Cristina Neacsu, Karen Daniels

1
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
www.aerohydro.com/products/rg/ papers/nasa_rg1/nas
af1.htm
www.cs.columbia.edu/ laza/heart3D/
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

• Covering Items
• set of spline bounded regions B B1, , Bj, ,
  Bm
• Target Item spline bounded region A
• Output a solution
  such that

Translated Bs cover A
Sample A and B
B2
A
B1
6
Approach
7
Previous Work - Covering

• Solving an apparel trim placement problem using
  a maximum cover problem approach (IIE
  Transactions 1999 Grinde, Daniels)
• Translational polygon covering using
  intersection graphs (CCCG01 Daniels, Inkulu)
• A Combinatorial Maximum Cover Approach to 2D
  Translational Geometric Covering (CCCG03
  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

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 ApproximationNumber 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),
  5568.
• D. Lutterkort and J. Peters. Linear Envelopes
  for Uniform B-Spline Curves. Curves and Surfaces
  Design Saint-Malo 1999. P. J. Laurent, et al
  (Eds.). Vanderbilt University Press, 2000, p.
  239246.
• D. Lutterkort and J. Peters. Tight Linear
  Envelopes for Splines. Numerische Mathematik 89,
  4 (2001), 735748.

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
Qjs
Triangles
Groups
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, CCCG03

15
Lagrangian Relaxation
1
Lagrange Multipliers
2
3
Lagrangian Relaxation LR(l)
Lagrangian Dual min LR(l), subject to l gt 0
4

• 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, CCCG03

16
Lagrangian Relaxation
Lagrangian Relaxation LR(l)
LR(l) is separable
SP1
SP2
Solve if (1-li) gt0 then set ti1 else set ti0
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
T
T
I
T
uncovered triangle
II
18
Results
AGENDA m of Bs n of points in As control
polygon l of points in As polygonal
approx. r total of points in Bs 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 C 0 7
2. 2 12 26 18 N C 0 17
3. 3 4 8 24 C C 8 21
4. 3 15 33 51 N N 33 1573
5. 3 10 22 33 N NC 22 452
6. 4 8 16 56 C C 31 1394

6.
4.
19
Results
AGENDA m of Bs n of points in As control
polygon l of points in As polygonal
approx. r total of points in Bs 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 36 1787
8. 4 12 28 68 N N 31 2187
9. 5 17 37 57 N NC 37 1984
10. 5 16 36 48 N C 36 2937
11. 5 13 27 67 N N 27 1084
12. 6 15 35 72 N C 37 1846

12.
8.
11.
20
Future Work

• Remove constraint of single inflection point
• Cusps, loops

21
Future Work continued

• Generalize further the covering problem

If I do not find an exact covering, can I modify
Qjs 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

Translated Q Covers P
Sample P and Q
P2
P1
Q1
Q2
Q3
NP-hard
Rigid, 2D, Exact, Polygonal Point, Translation
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.
p1
p1
p2
p3
p0
p4
p5
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
F3,4
F2,4
1
F2,4
F3,4
F1,4
F4,4
F1,4
F4,4
u
0
?u1
28
Backup
for open curves
Periodic cubic B-spline curves are well suited to
produce closed curves. In matrix notation
p1
p2
1
2
0
for closed curves
p3
p0
3
5
Periodic Cubic B-Spline Curves
4
p4
p5
29
Backup

• C0 continuity ensures that there are no gaps
  or breaks between a curves 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
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
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