Re: Quartic interpolation


Ignoramus11682 wrote:
This is about finding the best interpolation of a function given two
values and two slopes.

My problem (which is not a homework assignment, a real life need) is
as follows:

GIVEN: We are given two points: (x0, y0), and (x1, y1). We are also
given tangents at points x0 and x1: t0 and t1.

FIND OUT: Equation of a quartic function (4th degree polynomial) that
has value y0 in point x0, y1 in point x1, and derivatives t0 and t1 in
points x0 and x1 respectively, such that its graph has a shortest
length between points x0 and x1.

You can find a cubic satisfying these conditions. It
is called a "clamped spline". But there's only one cubic
solution. If you really want a family, then you're right that
you need at least degree 4.

Side note, we think that such a minimal length graph would be a
"nicest looking" interpolation without unnecessary extremums etc.

I think that I can find the family of possible coefficients (with one
parameter, since we find 5 coefficients given four values we have one
"degree of freedom") with a bit of linear algebra. (I even have it
written down)

Yes, that's a standard approach.

The length measure could be calculated also as a 7th degree polynomial
of that parameter. It would then need to be minimized numerically.

My question is this. Is the above correct and am I on the right
track.

"Minimum length spline" is not a common requirement
as far as I know, but it's certainly an approach you can
use. This sounds like an interesting variation on
interpolation. Might be a short paper in it, perhaps
to the ACM journal or the SIAM journal on numerical
methods.

- Randy

.

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