1. Spline Interpolation Method
Industrial Engineering Majors
Authors: Autar Kaw, Jai Paul
Transforming Numerical Methods Education for STEM Undergraduates

2. Spline Method of Interpolation http://numericalmethods.eng.usf.edu

3. What is Interpolation ?
Given (x0,y0), (x1,y1), …… (xn,yn), find the value of ‘y’ at a value of ‘x’ that is not given.

4. Interpolants Evaluate Differentiate, and Integrate.
Polynomials are the most common choice of interpolants because they are easy to:
Evaluate
Differentiate, and
Integrate.

5. Why Splines ?

6. Figure : Higher order polynomial interpolation is a bad idea
Why Splines ?
Figure : Higher order polynomial interpolation is a bad idea

7. Linear Interpolation

8. Linear Interpolation (contd)

9. Example
A curve needs to be fit through the given points to fabricate the cam. If the cam follows a straight line profile between x = 1.28 to x = 0.66, what is the value of y at x=1.1 ? Find using linear spline interpolation.
Point
x (in.)
y (in.) 1
2.20
0.00 2
1.28
0.88 3
0.66
1.14 4
1.20 5
–0.60
1.04 6
–1.04
0.60 7
–1.20 1 2 3 5 6 7 X Y 4

10. Linear Interpolation

11. Linear Interpolation Find the cam profile using linear splines.

12. Linear Interpolation

13. Quadratic Interpolation

14. Quadratic Interpolation (contd)

15. Quadratic Splines (contd)

16. Quadratic Splines (contd)

17. Quadratic Splines (contd)

18. Example
A curve needs to be fit through the given points to fabricate the cam. If the cam follows a straight line profile between x = 1.28 to x = 0.66, what is the value of y at x=1.1 ? Find using a quadratic spline interpolation.
Point
x (in.)
y (in.) 1
2.20
0.00 2
1.28
0.88 3
0.66
1.14 4
1.20 5
–0.60
1.04 6
–1.04
0.60 7
–1.20 1 2 3 5 6 7 X Y 4

19. Solution

20. Solution (contd)

21. Solution (contd)

22. Solution (contd)

23. Solution (contd)

24. Solution (contd)

25. Comparison

26. Additional Resources
For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit

27. THE END