Minkowski Sums and Offset Polygons

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Minkowski Sums and Offset Polygons

• Ron Wein

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Computing Minkowski Sums
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Planar Minkowski Sums

• Given two sets A and B in the plane, their
  Minkowski sum, denoted A ? B, is
• A ? B a b a ? A, b ? B

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The Sum Complexity (I)

• We are given two polygons P and Q with m and n
  vertices respectively. If both polygons are
  convex, the complexity of their sum is m
  n, and we can compute it in ?(m n) time
  using a very simple procedure.

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The Sum Complexity (II)

• If only one of the polygons is convex, the
  complexity of their sum is ?(mn).
• If both polygons are non-convex, the complexity
  of their sum is ?(m2n2).

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The Decomposition Method

• The prevailing method for computing the sum of
  two non- convex polygons Decompose P and Q
  into convex sub- polygons, compute the
  pair-wise sums of the sub-polygons and
  obtain the union of these sums.

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Convex Decomposition Schemes (I)
Naïve triangulation.
Optimal max-degree triangulation minimizing the
maximum degree.
Optimal sum-degrees triangulation minimizing the
sum of squared vertex degrees.
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Convex Decomposition Schemes (II)
Greedy convex decomposition add diagonals until
eliminating all reflex vertices.
Optimal polygon decomposition minimizing the
number of convex sub-polygons. Takes O(r2n log n)
time.
Optimal sum-degrees decomposition minimizing the
sum of squared vertex degrees.
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Convex Decomposition Schemes (III)(with Steiner
Points)
Slab decomposition add vertical segments from
each reflex vertex.
Angle bisection add an angle bisector for each
reflex vertex.
KD-decomposition adding vertical and horizontal
segments.
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Computing the Union

• Given a set of polygons S1, , SM with
  counterclockwise (positive) orientation,
  construct the arrangement of their edges.
• Set N(fu) 0 for the unbounded face fu. Then
  compute N(f ) for each other face using a
  simple BFS traversal.
• All faces with N(f ) gt 0 contribute to the
  union.

(1)
(2)
(0)
(0)
(1)
(1)
(2)
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Convolution of Planar Tracings(Guibas, Ramshaw
and Stolfi, 1983)

• A planar tracing T comprises a continuous set
  of points. Each point t ? T is associated
  with a direction dir(t).
• The convolution of two planar tracings S and T,
  denoted S ? T, comprises the pair-wise sum
  of all points s ? S and t ? T such that
  dir(s) dir(t).

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Convolution of Polygons (I)

• A polygonal tracing P consists of moves (going
  along an edge in a fixed direction), and
  turns (rotating at a vertex).
• For a vertex p of P, dir(p) is a continuous
  range of directions. The convolution P ? Q
  therefore contains the sum of each edge of
  the polygonal tracing Q, whose direction is in
  dir(p), with p.

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Convolution of Polygons (II)

• At the worst case, the convolution of two
  polygons P ? Q consists of ?(mn) line
  segments.
• Guibas and Seidel (1987) gave an
  output-sensitive algorithm for computing
  the convolution segments in O(m n K) time.
• The convolution segments form closed cycles.
  The Minkowski sum P ? Q contains all points
  whose winding number with respect to any of
  the convolution cycles is positive.

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The Case of a Single Cycle
Q
P
2
0
0
2
2
1
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The Case of Multiple Cycles

• We compute the first cycle, starting from the
  two bottommost vertices in P and Q.
• If P or Q are convex, we are done.
  Otherwise, while it is possible to locate two
  vertices p ? P and q ? Q that should be in
  the convolution, construct an additional
  convolution cycle.

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The Minkowski-Sum Package of CGAL
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The CGAL Arrangement Package

• Supports the construction of maintenance of 2D
  arrangements planar subdivisions induced
  by a set of planes.
• Handles a variety of curves. Each family of
  curves (line segments, poly-lines, circular
  arcs, conic arcs, etc.) is handled by a
  geometric traits-class.

• Handles degenerate cases robustly and
  accurately, relying on the exact
  computation paradigm.

Tangencypoints
Common intersectionpoints
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The Decomposition Method

• The Minkowski-sum package re-implements the
  robust algorithms for sum-computations
  using the convex polygon- decomposition
  method (Agarwal, Flato and Halperin, 2000).

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The Convolution Method

• The first software that robustly implements the
  convolution algorithms for polygons.
• Uses the same infrastructure for multi-way
  polygon union, used by the decomposition
  method. The same algorithm also works for
  the case of closed convolution cycles.

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Low-Dimensional Features

• The Minkowski sum of two polygons may contain
  low- dimensional features (isolated vertices
  and antennas).
• The package can treat them as follows
• discard them (output the regularized sum),
  or
• give access to them through the underlying
  arrangement.

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Experimental Input Sets (I)
stars
chain
fork
comb
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Experimental Input Sets (II)
cavity
random
knife
country
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Experimental Results Decomposition
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Experimental Results Convolution
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Polygon Offsetting
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Polygon Offsetting using the Convolution Method
Br
P
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Computing the convolution cycle
Computing the induced arrangement and the winding
numbers
0
0
2
2
1
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Related Traits Classes

• Computing the sum of two polygons (objects of
  the type Polygon_2ltKernelgt) is of course
  performed using the Arr_segment_traits_2ltKernelgt
  class and using exact rational arithmetic.

• In case of offsetting we should consider the
  following traits classes
• Arr_circle_segment_traits_2 handles line
  segments and circular arcs using exact rational
  arithmetic.
• Arr_conic_traits_2 handles bounded conic arcs
  using exact algebraic numbers (based on CORE).

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Offsetting a Polygon Edge (I)
? ? - 90º
p2 (x2, y2)
p1 (x1, y1)
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Offsetting a Polygon Edge (II)

• If the line supporting p1p2 is ax by c
  0, then the line supporting q1q2 is ax by
  (c lr) 0.

• Problem In the general case, offset arcs are
  not supported by rational lines!

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Our Approximation Scheme
p2
p1
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Computing Rational Angles (I)

• Using the half-angle formulae we have

• Observation If ? is rational, then sin(?) 2?
  / (1 ? 2) andcos(?) (1 - ? 2) / (1 ? 2) are
  also rational.

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Computing Rational Angles (II)
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The Approximation Quality

• Lemma For any polygon edge connecting (x1, y1)
  and (x2, y2) whose length is l and any ? gt 0, if
  we take an approximation of the edge length that
  satisfiesthen the approximation erroris
  bounded by ?.

q
q2
q1
p2
p1
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Running Times
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(Pentium IV 3 GHz, in milliseconds)
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Thank you!