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Spline Interpolation Method

Major: All Engineering Majors

Authors: Autar Kaw, Jai Paul

http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM

Undergraduates

Spline Method of Interpolation

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What is Interpolation ?

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

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Interpolants

Polynomials are the most common choice of interpolants because they are easy to:

EvaluateDifferentiate, and Integrate.

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Why Splines ?

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Why Splines ?

Figure : Higher order polynomial interpolation is a bad idea

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Linear Interpolation

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Linear Interpolation (contd)

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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

(s) (m/s)0 010 227.0415 362.7820 517.35

22.5 602.9730 901.67

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Linear Interpolation

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Quadratic Interpolation

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Quadratic Interpolation (contd)

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Quadratic Splines (contd)

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Quadratic Splines (contd)

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Quadratic Splines (contd)

Quadratic Spline InterpolationPart 1 of 2

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Quadratic Spline ExampleThe upward velocity of a rocket is given as a function of time. Using quadratic splinesa) Find the velocity at t=16 secondsb) Find the acceleration at t=16 secondsc) Find the distance covered between t=11 and t=16 seconds

t v(t)s m/s0 010 227.0415 362.7820 517.35

22.5 602.9730 901.67

Data and Plott v(t)s m/s0 010 227.0415 362.7820 517.35

22.5 602.9730 901.67

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Solution

Let us set up the equations

Each Spline Goes Through Two Consecutive Data Points

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t v(t)

s m/s

0 0

10 227.04

15 362.78

20 517.35

22.5 602.97

30 901.67

Each Spline Goes Through Two Consecutive Data Points

Derivatives are Continuous at Interior Data Points

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Derivatives are continuous at Interior Data Points

At t=10

At t=15

At t=20

At t=22.5

Last Equation

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Final Set of Equations

Coefficients of Spline

i ai bi ci

1 0 22.704 0

2 0.8888 4.928 88.88

3 -0.1356 35.66 -141.61

4 1.6048 -33.956 554.55

5 0.20889 28.86 -152.13

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END

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Quadratic Spline InterpolationPart 2 of 2

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Final Solution

Velocity at a Particular Pointa) Velocity at t=16

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Acceleration from Velocity Profileb) Acceleration at t=16

Acceleration from Velocity Profile

The quadratic spline valid at t=16 is given by

,

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Distance from Velocity Profilec) Find the distance covered by the rocket from

t=11s to t=16s.

Distance from Velocity Profile

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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

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Find a Smooth Shortest Path for a Robot

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Points for Robot Path

x y

2.00 7.24.5 7.15.25 6.07.81 5.09.20 3.510.60 5.0

Find the shortest but smooth path through consecutive data points

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Polynomial Interpolant Path

Spline Interpolant Path

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Compare Spline & Polynomial Interpolant Path

Length of pathPolynomial Interpolant=14.9

Spline Interpolant =12.9

THE END