The preceding section explains how
spline can be employed to
interpolate a function y of a scalar variable t, in the
case when y is a scalar. In this section we explain how to
perform more sophisticated interpolations. This includes
multidimensional interpolations, and interpolations that are splinings
of curves, rather than of functions.
spline can handle the case when y is a vector of
arbitrary specified dimensionality. The dimension can be specified with
the ‘-d’ option. For example, an input file could contain the
0.0 0.0 1.0 1.0 1.0 0.0 2.0 0.0 1.0
which are the coordinates (t,y) of the data points (0,0,1), (1,1,0), and (2,0,1). You would construct a spline (the graph of an interpolating function) passing through the points in this dataset by doing
spline -d 2 input_file > output_file
The option ‘-d 2’ is used because in this example, the dependent variable y is a two-dimensional vector. Each of the components of y will be interpolated independently, and the output file will contain points that lie along the graph of the resulting interpolating function.
When doing multidimensional splining, you may use any of the options that apply in the default one-dimensional case. For example, the ‘-f’ option will yield real-time cubic Bessel interpolation. As in the one-dimensional case, if the ‘-f’ option is used then the ‘-t’ option must be used as well, to specify an interpolation interval (a range of t values). The ‘-p’ option will yield a periodic spline, i.e., the graph of a periodic vector-valued function. For this, the first and last dataset y values must be the same.
spline can also be used to draw a curve through arbitrarily
chosen points in the plane, or in general through arbitrarily chosen
points in d-dimensional space. This is not the same as splining,
at least as the term is conventionally defined. The reason is that
`splining' refers to construction of a function, rather than the
construction of a curve that may or may not be the graph of a function.
Not every curve is the graph of a function.
The following example shows how you may `spline a curve'. The command
echo 0 0 1 0 1 1 0 1 | spline -d 2 -a -s | graph -T X
will construct a curve in the plane through the four points (0,0), (1,0), (1,1), and (0,1), and graph it on an X Window System display. The ‘-d 2’ option specifies that the dependent variable y is two-dimensional. The ‘-a’ option specifies that t values are missing from the input, and should be automatically generated. By default, the first t value is 0, the second is 1, etc. The ‘-s’ option specifies that the t values should be stripped from the output.
The same technique may be used to spline a closed curve. For example, doing
echo 0 0 1 0 0 1 0 0 | spline -d 2 -a -s -p | graph -T X
will construct and graph a closed, lozenge-shaped curve through the three points (0,0), (1,0), and (0,1). The construction of a closed curve is guaranteed by the ‘-p’ (i.e., ‘--periodic’) option, and by the repetition of the initial point (0,0) at the end of the sequence.
When splining a curve, whether open or closed, you may wish to substitute the ‘-A’ option for the ‘-a’ option. Like the ‘-a’ option, the ‘-A’ option specifies that t values are missing from the input and should be automatically generated. However, the increment from one t value to the next will be the distance between the corresponding values of y. This scheme for generating t values, when constructing a curve through a sequence of data points, is the scheme that is used in the well known FITPACK subroutine library. It is probably the best approach when the distances between successive points fluctuate considerably.
A curve through a sequence of points in the plane, whether open or closed, may cross itself. Some interesting visual effects can be obtained by adding negative tension to such a curve. For example, doing
echo 0 0 1 0 1 1 0 0 | spline -d 2 -a -s -p -T -14 -n 500 | graph -T X
will construct a closed curve through the three points (0,0), (1,0), and (0,1), which is wound into curlicues. The ‘-n 500’ option is included because there are so many windings. It specifies that 501 points should be generated, which is enough to draw a smooth curve.