Hermite Cubic Spline:
The major idea of the Hermite cubic spline is that a curve is spilt into segments. Each of the segments is approximated by an expression, say a parametric cubic function. The purpose for utilizing cubic functions to approximate the segments are :
- A cubic polynomial is the minimum-order polynomial function that produces C0, C1 & C2 continuity curves.
- A cubic polynomial is the lowest-degree polynomial that allows inflection in a curve segment and permits representation of non-planar (twisted) space curves.
- Higher-order polynomials contain some drawbacks; such like oscillation around control points, and are uneconomical in terms of holding information and computation.
The general form of a cubic function might be written as following
r = V (t ) = a0 + a1 t + a2 t 2 + a3 t 3
Here the point vector r of the cubic curve is represented by the parametric equation V (t). The segment represented by the equation contains highest-degree polynomial t3. Traditionally the parameter t is bounded through the parameter interval 0 < t < 1.
The general form shows the whole family of cubic curves. We might define any specific curve by specify the four coefficient vectors (i.e., a0, a1, a2, a3). But, the coefficients do not all contain direct physical significance and are not suitable "handles" for adjusting the segment shape or incorporating it into a composite curve.