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, ChangshaSubdecanal, ProfessorPh.D. in Central South University, 94CAGD, Mathematical Modeling

Previous work

Lyche, T., Winther, R., 1979. A stable recurrence relation for trigonometric B-splines. 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 significanceof 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) : :discontinuous) The support interval of bi(u): 3 segments to 4 − k segments Set : −1 < λ≤ 1, λ≠ -1

The case of multiple knots

λ = 0 (dashed lines) , λ = 0.5 (solid lines)

Trigonometric polynomial curvesQuadratic 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 curvesWhen a knot ui : multiplicity k (k=1,2,3)

the Trigonometric polynomial curves : continuity, at knot ui.

Open trigonometric curvesChoose 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 ellipsesWhen 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)(u ∈ [ui,ui+1])

decrease of ∆ui

fixed ∆ui-1, ∆ui+1

Merged with: Ti(0)Pi−1 ,Pi−1Ti(π/2).

Ti(ti)(u ∈ [ui,ui+1])

The edge of the given control polygon.Increase λ −1 < λ≤ 1

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 ApproximabilityThe 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 significanceof 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

λ= 0.5

The case of multiple knots

λ= 0λ= 0.5

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.

Examples:

λ=-0.3

λ=0

λ=0.6

λ=0

λ=-0.28

Cubic B-spline

The representation of ellipsesWhen 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

T(u) u∈[ui,ui+1]

Increase λ the edge Pi−2Pi−1

Parameter λ controls the shape of the curve T (u)

Examples:

ApproximabilityGiven: 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)

ApproximabilityGiven: : 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: a set of nodes :x1 < x2 < ··· < xm. Find: trigonometric function of the form Purpose: interpolate data given at the nodes

Goal:The interpolation matrix A = (Aij )m×m; Aij = Bj (xi ), i, j = 1, 2, . . . , m

A must be nonsingular.

Necessary condition Let: −0.5≤λ≤1 If 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 ?