Interpolating Splines: Which is the fairest of them all

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Interpolating SplinesWhich is the fairest of
them all?

• Raph Levien, Google
• Carlo Séquin, UC Berkeley

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Approaches to 2D Curve Design

• Bézier curves
• Approximating splines
• Useful if noisy data
• ? Interpolating splines

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What is the best interpolating spline?

• How do you define best?
• Fairness (smoothness)
• Locality (ripples and wiggles)
• Robustness (does it always converge?)
• Stability (perturbation ? small change)
• NOT Compute cost !

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Minimal Energy Curve

• Idealized thin elastic strip that goes through
  the data points
• The curve that minimizes bending energy

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Problems with MEC

• Lack of roundness

• Lack of convergence

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Does MEC optimize the wrong functional?

• Is there a better functional?
• Tweaks to fix roundness
• Scale-Invariant MEC
• MVC
• Space of all possible functionals is a pretty
  big zoo
• How to choose one?

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Properties of MEC worth preserving

• These hold for any sane variational spline
• Extensionality
• Adding an on-curve point preserves shape.
• Direct consequence of variational definition.
• G2-continuity
• Known splines with higher continuity have worse
  locality.
• G2 is fair enough (for drawn curves).

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2-Parameter Splines

• Each curve segment determined by a2-dimensional
  parameter space,
• Modulo scaling, rotation, translation
• Two parameters are tangent angles

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MEC is cut piecewise from a fixed curve

• Known as the rectangular elastica

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A Major New Result

• All 2-parameter, extensible splines have
  segments cut from a generator curve!
• (With scaling, rotation, translation to fit)

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Generator ? Extensional Spline

• Conversely, start with a curve and use it to
  generate a spline.
• Relationship between ?/?2 and ?/?3 must be
  single-valued.
• Preserve G2-continuity across points.

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Euler Spiral

• Curvature is linear in arc-length
• Aka Cornu spiral, Fresnel integrals, Clothoid,
  Railroad transition curve . . .

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Euler Spiral Spline

• A really good curve Euler spiral
• Fixes roundness problem
• Far more robust
• Some solution always seems to exist.
• Mentioned by Birkhoff de Boor, 1965
• Implemented by Mehlum, 70s
• Why is it not more popular?

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Which generating curve looks best?

• Much simpler than Which functional is best?
• We can employ empirical testing.
• Aesthetic curve family is promising.

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Empirical Study Aesthetic Curves
MECminimum
popularvotes
Exponent of Aesthetic Curve
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4-Parameter Splines (MVC)

• G4-continuity
• Locality is poorer
• 2-parameter is sparser (experience from font
  design)

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Fairness/Locality Tradeoff
Exponential falloff factor
Exponent of Aesthetic Curve
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Applications for Font Design

• Euler spiral spline
• Sparse control points
• Interactive editing
• G2 straight-to-curve transitions
• Several fonts drawn
• Inconsolata

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Efficient Implementation

• Can just use 2-D LUT to compute curvature from
  tangent angles.
• Newton solver to enforce G2 globally.
• Drawing is not much more expensive than de
  Casteljau.
• Can convert to concise Bézier curves.

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Bézier Representation of Font

• Euler spiral master
• Optimized conversion to Béziers
• Error tolerance lt 10-3
• Compatible with industry standard font formats

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Conclusions

• Best 2D spline is cut piecewise from some
  generating curve.
• Euler spiral is a very good choice.
• Aesthetic curves may be slightly better.
• Efficient and practical implementations.