Transcript Slide 1

Spline curves with a shape
parameter
Reporter: Hongguang Zhou
April. 2rd, 2008
Problem:



To adjust the shape of curves,
To change the position of curves.
Weights in rational Bézier , B-spline curves
are used.
Problem:


Spline has some deficiencies:
e.g. To adjust the shape of a curve,
but the control polygon must be
changed.
Motivation:


When the control polygons of splines
are fixed
Can rectify the shape of curves only by
adjusting the shape parameter.
Outline

Basis functions

Trigonometric polynomial curves with a shape
parameter

Approximability

Interpolation
References



Quadratic trigonometric polynomial curves with a shape
parameter
Xuli Han (CAGD 02)
Cubic trigonometric polynomial curves with a shape
parameter
Xuli Han (CAGD 04)
Uniform B-Spline with Shape Parameter
Wang Wentao, Wang Guozhao
(Journal of computer-aided design & computer graphics 04)
Quadratic trigonometric
polynomial curves with a
shape parameter
Xuli Han
CAGD. (2002) 503–512
About the author
Department of Applied Mathematics
and Applied Software, Central South
University, Changsha
Subdecanal, Professor
Ph.D. in Central South University, 94
CAGD, Mathematical Modeling
Previous work








Lyche, T., Winther, R., 1979. A stable recurrence relation for trigonometric Bsplines. J. Approx. Theory 25, 266–279.
Lyche, T., Schumaker, L.L., 1998. Quasi-interpolants based on trigonometric
splines. J. Approx. Theory 95, 280–309.
Peña, J.M., 1997. Shape preserving representations for trigonometric
polynomial curves. Computer Aided Geometric Design 14,5–11.
Schoenberg, I.J., 1964. On trigonometric spline interpolation. J. Math. Mech.
13, 795–825.
Koch, P.E., 1988. Multivariate trigonometric B-splines. J. Approx. Theory 54, 162–168.
Koch, P.E., Lyche, T., Neamtu, M., Schumaker, L.L., 1995. Control curves and knot
insertion for trigonometric splines. Adv. Comp. Math. 3, 405–424.
Sánchez-Reyes, J., 1998. Harmonic rational Bézier curves, p-Bézier curves
and trigonometric polynomials. Computer Aided Geometric Design 15, 909–
923.
Walz, G., 1997a. Some identities for trigonometric B-splines with application
to curve design. BIT 37, 189–201.
Construction of basis functions
Basis functions



For equidistant knots,
bi(u) : uniform basis functions.
For non-equidistant knots,
bi(u) : non-uniform basis functions.
For λ = 0, bi(u) : linear trigonometric
polynomial basis functions.
Uniform basis function
λ = 0 (dashed lines) , λ = 0.5 (solid lines).
Properties of basis functions

Has a support on the interval [ui,ui+3]:

Form a partition of unity:
The continuity of the basis
functions
 bi(u) has C1 continuity at each of the knots.
The case of multiple knots



knots are considered with multiplicity K=2,3
Shrink the corresponding intervals to zero;
Drop the corresponding pieces.
ui =ui+1 is a double knot
Geometric significance
of multiple knots

bi(u) has a knot of multiplicity k (k = 2 or 3) at a
parameter value u

At u, the continuity of bi(u) :

The support interval of bi(u): 3 segments to 4 − k
segments

Set : −1 < λ≤ 1, λ≠ -1
:discontinuous)
The case of multiple knots
λ = 0 (dashed lines) , λ = 0.5 (solid lines)
Trigonometric polynomial
curves
Quadratic trigonometric polynomial curve
with a shape parameter:
Given: points Pi (i = 0, 1, . . .,n) in R2 or R3 and a
knot vector U = (u0,u1, . . .,un+3).
When u ∈ [ui,ui+1], ui ≠ui+1 (2 ≤ i ≤ n)
The continuity of curves
When a knot ui : multiplicity k (k=1,2,3)
the Trigonometric polynomial curves :
at knot ui.
continuity,
Open trigonometric curves
Choose the knot vector:
T(U2)=Po, T(Un+1)=Pn;
Example:
Curves for λ = 0, 0.5, 1(solid lines) and the quadratic B-spline curves (dashed lines),
U = (0, 0, 0, 0.5, 1.5, 2, 3, 4, 5, 5, 5).
Closed trigonometric curves

Extend points Pi (i=0,1,…,n) by setting:
Pn+1=P0,Pn+2=P1
Let:Un+4=Un+3+∆U2, ∆U1= ∆Un+2,Un+5≥Un+4
bn+1(u) and bn+2(u) are given by expanding.

