Interpolation to Data Points
Lizheng Lu
Oct. 24, 2007
Problem
Interpolation VS.
Approximation
Interpolation
Approximation
Classification

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Curve
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
Constraint
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(piecewise) Bezier curves
B-spline curves
Rational Bezier/B-spline curves
Outline



Some classical methods
Some recent methods on geometric
interpolation
Estimate the tangent
C2k-1 Hermite Interpolation
Cubic Interpolation
C2 Cubic B-spline Interpolation
Given: A set of points
a knot sequence
Find: A cubic B-spline curve, s.t.
and
* *
 p 2   rs 
* * *   p   s 

 3   2
 * * *  p 4   s 3 

   
* * p5   re 

Geometric Hermite
Interpolation (GHI)
[de Boor et al., 1987]


Given: Planar points pi, with positions,
tangents and curvatures
Result: Piecewise cubic Bezier curves, having

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G2 continuity
6th order accuracy
Convexity preservation
Comments on GHI

Independent of parameterization
High accuracy

But, it usually includes nonlinear problems
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Questions on the existence of solution and
efficient implement
Difficult to estimate approximation order, etc…
References on GHI
High Order Approximation
of Rational Curves
[Floater, 2006]
Given: A rational curve
, where
f and g are of degree M and N, let k = M+N,
with parameters values
Find: A polynomial p of degree at most n+k-2,
and scalar values
satisfying the
2n interpolation conditions:
Geometric Interpolation by
Planar Cubic Polynomial Curves
Comp. Aided Geom. Des. 2007, 24(2): 67-78
Jernej Kozak
Marjeta Krajnc
FMF&IMFM
IMFM
Jadranska 19, Ljubljana, Slovenia
Problem
Given: six points
Find: a cubic polynomial parameter curve
which satisfies
An Alternative Solution:
Quintic Interpolating Curves
Find a quintic curve
s.t.,
where ti are chosen to be the uniform
and chord length parameterization.
Essential of Problem
Know: t0, t5, p0, p3
Unknown: t1, t2, t3, t4 , p1, p2
Equations: P3(ti) = Ti, i = 2, 3, 4
Solution of Problem
Know: t0, t5, p0, p3
Unknown: t1, t2, t3, t4 , p1, p2
Equations: P3(ti) = Ti, i = 2, 3, 4
Solved by Newton Iteration with initial values:
Existence of Solution


Provide two sufficient conditions
guaranteeing the existence
Summarize cases in a table which does
not allow a solution
Comparison
cubic
uniform
chord length
On Geometric Interpolation by Planar
Parametric Polynomial Curves
Mathematics of Computation 76(260): 1981-1993
Problem
Given: 2n points
Find: a cubic polynomial parameter curve
which satisfies
Main Results
If the data, sampled from a convex smooth
curve, are close enough, then
 equations that determine the interpolating polynomial
curve are derived for general n (Theorem 4.5)
 if the interpolating polynomial curve exists, the
approximation order is 2n for general n (Theorem 4.6)

the interpolating polynomial curve exists for n≤ 5
(Theorem 4.7)
On Geometric Interpolation of
Circle-like Curves
Comp. Aided Geom. Des. 2007, 24(4): 241-251
What is Circle-like Curve?
A circular arc of an arclength
is defined by
Suppose that a convex curve is parameterized by the
same parameter as . The curve will be called
circle-like, if it satisfies:
(1)
(2)
The Result
Outline



Some classical methods
Some methods on geometric
interpolation
Estimate the tangent
Tangent Estimation Methods
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FMill , 1974
Circle Method
Bessel
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[Ackland, 1915]
Akima, 1970
G. Albrecht, J.-P. Bécar, G. Farin, D.
Hansford, 2005, 2007
Problem
?
FMILL
Circle Method
Bessel
Parabola
f (t)
Bessel
Akima’s Method
Albrecht’s Method

Albrecht G., Bécar J.P.
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Farin G., Hansford D.
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Univ. de Valenciennes et du Hainaut–Cambrésis, France
Dep. Comp. Sci., Arizona State Univ.
Détermination de tangentes par l’emploi de coniques
d’approximation.
On the approximation order of tangent estimators.
CAGD, in press
Main Idea
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Method: Estimate the tangent by using the
interpolating conic of the given five points
Solution: solved by Pascal’s theorem in projective
geometry
Advantages
 Conic precision
 Less computations without computing the implicit
conic
Idea Derivation

[Farin, 2001]
Any conic section is uniquely determined by
five distinct points in the plane, pi=(xi, yi).
x2
xy
y2
x
y
x12
x22
f ( x, y )  2
x3
x1 y1
x2 y2
x3 y3
y12
y22
y32
x1
x2
x3
y1 1
y2 1
0
y3 1
x42
x52
x4 y4
x5 y5
y42
y52
x4
x5
y4 1
y5 1
1
Idea Derivation
[Pascal, 1640]
Projective Geometry
in CAGD

Express rational forms
R (u)   x(u), y (u), z (u),  (u) 
m
n
  Bi (u)R ij ,
i 0 j 0

Implicit representation
of rational forms
R (u) =  x(u), y (u), z (u)   (u)
Projective Geometry
in CAGD
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Express rational forms
Implicit representation
of rational forms

Chen, Sederberg
Line conics
Conic section
Projective Geometry
Projective Geometry

A line in
is represented by

The line joining the two points is
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The intersection of two lines is
Estimate the Tangent
Estimate the Tangent
Degenerate Cases
(b)
(a)
(c)
Examples
Experimental results
Non-convex Case
Conic method
Bessel
Akima
Circle method
Approximation order
Theoretical Analysis
Consider a planar curve:
Theoretical Analysis
Consider a planar curve:
Take five points:
Theoretical Analysis
Consider a planar curve:
Take five points:
Let:
Theoretical Analysis
Taylor expansion:
Exact tangent:
Exact norm:
Theoretical Analysis
For a point
, with the tangent:
Its corresponding tangent in the
projective space is:
Compute the
Approximation Order
To solve the k in:


Taylor expansion
Symbolic computation: MAPLE
Numerical Result (1)
Numerical Result (2)
Summary
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Obtain order four approximation for the
convex case, two for the inflection point
Estimate the approximation order with
theoretical justification
Estimate the direction of the tangent only,
not the vector!
Thank You!