Are Hermite Polynomials Always Cubic When Used for Interpolation?

[ice109]

are hermite interpolationg polynomials necessarily cubic even when used to interpolate between two points?

this page would have me believe so in calling it a "clamped cubic" :

http://math.fullerton.edu/mathews/n2003/HermitePolyMod.html

 

[Astronuc]

It's more the case that there exists a cubic polynomial of the form:

a x³ + b x² + c x + d, which satisfies the constraints at two points, (x₀, y₀) and (x₁, y₁), where

p(x₀) = f(x₀) = y₀

p(x₁) = f(x₁) = y₁

and

p'(x₀) = f'(x₀) = y'₀

p'(x₁) = f'(x₁) = y'₁

4 equations, and 4 unknowns (a, b, c, d)

This is the basis of the cubic spline.

 

[ice109]

i think given that argument for some groups of points with slopes the minimum curve that goes through both is a cubic.

 

[ice109]

anyone?