. He does not give formulas, but his remark that these 
produce occasional cusps indicates that the formulation is the same as 
Marckwardt's. 
Marckwardt remarks also that this formulation can produce unwarranted 
bulges in a contour line and suggests to prevent these by compressing each segment 
of the line in the direction of the chord by multiplication of the y-coordinate by 
a compression factor. To retain a smooth curve, the compression factor is made a 
variable which equals unity at the ends of the segment. 
A possibility which may deserve exploration is the use of the six- 
parameter quadratic and the nine-parameter biquadratic polynomials f(z,y) 7 h. 
Such polynomials could be made to fit at six or at nine consecutive points, 
respectively, and they could be used in the two-step procedure described below. 
10.4 Interpolation by parametric functions 
In the parametric representation of a contour line, the coordinates x 
and y are written as functions of a parameter s. If the discreet points have been 
recorded while tracing the contour line, this parameter may be the elapsed time or 
the distance along the line. In the latter case it will, at least in first 
approximation, be measured along the chords connecting the discreet points. 
Nakamura [62] computes a whole contour line at once by writing x and y as Fourier 
series of the path length along the contour line. In his examples, with relatively 
few discreet points, this gives very smooth contour lines. An accurate represen- 
tation of a contour line will usually require the use of a large number of discreet 
points and this will make Nakamura's procedure rather complicated. 
A simple procedure for obtaining smooth contour lines consists in using 
polynomials in two steps. First, at each of two successive discreet points the 
direction of the contour line is determined. Then, a curve is fitted between the 
two points. 
In the first step, polynomials of even degree (») 
x = ap tas t a28? ^ Qe. 
y 2 bg * b48 * bos? * ... (15) 
can be computed from the x,s and the y,s coordinates of a discreet point and the 
nearest % » points in each direction. From these, one computes the values of the 
first and second derivatives with respect to s at the discreet point: x',y',x",y". 
Koch [41] specifies here polynomials of the second or fourth degree; Marckwardt 
[58] prefers polynomials of at least the fourth degree. Obviously, the direction 
of the contour line at the discreet point is given by: 
dr/dy = z'/y' (16a) 
The curvature at this point is [58]: 
(xty"-y'a")/(at 24412) %2 (16b) 
In the second step, Koch and Marckwardt use polynomials (15) of odd 
degree to fit a curve between each two discreet points. Third degree polynomials 
produce a curve which has at its end points the computed values of z' and y'. 
They make the contour line smooth. Fifth degree polynomials result in a curve 
which has also the computed values of xz" and y" at the end points. They produce 
a contour line which has no discontinuous changes in curvature.