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., A stable recurrence relation for trigonometric B- splines. J. Approx. Theory 25, 266 – 279. Lyche, T., Schumaker, L.L., Quasi-interpolants based on trigonometric splines. J. Approx. Theory 95, 280 – 309. Pe ñ a, J.M., Shape preserving representations for trigonometric polynomial curves. Computer Aided Geometric Design 14,5 – 11. Schoenberg, I.J., On trigonometric spline interpolation. J. Math. Mech. 13, 795 – 825. Koch, P.E., Multivariate trigonometric B-splines. J. Approx. Theory 54, 162 – 168. Koch, P.E., Lyche, T., Neamtu, M., Schumaker, L.L., Control curves and knot insertion for trigonometric splines. Adv. Comp. Math. 3, 405 – 424. S á nchez-Reyes, J., Harmonic rational B é zier curves, p-B é zier 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, b i (u) : uniform basis functions. For non-equidistant knots, b i (u) : non-uniform basis functions. For λ = 0, b i (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 b i (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. u i =u i +1 is a double knot

Geometric significance of multiple knots b i (u) has a knot of multiplicity k (k = 2 or 3) at a parameter value u At u, the continuity of b i (u) : :discontinuous) The support interval of b i (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 curves Quadratic trigonometric polynomial curve with a shape parameter: Given: points P i (i = 0, 1,...,n) in R 2 or R 3 and a knot vector U = (u 0,u 1,...,u n+3 ). When u ∈ [u i,u i+1 ], u i ≠ u i+1 (2 ≤ i ≤ n)

The continuity of curves When a knot u i : multiplicity k (k=1,2,3) the Trigonometric polynomial curves : continuity, at knot u i.

Open trigonometric curves Choose the knot vector: T(U 2 )=P o, T(U n+1 )=P n ;

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 P i (i=0,1, …,n) by setting: P n+1 =P 0,P n+2 =P 1 Let:U n+4 =U n+3 +∆U 2, ∆U 1 = ∆U n+2,U n+5 ≥U n+4 b n+1 (u) and b n+2 (u) are given by expanding. T(u 2 )=T(U n+3 ), T ′ (U 2 )= T ′ (U n+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 ∈ [u i,u i+1 ], Origin:P i-1, unit vectors:P i-2 -P i-1, P i -P i-1 T (u) is an arc of an ellipse.

Approximability T i (t i ) (u ∈ [u i,u i+1 ]) decrease of ∆u i fixed ∆u i-1, ∆u i+1 Merged with: T i (0)P i−1,P i−1 T i (π/2). T i (t i ) (u ∈ [u i,u i+1 ]) The edge of the given control polygon. Increase λ −1 < λ≤ 1

Examples:

Approximability The associated quadratic B-spline curve: Given points P i ∈ R 2 or R 3 (i = 0, 1,...,n) and knots u 0 <u 1 < ··· <u n+3. u ∈ [u k,u k+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 u i (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., Quadratic trigonometric polynomial curves with a shape parameter. Computer Aided Geometric Design 19,503 – 512. Han, X., Piecewise quadratic trigonometric polynomial curves. Math. Comp. 72, 1369 – 1377.

Construction of basis functions

Basis functions For equidistant knots, B i (u) : uniform basis function,simple bi0=bi2=bi3=cio=ci1=ci3=0 For non-equidistant knots, B i (u) : non-uniform basis functions. For λ = 0, B i (u) : quadratic trigonometric polynomial basis functions.

Properties of basis functions Has a support on the interval [u i,u i +4]:  If −0.5 0 for u i <u<u i +4.  With a uniform knots vector, if −1 ≤ λ ≤ 1, B i (u) > 0 for u i <u<u i +4. Form a partition of unity:

The continuity of the basis functions With a non-uniform knot vector:  b i (u) has C2 continuity at each of the knots. With a uniform knot vector:  λ≠1, b i (u) has C3 continuity at each of the knots  λ=1, b i (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. u i =u i +1 is a double knot

Geometric significance of multiple knots b i (u) has a knot of multiplicity k (k = 2,3,4) at a parameter value u At u, the continuity of b i (u): discontinuous) The support interval of b i (u): 4 segments to 5 − k segments

The case of multiple knots λ= 0 λ= 0.5

The case of multiple knots λ= 0 λ= 0.5

Trigonometric polynomial curves Cubic trigonometric polynomial curve with a shape parameter: Given: points P i (i = 0, 1,...,n) in R 2 or R 3 and a knot vector U = (u 0,u 1,...,u n+4 ). When u ∈ [u i,u i+1 ], u i ≠ u i+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) ′.(1/4 λ+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 C 3-k continuity at ui  The curves have G 3 continuity at ui, k=1 With a uniform knot vector:  λ≠1, The curves have C 3 continuity at each of the knots  λ=1, The curves have C 5 continuity at each of the knots

Open trigonometric curves Choose the knot vector: T(U 0 )= T(U 3 )=P 0, T(U n+1 )= T(U n+4 )=P n ;

Closed trigonometric curves Extend points P i (i=0,1, …,n) by setting: P n+ 1=P 0,P n+2 =P 1,P n+3 =P 2 Let:∆U j = ∆U n+j+1, (j=1,2,3,4) B n+1 (u), B n+2 (u),B n+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 ellipses When the shape parameterλ = 0, u ∈ [u i,u i+1 ], P i−3 = (−a,−b), P i−2 = (−a, b), P i−1 = (a, b), P i = (a,−b), With a uniform knot vector, T (u) is an arc of an ellipse.

Trigonometric B é zier curve U ∈ [u i,u i+1 ], u i <u i+1, u i and u i+1 : triple points. (u 3 : quadruple point, u n+1 : quadruple point) -2 ≤λ≤1

Trigonometric B é zier curve

Examples: the cubic B é zier curve (dashed lines), the trigonometric B é zier curves with λ=−1 (dashdot lines) and λ = 0 (solid lines)

Approximability T(u) u ∈ [u i,u i+1 ] Increase λ the edge P i−2 P i−1 Parameter λ controls the shape of the curve T (u)

Examples:

Approximability Given: B(u): cubic B-spline curve with a knot vector U. T(u): cubic trigonometric polynomial curves, with λ Find: The relations of B(u) and T(u)

Approximability With a non-uniform knot vector U, λ = 0. T (u i ) = B(u i ) (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≥λ 0 ≈−

Approximability With a uniform knot vector, forλ= 0, With a uniform knot vector,forλ =λ 0, If λ 0 ≤λ≤0, then T (u) is close to B(u)

Approximability Given: : cubic B é zier curve T(u): trigonometric B é zier curve. (cubic trigonometric polynomial curves,with λ ) With the same control point P i-3,P i-2,P i-1,P i 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 = (A ij ) m×m ; A ij = B j (x i ), i, j = 1, 2,..., m A must be nonsingular.

Necessary condition Let: −0.5 ≤ λ ≤ 1 If the matrix A is nonsingular. Then A ii ≠ 0 (u i < x i <u i+4 ), i = 1, 2,...,m.

Sufficient condition Let: −0.5 ≤ λ ≤ 1 If  u i < x i ≤ u i+1 or u i+3 ≤ x i <u i+4, i = 1, 2,...,m,  If x i = u i+2 and 1 − 2a i+2 − 2d i+1 ≥ 0, i = 1, 2,...,m, Then A is nonsingular.

Method of Interpolation assign arbitrary value to P 0 and P m+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 ?