T(u2)=T(Un+3), T′(U2)= T′(Un+3)


Examples:
Closed curves for λ = 0, 0.5 (solid, dashed lines on the left), λ = 0.1, 0.3
(solid, dashed lines on the right) , quadratic B-spline curves (dotted lines)
The representation of ellipses
When the shape parameterλ = 0, u ∈ [ui,ui+1],
Origin:Pi-1, unit vectors:Pi-2-Pi-1, Pi-Pi-1
T (u) is an arc
of an ellipse.
Approximability
Ti(ti)
decrease of ∆ui
(u ∈ [ui,ui+1])
Merged with:
Ti(0)Pi−1 ,Pi−1Ti(π/2).
fixed ∆ui-1, ∆ui+1
Ti(ti)
Increase λ
(u ∈ [ui,ui+1])
−1 < λ≤ 1
The edge of the given
control polygon.
Examples:
Approximability

The associated quadratic B-spline curve:
Given points Pi ∈ R2 or R3 (i = 0, 1, . . .,n) and knots u0 <u1
< ···<un+3.
u ∈ [uk,uk+1]
Approximability
The relations of the trigonometric polynomial curves
and the quadratic B-spline curves:
Approximability
Conclusion of Approximability
The trigonometric polynomial curves intersect the quadratic B-spline
curves at each of the knots ui (i = 2, 3, . . . , n+1) corresponding to the
same control polygon.
For λ ∈ (−1, (√2−1)/2], the quadratic B-spline curves are closer to the
given control polygon;
For λ ∈ [(√2 − 1)/2,√5 − 2], the trigonometric polynomial curves are
very close to the quadratic B-spline curves;
For λ = (√2 − 1)/2 and λ = √5 − 2, the trigonometric polynomial
curves yield a tight envelope for the quadratic B-spline curves;
For λ ∈ [√5 − 2, 1], the trigonometric polynomial curves are closer to
the given control polygon.
Cubic trigonometric polynomial
curves with a shape parameter
Xuli Han
CAGD. (2004) 535–548
Related work:


Han, X., 2002. Quadratic trigonometric polynomial curves
with a shape parameter. Computer Aided Geometric Design
19,503–512.
Han, X., 2003. Piecewise quadratic trigonometric polynomial
curves. Math. Comp. 72, 1369–1377.
Construction of basis functions
Construction of basis functions
Basis functions



For equidistant knots,
Bi(u) : uniform basis function,simple
bi0=bi2=bi3=cio=ci1=ci3=0
For non-equidistant knots,
Bi(u) : non-uniform basis functions.
For λ = 0,
Bi(u) : quadratic trigonometric polynomial basis
functions.
Properties of basis functions


Has a support on the interval [ui,ui+4]:
 If −0.5<λ≤1, Bi(u) > 0 for ui <u<ui+4.
 With a uniform knots vector, if −1 ≤λ≤1,
Bi(u) > 0 for ui <u<ui+4.
Form a partition of unity:
The continuity of the basis
functions


With a non-uniform knot vector:
 bi(u) has C2 continuity at each of the knots.
With a uniform knot vector:
 λ≠1,bi(u) has C3 continuity at each of the knots
 λ=1, bi(u) has C5 continuity at each of the knots
The case of multiple knots



knots are considered with multiplicity K=2,3,4
Shrink the corresponding intervals to zero;
Drop the corresponding pieces.
ui =ui+1 is a double knot
Geometric significance
of multiple knots



bi(u) has a knot of multiplicity k (k = 2,3,4) at a
parameter value u
At u, the continuity of bi(u):
discontinuous)
The support interval of bi(u): 4 segments to 5 − k
segments
The case of multiple knots
λ= 0.5
λ= 0
The case of multiple knots
λ= 0.5
λ= 0
Trigonometric polynomial
curves


Cubic trigonometric polynomial curve with a
shape parameter:
Given: points Pi (i = 0, 1, . . .,n) in R2 or R3 and a
knot vector U = (u0,u1, . . .,un+4).
When u ∈ [ui,ui+1], ui ≠ui+1 (3 ≤ i ≤ n)
Trigonometric polynomial
curves

