1. http://numericalmethods.eng.usf.edu 1 Spline Interpolation Method Computer Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates

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

3. 3 What is Interpolation ? Given (x 0,y 0 ), (x 1,y 1 ), …… (x n,y n ), find the value of ‘y’ at a value of ‘x’ that is not given.

4. http://numericalmethods.eng.usf.edu4 Interpolants Polynomials are the most common choice of interpolants because they are easy to: Evaluate Differentiate, and Integrate.

5. http://numericalmethods.eng.usf.edu5 Why Splines ?

6. http://numericalmethods.eng.usf.edu6 Why Splines ? Figure : Higher order polynomial interpolation is a bad idea

7. http://numericalmethods.eng.usf.edu7 Linear Interpolation

8. http://numericalmethods.eng.usf.edu8 Linear Interpolation (contd)

9. http://numericalmethods.eng.usf.edu9 Example A robot arm with a rapid laser scanner is doing a quick quality check on holes drilled in a rectangular plate. The hole centers in the plate that describe the path the arm needs to take are given below. If the laser is traversing from x = 2 to x = 4.25 in a linear path, Find: the value of y at x = 4 using linear splines, the path of the robot if it follows linear splines, the length of that path. Figure 2 Location of holes on the rectangular plate.

10. http://numericalmethods.eng.usf.edu10 Linear Interpolation

11. http://numericalmethods.eng.usf.edu11 Linear Interpolation (contd) Find the path of the robot if it follows linear splines.

12. http://numericalmethods.eng.usf.edu12 Linear Interpolation (contd) Find the length of the path traversed by the robot following linear splines.

13. http://numericalmethods.eng.usf.edu13 Quadratic Interpolation

14. http://numericalmethods.eng.usf.edu14 Quadratic Interpolation (contd)

15. http://numericalmethods.eng.usf.edu15 Quadratic Splines (contd)

16. http://numericalmethods.eng.usf.edu16 Quadratic Splines (contd)

17. http://numericalmethods.eng.usf.edu17 Quadratic Splines (contd)

18. http://numericalmethods.eng.usf.edu18 Example A robot arm with a rapid laser scanner is doing a quick quality check on holes drilled in a rectangular plate. The hole centers in the plate that describe the path the arm needs to take are given below. If the laser is traversing from x = 2 to x = 4.25 in a linear path, Find: the length of the path traversed by the robot using quadratic splines and compare the answer to the linear spline and a fifth order polynomial result. Figure 2 Location of holes on the rectangular plate.

19. http://numericalmethods.eng.usf.edu19 Solution

20. http://numericalmethods.eng.usf.edu20 Solution (contd)

21. http://numericalmethods.eng.usf.edu21 Solution (contd)

22. http://numericalmethods.eng.usf.edu22 Solution (contd)

23. http://numericalmethods.eng.usf.edu23 Solution (contd) iaiai aiai aiai 10 −0.0444447.2889 2−1.0556 8.9278−11.777 30.68943 −9.394536.319 4−1.7651 28.945−113.40 53.2886 −64.042314.34 Solving the above 15 equations gives the 15 unknowns as

24. http://numericalmethods.eng.usf.edu24 Solution (contd)

25. http://numericalmethods.eng.usf.edu25 Solution (contd)

26. http://numericalmethods.eng.usf.edu26 Solution (contd)

27. http://numericalmethods.eng.usf.edu27 Comparison Compare the answer from part (a) to linear spline result and fifth order polynomial result.

28. http://numericalmethods.eng.usf.edu28 Comparison The absolute relative approximate error obtained between the results from the linear and quadratic spline is The absolute relative approximate error obtained between the results from the fifth order polynomial and quadratic spline is

29. 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 http://numericalmethods.eng.usf.edu/topics/spline_met hod.html

30. THE END http://numericalmethods.eng.usf.edu