1. 1 An Exact, Complete and Efficient Computation of Arrangements of Bézier Curves Iddo Hanniel – Technion, Haifa Ron Wein – Tel-Aviv University

2. 2 Planar Arrangements Given a collection C of curves in the plane, the arrangement A ( C ) is the subdivision of the plane into vertices, edges and faces induced by the curves in C.

3. 3 Applications of Arrangements  Robot Motion Planning.  Geographic Information Systems.  Assembly Planning.  Many other geometric problems can be mapped (through duality or otherwise) to problems on arrangements.  CAD – sketches, Boolean operations.

4. 4 Robustness Problems in Geometric Computing  Robustness in geometric computing is a difficult and well-known problem [Hoffman 89, 01][Yap 04].  Machine-precision arithmetic may yield incorrect results.

5. 5 The Exact Geometric Computation Paradigm  The Exact Geometric Computation Paradigm [Yap and Dubé 95] was successful in robustly solving many computational geometry problems.  It requires that all geometric operations within the algorithm are preformed correctly.  Makes use of exact arithmetic software libraries for multiple precision integers and rational numbers, extended floats [GMP], and algebraic numbers [CORE, LEDA].

6. 6 Our Contribution We present a software package for computing arrangements of Bézier curves that is:  Exact: Employs the exact computation paradigm (uses exact arithmetic).  Complete: Handles all degenerate cases.  Efficient: Makes use of geometric properties of Bézier curves for speed-up of the computations. Publicly available in the new CGAL 3.3 release.

7. 7 Previous Work Previous implementations of arrangements of algebraic curves are either:  Not exact: Most current solid modeling systems use non-robust floating point arithmetic.  Not complete: Do not handle all degenerate cases (MAPC [Keyser et al. 2000]). Handle only a specific family of curves (conics [Wein 2002], cubics [Eigenwillig et al. 2004]).  Not efficient: Computer algebra systems can be used for computing exact arrangements but are too slow.

8. 8 Preliminaries  CGAL’s Arrangement package [ www.cgal.org/, www.cs.tau.ac.il/CGAL/ ].  Exact rational and algebraic number types [CORE, LEDA].  Bézier curve properties for faster computation. Our work is based on three “building blocks”:

9. 9 CGAL’s Arrangement Package  The arrangement may either be constructed incrementally, or using a sweep-line algorithm.  It constructs the underlying data-structure and enables efficient traversal over the vertices, edges and faces of the arrangement.  Point-location queries are supported. CGAL’s arrangement package handles the topology of planar arrangements and separates it from the geometry of the curves:

10. 10 The package is designed to handle all sorts of degenerate inputs, such as: Robustness Issues Vertical segments Overlapping curves Several curves with common intersection Assuming correctness of the geometric methods.

11. 11 The Geometric Traits Functions  Comparing two points by their x-coordinates.  Divide a curve to x-monotone sub-curves.  Determine if a given point is above, below or on a sub-curve.  Compare the slope of two sub-curves next to their intersection.  Compute the intersection points (or overlaps) of two curves. The users should supply a traits class that handles the geometry of the curves. The geometric functions that need to be implemented by the traits class are:

12. 12 Bézier Curve Properties We Use  Convex hull property: B is entirely contained in the control polygon p 0...p n.  Variation diminishing property: the number of intersections of B with a line l is not greater than the intersection of l with the control polygon.  Derivative of a Bézier curve: is also a Bézier curve with control points. p0p0 p3p3 p1p1 p2p2 p4p4 p5p5 p6p6

13. 13 Bézier Curve Properties We Use The de Casteljau subdivision algorithm: P 2 (1) P 0 (3) p0p0 P 0 (1) p1p1 P 0 (2) P 1 (1) P 1 (2) p2p2 p0p0

14. 14 Exact Algebraic Number Types  Constructed from integers, rational numbers or floating point numbers.  Support exact {+, -, *, /, √..} operations.  Support the root_of_polynomial operator, but only for polynomials with rational coefficients.  Support exact comparisons!  Algebraic number types: CORE::Expr (www.cs.nyu.edu/exact/). LEDA::real (www.algorithmic-solutions.com/).

15. 15 Representing Geometric Types There are three geometric types in the traits class:  Point_2.  Curve_2 – represented by the input control points p 0 …p n. The representation of the polynomials X(t) and Y(t) in standard monomial basis is computed only when needed (lazy evaluation).  X_monotone_curve_2 – represented by its originating Curve_2 and its two endpoints.

