Hermite Cubic Spline:
The major idea of the Hermite cubic spline is that a curve is divided into segments. Each segment is approximated by an expression, say a parametric cubic function. The reasons for using cubic functions to approximate the segments are following:
- Any cubic polynomial is minimum-order polynomial function that generated C0, C1 and C2 continuity curves.
- A cubic polynomial is the lowest-degree polynomial which allows inflection in a curve segment and permits representation of non-planar (twisted) space curves.
- Higher-order polynomials have some disadvantages, such as oscillation regarding control points, and are uneconomical in terms of storing information and computation.
The general form of a cubic function may be written as following
r = V (t ) = a0 + a1 t + a2 t 2 + a3 t 3
where the point vector r of the cubic curve is described by the parametric equation V (t). The segment described by the equation has highest-degree polynomial t3. Traditionally the parameter t is bounded by the parameter interval 0 ≤t ≤ 1.
The general form represents the entire family of cubic curves. We may define any particular curve by specifying the four coefficient vectors (that is, a0, a1, a2, a3). Though, the coefficients do not all contain direct physical significance and are not convenient "handles" for adjusting the segment shape or incorporating it into a composite curve.