Dagstuhl Seminar on Geometric Model l ing , Germany , May 2008

1. Dagstuhl Seminar on Geometric Modelling, Germany, May2008 Curve subdivisionwith control of thearc-length …work in progress Durham University,UK ICIMAF, Havana, Cuba Ioannis Ivrissimtzis Victoria Hernández Jorge C. Estrada Silvio R. Morales

2. Outline • Motivation • The arc-length subdivision scheme • Definition • Convergence • Properties of the limit curve • Examples • Conclusions • Future work

3. Motivation • Uniform subdivision schemes produce a sequence of • piecewise linear functions with vertices at the dyadic • parameter points. • Schemes derived from local polynomial interpolation can be easily generalized to non-uniform parametrizations • Changing the position of the initial parameter values we get a different limit curve. The length of the limit curve depends on the parametrization that we use.

4. Motivation Classical 4-points subdivisionscheme: thelimit curve isveryclosetolongedges and veryfarfrom short edges This behavior is a consequence of the uniform parametrization: the same time is used to travel between two consecutive points of the starting polygon independently of the distance between them

5. Motivation 4-points subdivision using the chordal parametrization as initial parametrization The limit curve is very tight to short edges

6. Motivation • Studytheinfluence of theinitialparametrization in • theproperties of thelimit curve of the 4-points subdivisionscheme.

7. Howtomeasurethequality of parametrization? Farouki (1997), under the hypothesis that c(t) is differentiable, introduces functional J is a measure of closeness to arc-length parametrization J=1if c(t)isarc-lengthparametrized

8. Experiments with different initial parametrizations

10. Remarks • Theoptimalvalue of Jisobtainedwithchordalinitialparametrization ( b=1). • Ifwe use thecentripetalparametrization ( b=0.5) thenlength of the limit curve between two consecutive points of the starting polygon is directly proportional to the length of the corresponding edge.

11. Main Goal Propose a subdivision scheme where the length of the curve between two consecutive points of the starting polygon is controlled in all steps. Related work • Su, Li, Zhou, 2006, JCAM, select the tension parameters of the 4-point subdivision scheme in such a way that the length of the polygonal curve between two points of the initial polygon takes a prescribed value. • Dyn, Floater, Hormann, 2007, TechnicalReport: in each step of the 4-point subdivision schemetheparametrizationischangedbeforecomputing the interpolating polynomial.

12. The arc-length subdivision scheme initial polygon, 4 or more vertices are not collinear normalized direction of displacement magnitude of displacement

13. ...to select the magnitude of displacement in such a way that

14. The new pointisontheellipsewithfoci and semimajor axis

15. How does it work?

16. Convergence • Theorem:Assume that • The sequenceis convergent • ii) The new points are selected in a way that • for all i, where • Then the arc-length subdivision scheme converges and the limit curve is continuous.

17. Introducing the parametrization such that Proof where it is easy to prove that the sequence of piecewise linear functions interpolating the vertices of the polygon in each step j converges uniformly. the limit curve is continuous

18. The displacement direction Proposed by Yang, CAGD, 2006 convex case or its symmetric with respect to inflection case

19. Sequence …that guarantees the convergence of the scheme and the continuity of the limit curve is obtained solving the equations where is any sequence converging to 1

20. Properties of the limit curve In any step j, the length of the subdivision curve between two consecutive points of the starting polygon is proportional, with the same proportionality factor, to the length of the corresponding edge, If is finite, then

21. In any step j, the section of the subdivision curve between points is contained in the ellipse with foci and eccentricity If is finite, then

22. Examples

23. Examples

24. Conclusions • We have proposed a new interpolatory subdivision • scheme with control of arc length of the subdivision • curve in any step. • A sufficient condition that guarantee the convergence • of the scheme and the continuity of the limit curve • has been obtained. • We computed a bound for the Hausdorff distance between • the limit curve and the starting polygon.

25. Future work • Find sufficient conditions that guarantee the C1continuity • of the limit curve. • For each edge of the starting polygon use a different • factor for scaling the length of the subdivision curve • corresponding to that edge. • Given a real value a > 1, define a sequence such that the length of the limit curve is equal to a times the length of the starting polygon. • Prove that the limit curve is arc-length parametrized

26. Important References • N. Dyn, M. Floater, K. Hormann, Four point curve subdivision based on iterated chordal and centripetal parameterizations, TU Clausthal University, Technical Report, oct., 2007. • X. Yang, Normal based subdivision scheme for curve design, Computer Aided Geometric Design, 2006, 23, 243-260.