Cubic Hermite spline
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In the mathematical subfield of numerical analysis a cubic Hermite spline, named in honor of Charles Hermite (Hermite is pronounced air MIT), 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 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
- <math>\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</math>
where t ∈ [0, 1].
The four Hermite basis functions can be defined as
- <math>h_{00}(t) = 2t^3-3t^2+1 \,\!</math>
- <math>h_{10}(t) = t^3-2t^2+t \,\!</math>
- <math>h_{01}(t) = -2t^3+3t^2 \,\!</math>
- <math>h_{11}(t) = t^3-t^2 \,\!</math>
to give the polynomial as
- <math>\mathbf{p}(t) = h_{00}(t)\mathbf{p}_0 + h_{10}(t)\mathbf{m}_0 + h_{01}(t)\mathbf{p}_1 + h_{11}(t)\mathbf{m}_1.</math>
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: