# Cubic Hermite spline

In the mathematical subfield of numerical analysis a cubic Hermite spline, named in honor of Charles Hermite (pronounced air MEET), is a third-degree spline with each polynomial of the spline in Hermite form. The Hermite form consists of two control points and two control tangents on each for each polynomial.

On each subinterval, given a starting point p0 and an ending point p1 with starting tangent m0 and ending tangent m1, the polynomial can be defined by

[itex]\mathbf{p}(t) = (2t^3-3t^2+1)\mathbf{p}_0 + (t^3-2t^2+t)\mathbf{m}_0 + (-2t^3+3t^2)\mathbf{p}_1 +(t^3-t^2)\mathbf{m}_1[itex]

where t ∈ [0, 1].

The four Hermite basis functions can be defined as

[itex]H_0(t) = 2t^3-3t^2+1 \,\![itex]
[itex]H_1(t) = t^3-2t^2+t \,\![itex]
[itex]H_2(t) = -2t^3+3t^2 \,\![itex]
[itex]H_3(t) = t^3-t^2 \,\![itex]

to give the polynomial as

[itex]\mathbf{p}(t) = H_0(t)\mathbf{p}_0 + H_1(t)\mathbf{m}_0 + H_2(t)\mathbf{p}_1 + H_3(t)\mathbf{m}_1.[itex]

Since each subinterval must share tangents with neighboring subintervals, many techniques exist to determine values for shared tangents.

Some of the techniques for creating cubic Hermite splines include:

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