Lecture33

# Lecture33 - Chapter 16 Curve Fitting Splines Spline...

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Chapter 16 Curve Fitting: Splines

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Spline Interpolation There are cases where polynomial interpolation is bad Noisy data Sharp Corners (slope discontinuity) Humped or Flat data Overshoot Oscillations
Example : f ( x ) = sqrt ( abs ( x )) Interpolation at - 4, 3, 2, 1, 0, 1, 2, 3, 4

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Cubic interpolation 5 th -order 7 th -order Linear spline Spline Interpolation
Spline Interpolation Idea behind splines Use lower order polynomials to connect subsets of data points Make connections between adjacent splines smooth Lower order polynomials avoid oscillations and overshoots

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Spline Interpolation Use “piecewise” polynomials instead of a single polynomial Spline -- a thin, flexible metal or wooden lath Bent the lath around pegs at the required points Spline curves -- curves of minimum strain energy Piecewise Linear Interpolation Piecewise Quadratic Interpolation Piecewise Cubic Interpolation (cubic splines)
Drafting Spline Continuous function and derivatives Zero curvatures at end points

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Splines There are n -1 intervals and n data points s i ( x ) is a piecewise low-order polynomial
Spline fits of a set of 4 points

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Example: Ship Lines waterline

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Bow Stern
Original database Spline interpolation for submerged hull (below waterline)

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( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 n 1 n 1 n 1 n i i i i 2 2 2 2 1 1 1 1 x x b a x s x x b a x s x x b a x s x x b a x s - - - - - + = - + = - + = - + = b is the slope between points ( 29 ( 29 i 1 i i 1 i i 1 i i 1 i i i i x x f f
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Lecture33 - Chapter 16 Curve Fitting Splines Spline...

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