It is named after French mathematician Charles hermit
Moreover, It is an interpolating piecewise cubic polynomial with the specified tangent at each control points.
It adjusted locally because each curve section depends on its endpoints only.
Parametric cubic point function for any curve section then given by:
(0) = 𝑝𝑘
(1) = 𝑝𝑘+1
𝑝′(0) = 𝑑𝑝𝑘
𝑝′′(1) = 𝑑𝑝𝑘+1
Where dpk & dpk+1 are values of parametric derivatives at point pk & pk+1 respectively. // Hermit Interpolation //
Vector equation of cubic spline is:
(𝑢) = 𝑎𝑢3 + 𝑏𝑢2 + 𝑐𝑢 + 𝑑
Where x component of p is
𝑥(𝑢) = 𝑎𝑥𝑢3 + 𝑏𝑥𝑢2 + 𝑐𝑥𝑢 + 𝑑𝑥 and similarly y & z components
Matrix form of above equation is
Now substitute endpoint value of u as 0 & 1 in above equation & combine all four parametric equations in matrix form: // Hermit Interpolation //
Where 𝐻(u) for k=0, 1, 2, 3 referred to as blending functions because of that blend the boundary constraint values for curve section.
The shape of the four hermit blending function given below.
Fig. the hermit blending functions.
Moreover, Hermit Interpolation curves are used in digitizing application where we input the approximate curve slope means DPk & DPk+1.
But in an application where this input difficult to approximate at that place, we cannot use hermit curve.