Approximation with linear, biarc, conic and bihelix splinesgert/documents/2010/geometric... · 2010. 7. 14. · Approximation by linear splines Approximation by conic splines Approximation

Approximation by linear splinesApproximation by conic splines

Approximation by biarc and bihelix splines

Approximation with linear, biarc, conic andbihelix splines

Gert Vegter (joint work with Sunayana Ghosh)

Johann Bernoulli Institute for Mathematics and Computer ScienceUniversity of Groningen

The Netherlands

INRIA, Sophia Antipolis, February 4, 2010

Gert Vegter Approximation with linear, biarc, conic and bihelix splines

Splines in 2D

Linear spline Conic spline

0.0 0.5 1.0 1.5 2.0

-1.0

-0.5

0.5

1.0

Biarc spline

Approximation of smooth curves

Given: smooth (C2 or better) curve C in the plane, ε > 0

Goal: approximate C with spline(i) within Hausdorff-distance ε(ii) of low complexity

Smoothness Approximation Complexity(nr. of patches)

C2 linear spline O(ε−1/2)C3 biarc spline O(ε−1/3)C4 parabolic spline O(ε−1/4)C5 conic spline O(ε−1/5)C3 bihelix spline (3D) O(ε−1/3)

Orders of magnitude

Example: ε = 10−6

Number of patches:

linear spline ε−1/2 1000biarc spline ε−1/3 100parabolic spline ε−1/4 32conic spline ε−1/5 16bihelix spline (3D) ε−1/3 100

Constants: this talk

General strategy

-1.0 -0.5 0.5 1.0 1.5 2.0

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

Smooth curve α

General strategy

Spline: offset curve β(s) = α(s) + f (s) N(s), s ∈ I.Distance function f gives HD: δH(α, β) =|| f ||∞= maxs∈I |f (s)|Intersections: α(si) = β(si), s0 ≤ s1 ≤ . . . ≤ sn, so

f (s) = (s − s0) · · · (s − sn) [s0, . . . , sn, s]f︸︷︷︸divided difference

With σ = sn − s0 (and Hermite-Genocchi):

f (s) = 1(n+1)! f

(n+1)(s0) (s − s0) · · · (s − sn) + O(σn+2)

Goal: express f (n+1)(s0) in geometric invariants of α:(affine) curvature and its derivatives

General strategy

f (s) = 1(n+1)! f

(n+1)(s0) (s − s0) · · · (s − sn) + O(σn+2)

General strategy

f (s) = 1(n+1)! f

(n+1)(s0) (s − s0) · · · (s − sn) + O(σn+2)

General strategy

f (s) = 1(n+1)! f

(n+1)(s0) (s − s0) · · · (s − sn) + O(σn+2)

General strategy

f (s) = 1(n+1)! f

(n+1)(s0) (s − s0) · · · (s − sn) + O(σn+2)

1 Approximation by linear splines

2 Approximation by conic splines

3 Approximation by biarc and bihelix splines

Approximation by linear splines

Given: ε > 0.Question: Optimal size of linear spline approximating smooth curve towithin Hausdorff distance ε?

Optimal linear spline

Féjes Toth (1948) For convex C2-curve in R2 the minimal number Nof vertices of an inscribed, ε-approximating polygon is

N = 12√

2

(

∫ L

s=0κ(s)

12 ds

)

ε−1/2 + O(1).

Hausdorff distance

L

α

α(0)

α(σ)

Hausdorff distance

L

α

N(s)

f(s)

α(s)α(0)

α(σ)

T(s)

Hausdorff distance

L

α

N(s)

f(s)

α(s)α(0)

α(σ)

T(s)

Consider L as offset curve:

β(s) = α(s) + f (s) N(s)

Distance function f , with f (0) = f (σ) = 0

Hausdorff distance

L

α

N(s)

f(s)

α(s)α(0)

α(σ)

T(s)

Consider L as offset curve:

β(s) = α(s) + f (s) N(s)

Distance function f , with f (0) = f (σ) = 0

Hausdorff distance: δH(α, L) = max0≤s≤σ f (s) =: | | f | |∞

Frenet-Serret frame and curvature

α : [0, L] → R2: arc-length parametrization

α(s)

T(s) = α′(s)

N(s)

Frenet-Serret :

{T ′(s) = κ(s)N(s),

N ′(s) = −κ(s)T (s).

Approximation lemma

α : [0, σ] → R2: C2-curveChord L is offset curve

β(s) = α(s) + f (s) N(s)

with f (0) = f (σ) = 0.Distance function:

f (s) = s (s − σ) [0, σ, s] f = 12 s (s − σ) f′′(0) + O(σ3)

Hence:

|| f ||∞ = 12 max0≤s≤σ |s (s − σ) f′′(0)| + O(σ3)

= 18 |f′′(0)| σ2 + O(σ3)

Can show: f ′′(0) = κβ(0) − κα(0)Offset curve is line segment, so κβ(0) = 0

Hence: δH(α, β) = 18 |κα(0)| σ2 + O(σ3)

Approximation lemma

β(s) = α(s) + f (s) N(s)

f (s) = s (s − σ) [0, σ, s] f = 12 s (s − σ) f′′(0) + O(σ3)

Hence:

|| f ||∞ = 12 max0≤s≤σ |s (s − σ) f′′(0)| + O(σ3)

= 18 |f′′(0)| σ2 + O(σ3)

Hence: δH(α, β) = 18 |κα(0)| σ2 + O(σ3)

Approximation lemma

β(s) = α(s) + f (s) N(s)

f (s) = s (s − σ) [0, σ, s] f = 12 s (s − σ) f′′(0) + O(σ3)

Hence:

|| f ||∞ = 12 max0≤s≤σ |s (s − σ) f′′(0)| + O(σ3)

= 18 |f′′(0)| σ2 + O(σ3)

Hence: δH(α, β) = 18 |κα(0)| σ2 + O(σ3)

Approximation lemma

β(s) = α(s) + f (s) N(s)

f (s) = s (s − σ) [0, σ, s] f = 12 s (s − σ) f′′(0) + O(σ3)

Hence:

|| f ||∞ = 12 max0≤s≤σ |s (s − σ) f′′(0)| + O(σ3)

= 18 |f′′(0)| σ2 + O(σ3)

Hence: δH(α, β) = 18 |κα(0)| σ2 + O(σ3)

Approximation lemma

β(s) = α(s) + f (s) N(s)

f (s) = s (s − σ) [0, σ, s] f = 12 s (s − σ) f′′(0) + O(σ3)

Hence:

|| f ||∞ = 12 max0≤s≤σ |s (s − σ) f′′(0)| + O(σ3)

= 18 |f′′(0)| σ2 + O(σ3)

Hence: δH(α, β) = 18 |κα(0)| σ2 + O(σ3)

Approximation lemma

β(s) = α(s) + f (s) N(s)

f (s) = s (s − σ) [0, σ, s] f = 12 s (s − σ) f′′(0) + O(σ3)

Hence:

|| f ||∞ = 12 max0≤s≤σ |s (s − σ) f′′(0)| + O(σ3)

= 18 |f′′(0)| σ2 + O(σ3)

Hence: δH(α, β) = 18 |κα(0)| σ2 + O(σ3)

Approximation lemma

β(s) = α(s) + f (s) N(s)

f (s) = s (s − σ) [0, σ, s] f = 12 s (s − σ) f′′(0) + O(σ3)

Hence:

|| f ||∞ = 12 max0≤s≤σ |s (s − σ) f′′(0)| + O(σ3)

= 18 |f′′(0)| σ2 + O(σ3)

Hence: δH(α, β) = 18 |κα(0)| σ2 + O(σ3)

Complexity of linear spline

Hausdorff distance: ε

ε

l(s)α(s)

l(s) =

√

8εκ(s)

+ O(ε)

Féjes Toth (1948)

Nmin =∫L

s=0

dsl(s)

= 12√

2

(

∫L

s=0κ(s)

12 ds

)

ε−1/2 + O(1).

Higher dimensions

R. Schneider (1981)

For convex C3-hypersurface in Rd the minimal number of vertices ofan inscribed, ε-approximating polytope is

Nmin(ε) ≈ Cdε(1−d)/2∫

F

√K dF ,

with

Cd : constant depending only on dimension

K : Gaussian curvature of F (product of d − 1 main curvatures)

Introduction

Given a sufficiently smooth, plane curve α, with non vanishingcurvature and ε > 0.Goal

Complexity: Find the number of elements of a parbolic or conicspline approximating α.Algorithm: Approximate α with parabolic or conic spline.

Complexity

Theorem

Total number of elements of optimal :

parabolic spline: c1ε−1/4 + O(1)

c1 = 0.297∫L

0|k(s)|1/4κ(s)5/12ds

conic spline: c2ε−1/5 + O(1)

c2 = 0.186∫ L

0|k ′(s)|1/5κ(s)2/5ds

κ(s): (Euclidean) curvature

k(s): Affine curvature (constant for conics!)

Affine arc-length & curvature

γ : I → R2, smooth regular curve, r is its affine arc-lengthparameter if

[γ̇(r), γ̈(r)] = 1.

[γ̇(r),...γ(r)] = 0, so

...γ(r) + k(r)γ̇(r) = 0.

k(r): affine curvature at γ(r)

Affine Frenet-Serret frame:

γ̇ = t , ṫ = n, ṅ = −k t .

Curve uniquely determined by affine curvature up to equi-affinetransformations

[γ̇(r), γ̈(r)] = 1.

[γ̇(r),...γ(r)] = 0, so

...γ(r) + k(r)γ̇(r) = 0.

γ̇ = t , ṫ = n, ṅ = −k t .

[γ̇(r), γ̈(r)] = 1.

[γ̇(r),...γ(r)] = 0, so

...γ(r) + k(r)γ̇(r) = 0.

γ̇ = t , ṫ = n, ṅ = −k t .

[γ̇(r), γ̈(r)] = 1.

[γ̇(r),...γ(r)] = 0, so

...γ(r) + k(r)γ̇(r) = 0.

γ̇ = t , ṫ = n, ṅ = −k t .

[γ̇(r), γ̈(r)] = 1.

[γ̇(r),...γ(r)] = 0, so

...γ(r) + k(r)γ̇(r) = 0.

γ̇ = t , ṫ = n, ṅ = −k t .

Constant Affine Curvature

Conics are curves with constant affine curvature.

k > 0 ellipsek = 0 parabolak < 0 hyperbola

Osculating Conics

Osculating Conic: unique conic with contact order 4 (5-foldintersection, at point of non-zero Euclidean curvature on a planecurve)

Sextactic Point: point where osculating conic has contact oforder at least 5.

Affine curvature at a point = affine curvature osculating conic.

Affine Spirals (1)

Affine Spiral: curve with monotone affine curvature.

Any conic intersects an affine spiral in at most 5 points (countedwith multiplicity).

Affine Spirals (2)

Osculating conics of an affine spiral are disjoint, and do notintersect the spiral arc except at their point of contact.

Hausdorff Distance to bitangent parabolic arc

Theorem (part 1)

Given: α, a smooth spiral of length σHausdorff distance to a bitangent parabolic arc β:

δH(α, β) =1

128 |k0|κ5/30 σ

4 + O(σ5)

κ0: Euclidean curvature of α (e.g., at endpoint)

k0: affine curvature of α

Proof: sketch

Strategy: consider bitangent parabolic arc as offset curveβ : [0, ρ] → R2, and use kβ ≡ 0.

α : [0, ρ] → R2, affine arc-length parametrized.Parabolic arc as offset curve: β(u) = α(u) + f (u) N(u)

δH(α, β) =|| f ||∞= 124 4! |f(4)(0)| ρ4 + O(ρ5)

Can show: f (4)(0) = 3 κα(0)1/3 (kβ(0) − kα(0))

Offset curve is parabola, so kβ(0) = 0

ρ = κα(0)1/3σ + O(σ2)

δH(α, β) =1

128κα(0)5/3 |kα(0)| σ4 + O(σ5)

Proof: sketch

δH(α, β) =|| f ||∞= 124 4! |f(4)(0)| ρ4 + O(ρ5)

Can show: f (4)(0) = 3 κα(0)1/3 (kβ(0) − kα(0))

ρ = κα(0)1/3σ + O(σ2)

δH(α, β) =1

128κα(0)5/3 |kα(0)| σ4 + O(σ5)

Proof: sketch

δH(α, β) =|| f ||∞= 124 4! |f(4)(0)| ρ4 + O(ρ5)

Can show: f (4)(0) = 3 κα(0)1/3 (kβ(0) − kα(0))

ρ = κα(0)1/3σ + O(σ2)

δH(α, β) =1

128κα(0)5/3 |kα(0)| σ4 + O(σ5)

Proof: sketch

δH(α, β) =|| f ||∞= 124 4! |f(4)(0)| ρ4 + O(ρ5)

Can show: f (4)(0) = 3 κα(0)1/3 (kβ(0) − kα(0))

ρ = κα(0)1/3σ + O(σ2)

δH(α, β) =1

128κα(0)5/3 |kα(0)| σ4 + O(σ5)

Proof: sketch

δH(α, β) =|| f ||∞= 124 4! |f(4)(0)| ρ4 + O(ρ5)

Can show: f (4)(0) = 3 κα(0)1/3 (kβ(0) − kα(0))

ρ = κα(0)1/3σ + O(σ2)

δH(α, β) =1

128κα(0)5/3 |kα(0)| σ4 + O(σ5)

Proof: sketch

δH(α, β) =|| f ||∞= 124 4! |f(4)(0)| ρ4 + O(ρ5)

Can show: f (4)(0) = 3 κα(0)1/3 (kβ(0) − kα(0))

ρ = κα(0)1/3σ + O(σ2)

δH(α, β) =1

128κα(0)5/3 |kα(0)| σ4 + O(σ5)

Bitangent Conics (1)

α : [0, σ] → R2, affine spiral arc

Pencil of conics, tangent at endpoints, and intersecting at onemore point.

−3 0−2

2.0

0.0

−1

1.5

0.5

1.0

Bitangent conics, Cu and Cu ′ , u 6= u ′, have no other intersection.

Cu is tangent to α at endpoints, intersects it at α(u) and at noother point.

Bezout’s theorem: two conics intersect at 4 points.

Cu and Cu ′ intersect with multiplicity 2.

Equioscillation Property (1)

0.1

0.2

2.0−0.1

0.4

3.0

0.0

1.0

t

2.51.5

0.3

−0.2

α(u) divides α into two parts: α−u and α+u

C−u and C+u corresponding parts of the conic arc Cu.

0.1

0.2

2.0−0.1

0.4

3.0

0.0

1.0

t

2.51.5

0.3

−0.2

α(u) divides α into two parts: α−u and α+u

C−u and C+u corresponding parts of the conic arc Cu.

−3 0−2

2.0

0.0

−1

1.5

0.5

1.0

0.5

0.0

3.0

−0.25

2.01.0

0.25

−0.75

2.51.5

−0.5

t

Cu∗ unique conic, such that du∗ = min0≤u≤σ δH(α, Cu)

du∗ = δH(α−u∗ , C

−u∗) = δH(α

+u∗ , C

+u∗)

−3 0−2

2.0

0.0

−1

1.5

0.5

1.0

0.5

0.0

3.0

−0.25

2.01.0

0.25

−0.75

2.51.5

−0.5

t

Cu∗ unique conic, such that du∗ = min0≤u≤σ δH(α, Cu)

du∗ = δH(α−u∗ , C

−u∗) = δH(α

+u∗ , C

+u∗)

Hausdorff Distance

Theorem

Given: α, a smooth spiral of length σ

β: parabolic arc, asymptotic expansion of the Hausdorff distance:

δH(α, β) =1

128 |k0|κ5/30 σ

4 + O(σ5)

β: conic arc, asymptotic expansion of the Hausdorff distance:

δH(α, β) =1

2000√

5|k ′0 |κ

20σ

5 + O(σ6)

Hausdorff Distance

Theorem

Given: α, a smooth spiral of length σ

β: parabolic arc, asymptotic expansion of the Hausdorff distance:

δH(α, β) =1

128 |k0|κ5/30 σ

4 + O(σ5)

β: conic arc, asymptotic expansion of the Hausdorff distance:

δH(α, β) =1

2000√

5|k ′0 |κ

20σ

5 + O(σ6)

Proof: Brief Sketch

Strategy: Use equioscillation propery to characterize bitangent conicminimizing Hausdorff distance.

α : [0, ρ] → R2, affine arc-length parametrized.

Conic arc: β(u, v) = α(u) + u2 (u − v)(u − ρ)2 D(u, ρ)︸︷︷︸

d(u,v,ρ)

N(u)

Equioscillation property:

min0≤v≤ρ max0≤u≤ρ |d(u, v , ρ)| occurs for v = 12ρ,

and for two values of u, symmetric w.r.to v = 12ρ.

Rest of proof as for parabolic spline

Proof: Brief Sketch

d(u,v,ρ)

N(u)

Proof: Brief Sketch

d(u,v,ρ)

N(u)

Proof: Brief Sketch

d(u,v,ρ)

N(u)

Monotonicity of the Hausdorff Distance

α : I → R2, affine spiral curve.

Hausdorff distance between the affine spiral and its optimalbitangent conic arc is monotone.

−3

−0.5

1.5

0−2

0.0

1.0

0.5

−1

Yields bisection algorithm!

Monotonicity of the Hausdorff Distance

α : I → R2, affine spiral curve.

Hausdorff distance between the affine spiral and its optimalbitangent conic arc is monotone.

−3

−0.5

1.5

0−2

0.0

1.0

0.5

−1

Yields bisection algorithm!

Algorithm (1)

Spiral Curve: α(t) = (t cos(t), t sin(t)), with π6 ≤ t ≤ 2π.

ε 10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8

Complexity (Exp./Theor.)Parab. 5 9 15 26 46 82 145 257Conic 3 4 6 9 13 21 32 51

Algorithm (2)

Biarc

0.0 0.5 1.0 1.5 2.0

-1.0

-0.5

0.5

1.0

Bitangent G1-curve, consisting of two circular arcs

Tangent continuity at junction point

Biarc splines

Biarc: two circular arcs, with tangent continuous junction

Biarc spline: tangent continuous, elements are biarcs

Goal:determine complexity of ε-accurate approximating biarc spline

Related work:Meek-Walton ’94Held, Eibl ’04Drysdale, Rote, Sturm ’08

Biarc splines

Biarc

Biarcs

Biarcs: junction circle

-0.5 0.5 1.0

-1.0

-0.5

0.5

1.0

Locus of junction points is a circle

Approximation Lemma (1)

0.2 0.4 0.6 0.8 1.0

0.002

0.004

0.006

If f : [0, σ] → R is C1, C3 on [0, uσ] and [uσ, σ]1 f (0) = f ′(0) = 0, and f (σ) = f ′(σ) = 02 f ′′′(σ) = f ′′′(0) + O(σ)

Then| | f | |∞≥ 1324 |f

′′′(0)| σ3 + O(σ4)

Approximation Lemma (2)

0.2 0.4 0.6 0.8 1.0

-0.02

-0.01

0.01

0.02

| | f | |∞≥ 1324 |f′′′(0)| σ3 + O(σ4)

Equality: junction point uσ = 12σ + O(σ2)

Derivative at junction point: f ′( 12σ) =124 f

′′′(0)σ2 + O(σ3)

Application: optimal biarc

α : [0, σ] → R2: C3-curve

Bitangent biarc: offset curve β(s) = α(s) + f (s) N(s)

Hausdorff distance: δH(α, β) ≥| | f | |∞= 1324 |f ′′′(0)| σ3 + O(σ4)Can show: f ′′′(0) = κ ′β(0) − κ

′α(0)

Offset curve is piecewise circular, so κ ′β(0) = 0

Hence: δH(α, β) ≥ 1324 |κ ′α(0)| σ3 + O(σ4)Equality iff junction point of biarc is midpoint of α – up to O(σ4)

α : [0, σ] → R2: C3-curve

′α(0)

α : [0, σ] → R2: C3-curve

′α(0)

α : [0, σ] → R2: C3-curve

′α(0)

α : [0, σ] → R2: C3-curve

′α(0)

α : [0, σ] → R2: C3-curve

′α(0)

α : [0, σ] → R2: C3-curve

′α(0)

Complexity of optimal biarc

Hausdorff distance to optimal biarc:

δH(α, β) =1

324 |κ′0| σ

3 + O(σ4)

Complexity of optimal biarc-spline:

N ∼1

3 3√

12

(

∫L

s=0|κ ′(s)|1/3 ds

)

ε−1/3

Complexity of optimal biarc

Hausdorff distance to optimal biarc:

δH(α, β) =1

324 |κ′0| σ

3 + O(σ4)

Complexity of optimal biarc-spline:

N ∼1

3 3√

12

(

∫L

s=0|κ ′(s)|1/3 ds

)

ε−1/3

Bihelix splines (1)

Problem: approximation of C3 space-curves (3D)

Bihelix: two helical arcs, with tangent continuous junction

Bihelix spline: tangent continuous, elements are bihelices

Goal:determine complexity of ε-accurate approximating bihelix spline

Bihelix splines (1)

Bihelix splines (2)

Bihelical arcs withEndpoints p0 and p1End tangents T0 and T1 (unit vectors)

Junction points form one-parameter family of cylinders

(through endpoints of curve segment)

Direction of cylinder axes: all unit vectors perpendicular toT1 − T0 (one-parameter family)

Bihelix splines (2)

Approximation Lemma

Curve α : [0, σ] → R3: smooth space curve

Offset curve β(s) = α(s) + f (s) N(s) + g(s) B(s):

smooth on [0, uσ] and on [uσ, σ]

bitangent to α

κ ′β(0) = 0 and κ′β(σ) = 0 (e.g., if β is bi-helix)

Hausdorff distance

δH(α, β) ≥ 1324 |κ ′0| σ3 + O(σ4)Equality iff

u = 12 + O(σ), and

α and β have the same Frenet-Serret frame at endpoints(up to O(σ2))

Approximation Lemma

bitangent to α

Hausdorff distance

u = 12 + O(σ), and

Approximation Lemma

bitangent to α

Hausdorff distance

u = 12 + O(σ), and

Approximation Lemma

bitangent to α

Hausdorff distance

u = 12 + O(σ), and

Approximation Lemma

bitangent to α

Hausdorff distance

u = 12 + O(σ), and

Complexity of optimal bihelix splines

Hausdorff distance to optimal bihelix:

δH =1

324 |κ′0| σ

3 + O(σ4)

Complexity of optimal bihelix-spline:

N ∼1

3 3√

12

(

∫L

s=0|κ ′(s)|1/3 ds

)

ε−1/3

Complexity of optimal bihelix splines

Hausdorff distance to optimal bihelix:

δH =1

324 |κ′0| σ

3 + O(σ4)

Complexity of optimal bihelix-spline:

N ∼1

3 3√

12

(

∫L

s=0|κ ′(s)|1/3 ds

)

ε−1/3

Open problems

Approximation of curves in space with helical arcs (ongoing work)

Approximation of curves in higher dimensions (ongoing work)

Good algorithm(s) for optimalpolytopes inscribing convex hypersurfaces (arbitrary dimension)

cf. Schneider (1981), Gruber (1993)polyhedral surfaces approximating general hypersurfaces

cf. Clarkson (2006)polyhedral surfaces approximating submanifolds (arbitrarycodimension)

Open: Approximation of (convex) hypersurfaces with piecewisequadrics (envelope surfaces?).

Open problems

Approximation by linear splinesApproximation by conic splinesApproximation by biarc and bihelix splines

Documents

Approximation with linear, biarc, conic and bihelix splinesgert/documents/2010/geometric... · 2010. 7. 14. · Approximation by linear splines Approximation by conic splines Approximation