Interpolation on a single interval

Unit breach (0, 1)

On the assemblage breach , accustomed a starting point p0 at and an catastrophe point p1 at with starting departure m0 at and catastrophe departure m1 at , the polynomial can be authentic by

The four Hermite base functions. The interpolant in anniversary subinterval is a beeline aggregate of these four functions.

where t ∈ 0, 1.

editInterpolation on (xk, xk+1)

Interpolating in the breach can now be done with the formula

with and refers to the base functions, authentic below. Note that the departure ethics accept been scaled by compared to the blueprint on the assemblage interval.

editUniqueness

The formulae defined aloft are affirmed to aftermath a altered aisle amid the two points.

Proof:

Let be addition third amount polynomial acceptable the accustomed abuttals conditions. Define . Since both and are third amount polynomials, is at a lot of a third amount polynomial. Furthermore:

(We accept both and amuse the abuttals conditions)

So accept to be of the form:

We apperceive along that:

Putting and together, we deduce that and , appropriately

editRepresentations

We can address the departure polynomial as

where are Hermite base functions. These can be accounting in altered ways, anniversary way absolute altered properties.

expanded factorized Bernstein

The "expanded" cavalcade shows the representation acclimated in the analogue above. The "factorized" cavalcade shows immediately, that and are aught at the boundaries. You can added achieve that and accept a aught of complication 2 at 0 and and accept such a aught at 1, appropriately they accept abruptness 0 at those boundaries. The "Bernstein" cavalcade shows the atomization of the Hermite base functions into Bernstein polynomials of adjustment 3:

Using this affiliation you can accurate cubic Hermite departure in agreement of cubic Bézier curves with account to the four ethics and do Hermite departure application the de Casteljau algorithm. It shows that in a cubic Bézier application the two ascendancy credibility in the average actuate the tangents of the departure ambit at the corresponding alien points.