Interpolation to Data Points Lizheng Lu Oct. 24, 2007.
Transcript of Interpolation to Data Points Lizheng Lu Oct. 24, 2007.
Classification
Curve
Constraint
(piecewise) Bezier curves B-spline curves Rational Bezier/B-spline curves
C2 Cubic B-spline Interpolation
Given: A set of points and a knot sequence Find: A cubic B-spline curve, s.t.
2
3 2
4 3
5
* *
* * *
* * *
* *
s
e
p r
p s
p s
p r
Geometric Hermite Interpolation (GHI)
Given: Planar points pi, with positions, tangents and curvatures
Result: Piecewise cubic Bezier curves, having G2 continuity 6th order accuracy Convexity preservation
[de Boor et al., 1987]
Comments on GHI
Independent of parameterization High accuracy
But, it usually includes nonlinear problems Questions on the existence of solution and
efficient implement Difficult to estimate approximation order,
etc…
High Order Approximationof Rational Curves
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:
[Floater, 2006]
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
An Alternative Solution:Quintic Interpolating CurvesFind 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
Solved by Newton Iteration with initial values:
Know: t0, t5, p0, p3
Unknown: t1, t2, t3, t4 , p1, p2
Equations: P3(ti) = Ti, i = 2, 3, 4
Existence of Solution
Provide two sufficient conditions guaranteeing the existence
Summarize cases in a table which does not allow a solution
On Geometric Interpolation by Planar Parametric Polynomial
Curves
Mathematics of Computation 76(260): 1981-1993
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)
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 calledcircle-like, if it satisfies:(1)
(2)
Tangent Estimation Methods
FMill , 1974 Circle Method Bessel
[Ackland, 1915] Akima, 1970 G. Albrecht, J.-P. Bécar, G. Farin, D. Ha
nsford, 2005, 2007
Albrecht’s Method Albrecht G., Bécar J.P.
Univ. de Valenciennes et du Hainaut–Cambrésis, France Farin G., Hansford D.
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 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 Any conic section is uniquely determined by f
ive distinct points in the plane, pi=(xi, yi).
[Farin, 2001]
2 2
2 21 1 1 1 1 12 22 2 2 2 2 22 23 3 3 3 3 32 24 4 4 4 4 42 25 5 5 5 5 5
1
1
1( , ) 0
1
1
1
x xy y x y
x x y y x y
x x y y x yf x y
x x y y x y
x x y y x y
x x y y x y
Projective Geometry in CAGD
Express rational forms
Implicit representation of rational forms
0 0
( ) ( ), ( ), ( ), ( )
( ) ,
( ) ( ), ( ), ( ) ( )
m n
i iji j
x y z
B
x y z
R
R
R
u u u u u
u
u = u u u u
Projective Geometry in CAGD
Express rational forms
Implicit representation of rational forms Chen, Sederberg
Conic sectionLine conics
Projective Geometry
A line in is represented by
The line joining the two points is
The intersection of two lines is
Theoretical Analysis
For a point , with the tangent:
Its corresponding tangent in the projective space is:
Summary
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!