If an interpolating curve follows very closely to the data polygon, the
length of the curve segment between two adjacent data points would be very
close to the length of the *chord* of these two data points, and the
the length of the interpolating curve would also be very close to the total
length of
the data polygon. In the figure below, each curve segment of an interpolating
polynomial is very close to the length of its supporting chord, and the
length of the curve is close to the length of the data polygon. Therefore,
if the domain is subdivided according to the distribution of the chord
lengths, the parameters will be an approximation of the
**
arc-length parameterization**. This is
the merit of the *chord length* or *chordal* method.

Suppose the data points are **D**_{0}, **D**_{1}, ...,
**D**_{n}. The length between **D**_{i-1}
and **D**_{i} is
|**D**_{i}-**D**_{i-1}|, and the
length of the data polygon is the sum of the lengths of these chords:

Therefore, the ratio of the chord length from data point
**D**_{0} to data point **D**_{k}, denoted as
*L*_{k}, over the length of the data polygon is

If we prefer to have an arc-length parameterization of the
interpolating curve, the domain has to be divided according to the ratio
*L*_{k}. More precisely, if the domain is [0,1],
then parameter *t*_{k} should be located at the value of
*L*_{k}:

where ** L** is the length of the data polygon. In this way, the
parameters divide the domain into the ratio of the chord lengths.

Let us look at an example. Suppose we have four data points (*n* = 3):
**D**_{0} = < 0,0 >,
**D**_{1} = < 1,2 >,
**D**_{2} = < 3,4 > and
**D**_{3} = < 4,0 >.
The length of each chord is

and the total length is

Finally, we have the corresponding parameters:

The following figures show the data points and the parameter distributions of the uniformly spaced method and the chord length method.

What if the domain is [*a*,*b*] rather than [0,1]? Note that
*L*_{k} is a ratio between 0 and 1.
Since the length of [*a*, *b*] is *b*-*a*,
*L*_{k}(*b*-*a*),
0 <= *k* <= *n*, divide
[0,*b*-*a*] into the same ratio as we did for [0,1]. Therefore,
the following parameters divide [*a*,*b*] according to
the chord lengths:

The chord length method is widely used and usually performs well. Since
it is known (proved by R. Farouki and also well-known in geometry) that polynomial curves cannot be
parameterized to have unit speed (*i.e.*, arc-length parameterization),
the chord length can only be an approximation. Sometimes, a longer
chord may cause its curve segment to have a bulge bigger than necessary.
In the figure below, the black and blue curves both interpolate 7 data points.
As you can see, both curves have very similar shape, except for the last
segment, and the one computed with chord length method wiggling a little.
The last curve segments are very different and the curve using the chord
length method has a large bulge and twists away from the red curve produced
by the uniformly spaced method.
This is a commonly seen problem with the chord length method.