16. 16 Representing Point_2  We store a reference to the originating curves of the point coupled with a corresponding small control polygon bounding the point.  The bounding polygons induce a rational bounding box on the point and a rational bounding interval on the parameter of the point in the originating curve.  The exact algebraic intersection/x-tangency parameters (and corresponding (x,y) coordinates) are computed only if necessary. Originating curves Algebraic coords Rational BBox

17. 17 Computing Intersection Points Exactly  Computing exactly – the roots of the polynomial system: X 1 (t)-X 2 (s)=0 Y 1 (t)-Y 2 (s)=0  Can be computed exactly as the roots of the resultants: Res s (X 1 (t)-X 2 (s), Y 1 (t)-Y 2 (s)) Res t (X 1 (t)-X 2 (s), Y 1 (t)-Y 2 (s))  Pairing (t,s) pairs and computing the corresponding (x,y) coordinates by assigning in original curve. Problem: the high degree of the resultants (d 1  d 2 ). Thus, we wish to avoid exact computation when possible.

18. 18 Isolating Intersection Points – Geometric Filtering  Ensuring there is at most one intersection – the tangent bounding cones of the curves are disjoint.  Ensuring there is at least one intersection – the skewed bounding boxes intersect fully, with endpoints on opposite sides of the intersection. B1B1 B2B2 C(B 1 )

19. 19 Comparing Points and Slopes  Comparing x-coordinates of two points: Exact – comparison of algebraic coordinates (we wish to prevent this whenever possible). Geometric filtering – compare the rational bounding boxes, refine the bounding polygons (and hence the bounding boxes) if needed.  Comparing slope of curves to the right of their intersection point: Exact – compare the derivative at the algebraic coordinates (we wish to prevent this whenever possible). Geometric filtering – if the intersection point is isolated then the tangent cones are disjoint – compare any two tangents.

20. 20 Determining Point-Curve Position  If we know the point is not on the curve we can subdivide until the bounding boxes are disjoint.  How do we determine if a point is on a curve? Combinatorial filtering – if the curve is one of the point’s originating curves we are done. Geometric filtering – compare bounding boxes and do some refining (in most cases this is enough to verify the point is not on the curve). Exact – if point p=(x 0,y 0 ) is rational, evaluate the polynomial Y(t 0 ) at the root t 0 of x 0 =X(t) (which is a polynomial with rational coordinates). If point p is algebraic, intersect the curve with one of p’s originating curves and compare intersection parameter.

21. 21 Experimental Results Examples of Degeneracy 4 curves intersecting at a single point (~0.1 sec) Intersection with self intersection (~0.1 sec)

22. 22 Experimental Results – Random Parabolas Input fileNumber of vertices IRIT floating point traits Exact Conics traits Exact Bézier traits Filtered Bézier traits Parabolas_10 570.00720.1490.3320.240 Parabolas_20 1250.01257.4071.0600.632 Parabolas_30 2380.023126.0242.3801.576 Parabolas_40 422N/A232.3154.3853.752 Arrangement construction time for different inputs

23. 23 Experimental Results – Higher Degrees Input fileNumber of vertices IRIT floating point traits Exact Bézier traits Filtered Bézier traits Degree_3 400.0041.2200.112 Degree_4 520.0167.9160.192 Degree_5 500.02051.0790.244

24. 24 Experimental Results - BOPS Char. #1 n1n1 Char. #2 n2n2 Exact traitsFiltered traits T init T comp T init T comp A23V200.10517.40.0280.154 H48O160.195174.90.0320.325 S35Z150.382367.70.0300.502 m52g430.402543.60.0720.532 g43s290.360737.80.0690.734

25. 25 Future Work  Test other representations and number types for the geometric filtering (interval arithmetic, extended floats).  Further research of the algorithms under the exact computation paradigm and try to incorporate separation bounds.  Extend the implementation to rational Bézier curves.  Extend the implementation to B-spline and NURBS curves – either by repeated knot insertion or using their similar properties.

26. 26 Thank you! This research has been supported by: European FP6 NoE grant 506766 (AIM@SHAPE). Israel Science Foundation (grant No. 857/04). IST Programme of the EU as Shared-cost RTD (FET Open) Project Contract No IST-006413 (ACS - Algorithms for Complex Shapes). Israel Science Foundation (grant no. 236/06). The Hermann Minkowski-Minerva Center for Geometry at TAU.