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Spline Interpolation Method

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

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Spline Interpolation Method

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

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

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

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

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

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

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

  9. 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. http://numericalmethods.eng.usf.edu

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

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

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

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

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

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

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

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

  18. 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. http://numericalmethods.eng.usf.edu

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

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

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

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

  23. Solution (contd) Solving the above 15 equations gives the 15 unknowns as http://numericalmethods.eng.usf.edu

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

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

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

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

  28. 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 http://numericalmethods.eng.usf.edu

  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_method.html

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

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