Curve fitting and fairing using conic splines

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Curve fitting and fairing using conic splines

• Yang Xunnian
• Zhejiang University

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Overview

• Introduction
• Related works
• G2 interpolation by conics
• Curve fitting and fairing
• Examples

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Introduction

• Definition
• Intersection with a cone
• Representation
• - Parametric curve
• - Algebraic curve

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P1 G2 conic spline

• Local, single conic spline, explicit solution

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Related works

• Globle G2 conic spline
• (Farin89)
• Local biconic interpolation
• (Pottman91)
• Nonlinear G2 interpolation
• (Schaback93)

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Fairness of a conic (parabola)

• Sapidis, N and Frey, WH. 1992

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Fairness of a conic (general)

• Frey,Field00
• Ahn,Kim98

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P2 Fairness of a conic spline

• With G2 continuity
• Control the curvature plot

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G1 conic spline

• For sparse data
• Quadric Bézier spline

• For dense noisy data
• Fitting by arc spline (Yang01)
• Data reduction for arcs (Yang02)

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Construction of G2 conic spline

• P2i-2, P2i, P2i2 are fixed
• Tangents at P2i-2, P2i2 are not changed
• Curvatures at P2i-2, P2i2 are not changed

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Equation and solution
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Comparison

• Advantage of new method
• Local method
• With minimum conics
• With explicit solution

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Curve fairing joint move

• k(P2i) is an unwanted local extremum
• Rule Ci and Ci1 are fair

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Curve fairing weight scaling

• Unwanted curvature extremum in the conic
• Rule the current conic is fair

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The algorithm

• Move a joint point
• Scaling the weights
• G2 conic construction
• Repeat and convergence

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Examples interpolation

• Dashed G1 conic spline
• Solid G2 conic spline

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Curvature plots interpolation

• Left for G1 spline (dashed), global G2 spline
  (solid)
• Right for local G2 spline

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Examples approximation

• Left noisy data and fitting curves
• Right conic spline with control polygon

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Curvature plots

• Left curvature plots of arc spline
• Right curvature plots of conic spline

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Mouse section

• Conic spline with control polygon
• A local feature

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Curvature plots

• Left curvature plots of arc spline
• Right curvature plots of conic spline

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The end

• Thank you!