With a uniform knot vector,
T(u)=(f0(t),f1(t),f2(t),f3(t)) . (Pi-3,Pi-2,Pi-1,P1)′ .(1/4λ+6)
t∈[0,Π/2]
The continuity of the curves


With a non-uniform knot vector, ui has multiplicity k
(k=1,2,3,4)
 The curves have C3-k continuity at ui
 The curves have G3 continuity at ui, k=1
With a uniform knot vector:
 λ≠1, The curves have C3 continuity at each of
the knots
 λ=1,
The curves have C5 continuity at each of
the knots
Open trigonometric curves

Choose the knot vector:
T(U0)= T(U3)=P0, T(Un+1)= T(Un+4)=Pn;
Closed trigonometric curves
 Extend points Pi (i=0,1,…,n) by setting:
Pn+1=P0,Pn+2=P1,Pn+3=P2
 Let:∆Uj= ∆Un+j+1, (j=1,2,3,4)
 Bn+1(u), Bn+2(u),Bn+3(u) are given by expanding.
λ=0
Examples:
λ=0.
6
λ=-0.28
λ=0.3
λ=0
Cubic Bspline
The representation of ellipses
When the shape parameterλ = 0, u ∈ [ui,ui+1],
Pi−3 = (−a,−b), Pi−2 = (−a, b), Pi−1 = (a, b), Pi = (a,−b),
With a uniform knot vector,
T (u) is an arc of
an ellipse.
Trigonometric Bézier curve



U∈ [ui,ui+1], ui <ui+1,
ui and ui+1 : triple points.
(u3 : quadruple point , un+1 : quadruple point)
-2≤λ≤1
Trigonometric Bézier curve
Examples:
the cubic Bézier curve (dashed lines) , the trigonometric Bézier curves
with λ=−1 (dashdot lines) and λ = 0 (solid lines)
Approximability
Parameter λ controls the shape of the curve T (u)
T(u)
u∈[ui,ui+1]
Increase λ
the edge
Pi−2Pi−1
Examples:
Approximability
Given:
B(u): cubic B-spline curve with a knot vector U.
T(u): cubic trigonometric polynomial curves, withλ
Find:
The relations of B(u) and T(u)
Approximability

With a non-uniform knot vector U,
λ = 0.

T (ui ) = B(ui) (i = 3, 4, . . . , n+1)

Approximability

With a uniform knot vector
−1≤λ ≤1,
g(λ) ≤ 1 if and only if λ≥0;
h(λ) ≤ 1 if and only if λ≥λ0≈−0.2723.
Approximability

With a uniform knot vector , forλ= 0,

With a uniform knot vector ,forλ =λ0,
If λ0 ≤λ≤0,
then T (u) is close to B(u)
Approximability
Given:
: cubic Bézier curve
T(u): trigonometric Bézier curve.
(cubic trigonometric polynomial curves,withλ )
With the same control point Pi-3,Pi-2,Pi-1,Pi
Find:
The relations of
and T(u)
Approximability
T(u) is close to
, when λ≈−0.65.
Interpolation

Given:

Find: trigonometric function of the form

Purpose: interpolate data given at the nodes
a set of nodes :x1 < x2 < ··· < xm.
Goal:
The interpolation matrix
A = (Aij )m×m;
Aij = Bj (xi ), i, j = 1, 2, . . . , m
A must be nonsingular.
Necessary condition


Let:
If
−0.5≤λ≤1
the matrix A is nonsingular.

Then
Aii ≠0 (ui < xi <ui+4) , i = 1, 2, . . .,m.
Sufficient condition


Let: −0.5≤λ≤1
If
ui < xi ≤ui+1 or ui+3≤xi<ui+4 , i =
1, 2, . . .,m,
If xi = ui+2 and 1 − 2ai+2 − 2di+1≥0 , i = 1,
2, . . .,m,



Then
A is nonsingular.
Method of Interpolation
assign arbitrary value to P0 and Pm+1, then solve the equations
Uniform B-Spline with
Shape Parameter
Conclusions:

Properties of trigonometric polynomial curves

Shape parameter controls the shape of the curves

Compare with B-spline, Bézier in some aspects.
Thank you
Questions ?