Spline Interpolation
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Spline interpolation is a technique for generating new data points within the range of a discrete set of known data points. It uses piecewise polynomials, typically cubic polynomials, to fit curves to these data points. The document discusses linear and quadratic spline interpolation and provides an example of using quadratic splines to interpolate the velocity of a rocket at different times and calculate the velocity, distance, and acceleration at t=16 seconds.
Spline Interpolation
- 1. Interpolation Topic: Spline Interpolation Method Major: General
- 2. 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.
- 3. Interpolants Polynomials are the most common choice of interpolants because they are easy to: Evaluate Differentiate, and Integrate .
- 5. Why Splines ? Figure : Higher order polynomial interpolation is a bad idea
- 8. Example The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using linear splines. Table : Velocity as a function of time Figure : Velocity vs. time data for the rocket example 901.67 30 602.97 22.5 517.35 20 362.78 15 227.04 10 0 0 m/s s v(t) t
- 15. Example The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using quadratic splines. Table : Velocity as a function of time Figure : Velocity vs. time data for the rocket example 901.67 30 602.97 22.5 517.35 20 362.78 15 227.04 10 0 0 m/s s v(t) t
- 16. Solution
- 17. Solution (contd)
- 18. Solution (contd)
- 19. Solution (contd)
- 20. Solution (contd)
- 21. Solution (contd)
- 22. Distance from Velocity Profile Find the distance covered by the rocket from t=11s to t=16s
- 23. Acceleration from Velocity Profile , Find the acceleration of the rocket at t=16s. The quadratic spline valid at t=16 is given by