Cubic Spline Interpolation Methods
- The Cubic Spline Interpolation Methods are mostly used for representing the path of moving object or existing object shape or drawing.
- Sometimes it also used to design the object shapes.
- Cubic spline gives reasonable computation on as compared to higher order spline. And more stable compared to lower order polynomial spline. So it is often used for modeling curve shape.
- Fig. A piecewise continuous Cubic Spline Interpolation Methods of n+1 control points. 𝑝𝑘 = (𝑥𝑘,𝑘,𝑧𝑘) Where, k=0, 1, 2, 3 …, n
- Cubic interpolation splines obtained by fitting the input points with the piecewise cubic polynomial curve that passes through every control point.
- Parametric cubic polynomial for this curve given by
(𝑢) = 𝑎𝑥𝑢3 + 𝑏𝑥𝑢2 + 𝑐𝑥𝑢 + 𝑑𝑥
Moreover, (𝑢) = 𝑎𝑦𝑢3 + 𝑏𝑦𝑢2 + 𝑐𝑦𝑢 + 𝑑𝑦
(𝑢) = 𝑎𝑧𝑢3 + 𝑏𝑧𝑢2 + 𝑐𝑧𝑢 + 𝑑𝑧 𝑤ℎ𝑒𝑟𝑒( 0 ≤ 𝑢 ≤ 1)
- For above equation, we need to determine for constant a, b, c, and d the polynomial representation for each of n curve section.
- This obtained by settling proper boundary condition at the joints.
- Now we will see the common method for settling this condition.
Natural Cubic Splines [ Cubic Spline Interpolation Methods ]
- A natural cubic splines a mathematical representation of the original drafting spline.
- We consider that curve is in 𝑐2 continuity means first and second parametric derivatives of adjacent curve section are same as a control point.
- For the ‘’n+1’’ control point, we haven curve section and 4n polynomial constants to find.
- For all interior control points, we have four boundary conditions. The two curve section on either side of control point must have same first & second order derivative at the control points and each curve passes through that control points.
- We get other two conditions as 𝑝0 (first control points) starting & 𝑝(last control point) is an endpoint of the curve.
- We still required two conditions for obtaining coefficient values.
- One approach is to set up the second derivative at 𝑝0 & 𝑝𝑛 to be 0. Another approach is to add one extra dummy point at each end. I.e. we add 𝑝−1 & 𝑝𝑛+1 then all original control points are interior and we get 4n boundary condition.
- Although it is a mathematical model it has a major disadvantage is the change in the control point entire curve changed.
- So it not allowed for local control and we cannot modify part of the curve.