1. Cristina Neacsu, Karen Daniels University of Massachusetts Lowell
Translational Covering of
Closed Planar Cubic B-Spline Curves
Cristina Neacsu, Karen Daniels
University of Massachusetts Lowell

2. B-Spline Curves & Surfaces
Used in geometric modeling, graphics
Gerald Farin. Curves and Surfaces for Computer Aided Geometric Design, Academic Press, 1988
Theo Pavlidis. Algorithms for Graphics and Image Processing, ComputerScience Press, 1982
papers/nasa_rg1/nasaf1.htm
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3. B-Spline Curves B-Splines are defined by control polygons
Periodic Cubic B-Spline Curves
Use a special set of cubic blending functions that have only local influence and each curve segment depends only on the 4 neighboring control points.
Current work assumption:
non-self-intersecting control polygons (at most one inflection point per curve segment).

4. Sample Application Areas
Sensors
Locate, Identify,
Track, Observe
Lethal Action
CAD
Sensor Coverage
Targeting

5. Spline Covering Problem
NP-Hard
Covering Items:
set of spline bounded regions B = {B1, …, Bj, …, Bm}
Target Item: spline bounded region A
Output: a solution such that
Sample A and B
Translated Bs cover A A B1 B2

6. Approach Combinatorial Covering Algorithm Original instance feasible
Construct
Inner polygonal approximation to B
Outer polygonal approximation to A
Combinatorial Covering Algorithm
feasible
infeasible
Original instance feasible
Construct Cover ?
Construct
Outer polygonal approximation to B
Inner polygonal approximation to A
Combinatorial Covering Algorithm
feasible
infeasible ?
Original instance infeasible

7. Previous Work - Covering
13 {3}
1 {1}
12 {3}
6 {1}
5 {3}
4 {2}
3 {2}
2 {1}
11 {2}
10 {2}
9 {3}
8 {1}
7 {3} P
Solving an apparel trim placement problem using a maximum cover problem approach (IIE Transactions 1999 Grinde, Daniels)
Translational polygon covering using intersection graphs (CCCG’01 Daniels, Inkulu)
A Combinatorial Maximum Cover Approach to 2D Translational Geometric Covering (CCCG’03 Daniels, Mathur, Grinde)
Rigid, 2D, Exact, Polygonal & Point, Translation

8. Previous Work - Splines
Convex hull property & variation diminishing property
[Gerald Farin. Curves and
Surfaces for CAD. Academic
Press 1988]
Trisection method
[Theo Pavlidis. Algorithms
for Graphics and Image
Processing. Computer Science
Press 1982]
Monotone partitioning of spline-bounded shapes [Daniels 92]
Estimate on the distance between a B-spline curve and its control polygon [D. Lutterkort and J. Peters. Linear Envelopes for Uniform B-Spline Curves. 2000] L M A B C D P1 P2 P3 P4 l1 l2

9. Curve Polygonal Approximation
Monotone partitioning of spline-bounded shapes [Daniels 92]
Piecewise-linear approximations using midpoints, inflection points, starting points for each curve segment, tangents at these points, and intersection between these tangents.

10. Polygonal Approximation Number of Vertices
The total number  of vertices of the polygonal approximation is:
where n is the number of control points of the control polygon, nN is the number of curve segments with an N-shaped control polyline, nV is the number of curve segments with a V-shaped control polyline, and nS is the number of curve segments with an S-shaped control polyline.

11. Polygonal Approximations
Monotone envelope N (shaded) and rectangle-based envelope H (boldfaced) for non-N-shaped control polyline
Monotone envelope N (shaded) and rectangle-based envelope H (boldfaced) for N-shaped control polyline
K. Daniels, R. D. Bergeron, G. Grinstein. Line-monotonic Partitioning of Planar Cubic B-Splines. Computer and Graphics 16, 1 (1992), 55:68.
D. Lutterkort and J. Peters. Linear Envelopes for Uniform B-Spline Curves. Curves and Surfaces Design: Saint-Malo P. J. Laurent, et al (Eds.). Vanderbilt University Press, 2000, p. 239:246.
D. Lutterkort and J. Peters. Tight Linear Envelopes for Splines. Numerische Mathematik 89, 4 (2001), 735:748.

12. Polygonal Approximation
Curve segment with non-N-shaped control polyline
Theorem: For a non-N-shaped control polyline, X1 and X2 cannot both be outside C.
C is the convex hull
of SP2P3E
H = {H1, H2},
where H1 is SX1M and
H2 is MX2E
Corollary: For a non-N-shaped control polyline, if X1 or X2 is outside C, then H exits C through P2P22.

13. Polygonal Approximation
Curve Segment with nondegenerate-N-shaped control polyline
H = N
Theorem: For a nondegenerate N-shaped control polyline, X1 and X2 cannot both be outside C.
Corollary: For a nondegenerate-N-shaped control polyline, H can exit C only through SP3 if X2 is outside C, or through EP2, if X1 is outside C.

