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
Approximation by linear splinesApproximation by conic splines
Approximation by biarc 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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
1 Approximation by linear splines
2 Approximation by conic splines
3 Approximation by biarc and bihelix splines
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
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?
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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).
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
Hausdorff distance
L
α
α(0)
α(σ)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
Hausdorff distance
L
α
N(s)
f(s)
α(s)α(0)
α(σ)
T(s)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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 | |∞
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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).
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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).
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
1 Approximation by linear splines
2 Approximation by conic splines
3 Approximation by biarc and bihelix splines
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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.
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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!)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
Constant Affine Curvature
Conics are curves with constant affine curvature.
k > 0 ellipsek = 0 parabolak < 0 hyperbola
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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.
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
Affine Spirals (1)
Affine Spiral: curve with monotone affine curvature.
Any conic intersects an affine spiral in at most 5 points (countedwith multiplicity).
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
Affine Spirals (2)
Osculating conics of an affine spiral are disjoint, and do notintersect the spiral arc except at their point of contact.
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
Bitangent Conics (2)
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.
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
Bitangent Conics (2)
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.
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
Bitangent Conics (2)
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.
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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.
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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.
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
Equioscillation Property (2)
−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∗)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
Equioscillation Property (2)
−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∗)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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!
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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!
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
Algorithm (2)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
1 Approximation by linear splines
2 Approximation by conic splines
3 Approximation by biarc and bihelix splines
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
Biarc
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
Biarc
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
Biarcs
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
Biarcs: junction circle
-0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
Locus of junction points is a circle
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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)
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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))
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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))
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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))
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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))
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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))
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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?).
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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?).
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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?).
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splines
Approximation by biarc and bihelix splines
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?).
Gert Vegter Approximation with linear, biarc, conic and bihelix splines
Approximation by linear splinesApproximation by conic splinesApproximation by biarc and bihelix splines