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
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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
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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)
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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
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Spline fits of a set of 4 points
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Example: Ship Lines waterline
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Bow Stern
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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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