14. Combinatorial Covering Procedure Lagrangian-Cover
Triangles:
Groups:
Qj’s:
Group choices:
G1 for Q1
G2 for Q2
and group k has been assigned to some Qj T1 G1 Q1 T2 T3 G2 T4 Q2 T5 G3
Grinde, Daniels. Solving an apparel trim placement problem using a maximum cover problem approach, IIE Transactions, 1999
Daniels, Mathur, Grinde. A Combinatorial Maximum Cover Approach to 2D Translational Geometric Covering, CCCG’03

15. Lagrangian Relaxation1 2
Lagrange Multipliers 1 2 3
maximize
l>=0 and subtracting term < 0
removing constraints 4
minimize
Lower bounds come from any feasible solution to 3
Lagrangian Relaxation LR(l) 4
Lagrangian Dual: min LR(l), subject to l >= 0
Grinde, Daniels. Solving an apparel trim placement problem using a maximum cover problem approach, IIE Transactions, 1999
Daniels, Mathur, Grinde. A Combinatorial Maximum Cover Approach to 2D Translational Geometric Covering, CCCG’03

16. Lagrangian Relaxation
Lagrangian Relaxation LR(l)
LR(l) is separable
SP1
SP2
Solve: if (1-li) >=0
then set ti=1
else set ti=0
Solve: Redistribute:
Solve j sub-subproblems
- compute gkj coefficients
- set to 1 gkj with largest coefficient
For candidate l values, solve SP1, SP2

17. Triangle Subdivision Invariant: T is a triangulation of P I II P T T’
uncovered triangle II

18. Results C++, LEDA library 450 MHz SPARC Ultra # m n l r A B k Time 1.
AGENDA
m = # of Bs
n = # of points in A’s control polygon
l = # of points in A’s polygonal approx.
r = total # of points in B’s polygonal app.
C = convex
N = non-convex
k = # points in subdivision # m n l r A B k
Time 1. 2 6 12 26 C 7 2. 18 N 17 3. 3 4 8 24 21 4. 15 33 51
1573 5. 10 22
N&C
452 6. 16 56 31
1394
C++, LEDA library
450 MHz SPARC Ultra 6. 4.

19. Results # m n l r A B k Time 7. 4 15 36 40 N C 1787 8. 12 28 68 31
AGENDA
m = # of Bs
n = # of points in A’s control polygon
l = # of points in A’s polygonal approx.
r = total # of points in B’s polygonal app.
C = convex
N = non-convex
k = # points in subdivision # m n l r A B k
Time 7. 4 15 36 40 N C
1787 8. 12 28 68 31
2187 9. 5 17 37 57
N&C
1984
10. 16 48
2937
11. 13 27 67
1084
12. 6 35 72
1846* 8.
12.
11.

20. Future Work Remove constraint of single inflection point Cusps, loops
Construct
Inner polygonal approximation to B
Outer polygonal approximation to A
Combinatorial Covering Algorithm
feasible
infeasible
Original instance feasible
Construct Cover ?
Outer polygonal approximation to B
Inner polygonal approximation to A
Original instance infeasible

21. Future Work – continued
Generalize further the covering problem
Rigid, 2D, Exact, Spline, Translation
this work:
flexible
approximate 3D
rotation
future work:
If I do not find an exact covering, can I modify Qj’s control polygons such that they can cover P?

22. Thank you!

23. Backup – Polygonal Covering
Input:
Covering Items: Q = {Q1, Q2 , ... , Qm}
Target Items: P = {P1, P2 , ... , Ps}
Subgroup G of
Output: a solution g = {g 1, …,g j , ... , g m}, , such that
Sample P and Q
Translated Q Covers P P2 P1 Q1 Q2 Q3
Rigid, 2D, Exact, Polygonal & Point, Translation
NP-hard

24. Backup
Subdivision

25. Backup
Number of points in polygonal approximation

26. Backup
Each segment of a cubic B-spline curve is influenced by only 4 control points, and conversely each control point influences only 4 curve segments. pi 1 i
pi+2
pi-1
pi+1
p1* p1 p2 p0 p3 p5 p4

27. Backup For segment i of a periodic cubic B-spline curve we have: with
for open curves
and where Fj,4 are called blending functions and they are: F 1 u
F3,4
F2,4 1
F3,4
F2,4
F1,4
F4,4
F4,4
F1,4 u
u=1

28. Periodic Cubic B-Spline Curves
Backup
for open curves
Periodic cubic B-spline curves are well suited to produce closed curves. In matrix notation: p1 p2 1 2
for closed curves p0 p3 3 5 4
Periodic Cubic B-Spline Curves p5 p4

29. Backup
C0 continuity ensures that there are no gaps or breaks between a curve’s beginning and ending points.
C1 continuity between two curve segments requires a common tangent line at their joining point; C1 continuity also requires C0 continuity.
C2 continuity requires that the two curves possess equal curvature at their joint.

30. Backup Variables: Parameters:
Lagrangian Relaxation is used as a heuristic since optimal value of Lagrangian Dual is no better than Linear Programming relaxation.
exactly 1 group chosen for each Qj
Brought into objective function for Lagrangian Relaxation
value of 1 contributed to objective function for each triangle covered by a Qj, where that triangle is in a group chosen for that Qj
Variables:
Parameters: