Re: Quadratic Splines

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Hi,

I find it strange that you can find a lot about Cubic Splines on the Web,
while (almost) nothing can be found about Quadratic Splines.

A cubic spline in 2-D can be defined by:

   z(t) = (1-t)^3.z_0 + 3.t.(1-t)^2.z_1 + 3.t^2.(1-t).z_3 + t^3.z_4

Quite analogously, I could write for a quadratic spline:

  z(t) = (1-t)^2.z_0 + 2.t.(1-t).z_1 + t^2.z_3

Why is it that I can find anything about the former, 
but not about the latter ? 

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Han de Bruijn wrote:

> Hi,
>
> I find it strange that you can find a lot about Cubic Splines on the Web,
> while (almost) nothing can be found about Quadratic Splines.
>
> A cubic spline in 2-D can be defined by:
>
>    z(t) = (1-t)^3.z_0 + 3.t.(1-t)^2.z_1 + 3.t^2.(1-t).z_3 + t^3.z_4
>
> Quite analogously, I could write for a quadratic spline:
>
>   z(t) = (1-t)^2.z_0 + 2.t.(1-t).z_1 + t^2.z_3
>
> Why is it that I can find anything about the former, but not about the
> latter ?

I don't know why you've written them the way you have.  Write the cubic as
a_0 + a_1 t + a_2 t^2 + a_3 t^3.  You have 4 parameters to work with.  If you
just want to match values at the endpoints, you need two parameters (linear
interpolation).  If you also want to match derivatives, you need two more
parameters (cubic splines).  Quadratic splines only give you three
parameters, which doesn't allow you to match derivatives at both ends.

Jon Miller

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Han de Bruijn wrote:
> 
> Hi,
> 
> I find it strange that you can find a lot about Cubic Splines on the Web,
> while (almost) nothing can be found about Quadratic Splines.
> 
> A cubic spline in 2-D can be defined by:
> 
>    z(t) = (1-t)^3.z_0 + 3.t.(1-t)^2.z_1 + 3.t^2.(1-t).z_3 + t^3.z_4
> 
> Quite analogously, I could write for a quadratic spline:
> 
>   z(t) = (1-t)^2.z_0 + 2.t.(1-t).z_1 + t^2.z_3
> 
> Why is it that I can find anything about the former,
> but not about the latter ?

Because cubics are more useful. They give a more natural
appearing curve. Quadratics have too few degrees of freedom
and the functional form constrains them to do things not
implied by the data. Most importantly, they can't have
an inflection point between two data points.

         - Randy

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Han de Bruijn wrote:
> 
> Hi,
> 
> I find it strange that you can find a lot about Cubic Splines on the Web,
> while (almost) nothing can be found about Quadratic Splines.
> 
> A cubic spline in 2-D can be defined by:
> 
>    z(t) = (1-t)^3.z_0 + 3.t.(1-t)^2.z_1 + 3.t^2.(1-t).z_3 + t^3.z_4
> 
> Quite analogously, I could write for a quadratic spline:
> 
>   z(t) = (1-t)^2.z_0 + 2.t.(1-t).z_1 + t^2.z_3
> 
> Why is it that I can find anything about the former,
> but not about the latter ?

I have always thought it was for the following reason.  You match up the
end-points, and also the 1st derivatives at the interior points.  See
how many equations, and how many unknowns, and you are left with one
more equation to come up with.  Where will that come from?

With cubic splines, you have need of two new equations.  Since there are
two ends of the data points, that is a natural place to look.

For this reason, it would make sense to me to use polynomials of odd
degree when doing splines.

-- 
Stephen Montgomery-Smith
stephen@math.missouri.edu
http://www.math.missouri.edu/~stephen

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