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Constrained Spline Curves

Constrained Spline Curves. Reporter: Jun Chen Dec 7, 2006. References. A constrained guided G 1 continuous spline curve (CAD,2003) D.S. Meek, B.H. Ong, D.J. Walton

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Constrained Spline Curves

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  1. Constrained Spline Curves Reporter: Jun Chen Dec 7, 2006

  2. References A constrained guided G1 continuous spline curve (CAD,2003) D.S. Meek, B.H. Ong, D.J. Walton A smooth, obstacle-avoiding curve (Computers & Graphics,2006) Z. Li, D.S. Meek, D.J. Walton

  3. A constrained guided G1continuous spline curve D.S. Meek, B.H. Ong, D.J. Walton CAD, 2003, 35: p591-599

  4. About the authors Dereck S. Meek Desmond J. Walton

  5. About the authors • They are Professors of Computer Science at the University of Manitoba. They obtained a PhD from the University of Manitoba. • Recent Research Publications(1989--1999): Arc splines, Clothoids, Cubic B-splines and cubic Bezier curves, Spirals, Surfaces and 3D modelling, Visualization, etc. (CAD, CAGD, Computers & Graphics, Journal of Computational and Applied Mathematics)

  6. About the authors • Boon-Hua Ong is an associate professor in the School of Mathematical Sciences, Universiti Sains Malaysia, Malaysia. She received a PhD from the University of California at Berkeley, USA in Mathematics. • Her current research interest is in Approximation theory, CAGD.

  7. Work of this paper

  8. Work of this paper

  9. Background • Designing a shape to be cut from a flat sheet of material. • Designing a smooth robot path that avoids obstacles.

  10. Previous work • Nowacki H, Liu D, Lu X. Fairing Bezier curves with constraints. Comput Aid Geomet Des 1990;7:43–55. • Goodman TNT, Ong BH, Unsworth K. Constrained interpolation using rational cubic splines. In: Farin G, editor. NURBS for curve and surface design. Philadelphia: SIAM; 1991. p. 59–74. • Duan Q, Xu G, Liu A, Wang X, Cheng F. Constrained interpolation using rational cubic spline curves with linear denominators. Korean J Comput Appl Math 1999;6:203–15. • Zhang C, Cheng F. Constrained shape-preserving parametric curve interpolation with minimum curvature. Preprint.

  11. Advantage • The boundary consists of straight line segments and circular arcs. • The curve here is quadratic, most other results use cubics. • The curve here is guided by control points rather than by the more commonly used interpolation points.

  12. Outline 1.单段曲线的构造－－临界点条件 2.组合曲线的构造

  13. Def. of G1 • A curve is G1continuous if it is continuous and has a continuous unit tangent vector (equivalently, the tangent direction is continuous).

  14. Rational quadraticBezier curves

  15. Rational quadraticBezier curves Result: The points on a curve (1) are a weighted average of the control points B0, B1, and B2. With the restrictions on t and w in Eq. (1), all of the weights are positive, so all points B(t, w) are strictly inside the control triangle B0B1B2.

  16. Lemma 1 Lemma 1. If P is strictly inside the Bezier control triangle B0B1B2, then there is a unique curve (1) that passes through P. If P is not inside the control triangle, then there is no curve (1) that passes through P.

  17. Lemma 1 stands for the scalar cross product of two dimensional vectors. Def: It is positive when the counterclockwise angle from a to b is greater than 0. Property:

  18. Lemma 1 r

  19. Lemma 1 The direction of this vector is constant with respect to w, so the locus of points B(t, w) with fixedt and increasing w is a straight line fromB(t, 0) toB1.

  20. Lemma 1

  21. Lemma 2 Lemma 2. For the distinct points S0 and S1, if conditions below are satisfied, then there is a unique curve (1) tangent to the line segment S0S1. Otherwise, there is no curve (1) that is tangent to line segment S0S1.

  22. Lemma 2 Set the infinite line L through the points S0 and S1 . L must • pass through the control triangle for a tangent to be possible, • cannot cut B0B2, • cannot pass through B1 .

  23. Lemma 2 B(t, w) must be on L, and are collinear: Curve (1) is tangent to L , and are co-line:

  24. Lemma 2 It shows that this quadratic has a double root.

  25. Lemma 2 B1 cannot be on line L.

  26. Lemma 2 Remark:

  27. Lemma 2 The point of tangency P= B(t, w), may not be in the line segment S0S1, so a final check is needed:

  28. Lemma 3 Lemma 3.S0 and S1 are distinct endpoints of an arc, the sweep angle of the arc θ is non-zero, if conditions below are satisfied, then there exist one or more curves (1) that are tangent to the arc from S0 to S1. Otherwise, no curve (1) exists that is tangent to the arc from S0to S1 .

  29. Lemma 3 By analogy with Lemma 2: has a double root in t. Remark:The condition for the equation to have a double root is showed in the Appendix A. A check is also made to see if the point is in the arc, and there may be several curves (1) tangent to the arc.

  30. Solution of the problem The solution is based on a quadratic B-spline with uniform knots. each Bezier segment can be considered separately because overall G1 continuity is guaranteed from the Bezier control polyline.

  31. Solution of the problem

  32. Case 1 Boundary does not enter the control triangle B0B1B2.

  33. Case 2 Boundary enters the control triangle B0B1B2 by crossing line segment B0B2.

  34. Special case • To force the curve to be tangent to a straight line segment of the boundary at a given point. • To run the curve along a straight line segment of the boundary. • To force a cusp at a boundary point.

  35. Result

  36. Control polyline crossing the boundary

  37. Result

  38. Future work QuadraticCubic G1 G2

  39. Constrained interpolation with rational cubics D.S. Meek, B.H. Ong, D.J. Walton CAGD, 2003, 20: p253-275

  40. Rational cubic

  41. Result

  42. A smooth, obstacle-avoiding curve Z. Li, D.S. Meek, D.J. Walton Computers & Graphics,2006,30,581-587

  43. About the authors (李重) • 1994.9-1998.6杭州大学(现浙江大学)计算机系,获学士学位. • 1998.9-2003.6浙江大学数学系计算数学专业,硕博连读,获博士学位,指导老师王兴华教授和韩丹夫教授. • 2003.6-2003.12加拿大 Manitoba 大学计算机系,学术访问,指导老师D.S. Meek 教授和D. J. alton教授. • 2004.12-2006.12上海交通大学计算机系,博士后研究,指导老师马利庄教授. • 研究领域和方向:计算机图形学,数字媒体设计,图像处理,计算机仿真等.

  44. Work of this paper • It is shown that the spline curve can designed to be as close as desired to the guiding polyline and thus a spline curve that avoids the obstacles can always be found.

  45. Work of this paper

  46. Previous work and comparison • Constrained interpolation with rational cubics. (References [5]) • Comparison: The method proposed here solves a similar problem with simpler curves (polynomials instead of rationals) that are easier to find (quadratic instead of quartic equations).

  47. Previous work and comparison • References [6], [7], [8].(Useing clothoids to smooth polyline paths.) • Comparison: A disadvantage of clothoids is that the clothoid is a transcendental curve and thus cannot easily be incorporated into standard graphics packages as a NURBS curve.

  48. Previous work and comparison • References [1], [11], [12] • Comparison: Ref. [1] uses a divide-and-conquer iteration, Ref. [11], [12] requires linear programming, both of them require much more computation than the method proposed here.

  49. Curves and Surfaces for CAGD: A Practical Guide (5e) Gerald Farin Morgan Kaufmann, 2001

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