The COMMUNITY MATHEMATICS CENTRE, COMAC
(2017)
COnnecting
the dots...
The art and science of
interpolation and extrapolation.
At Right Angles, 6 (2).
pp. 1216.
Abstract
he following is an extremely common scenario in
the sciences: two variables y and x are connected by a
functional relation, y = f(x), but f is unknown and our
task is to find it. The only actions available to us are to perform
experiments and find the values of y corresponding to selected
values of x. After doing these experiments, we obtain the
following n pairs of values of x and y:
M α C
er and compass is part of the standard geometry syllabus at the
is a standard procedure for doing We
the may
job,
is so
plot and
these it
points
on a simple
sheet of graph paper and get
something
which
like this: if not
t to think of an alternative to it that
is just
as looks
simple,
procedure, announced in a Twitter post [1].
A
F
Figure 1
D
Angle bisector
Armed with only these data points, can we determine the
unknown function f? Another way of expressing this question
is the following: Can we fit a definite, unique curve to these data
points? Note that the curve must pass through all the points.
(So this is not a problem of finding the “line of best fit” or the
“curve of best fit”).
I
A moment's reflection will tell us that the answer is No. The
question is too broad to admit an answer when stated this way.
Even if we were only interested in polynomial functions f (this
is very often the case), it is still not possible for the data to yield
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