namespace CGAL {
/*!

\mainpage User Manual
\anchor Chapter_2D_Minkowski_Sums

\anchor chapterMinkowskisum2

\cgalAutoToc
\authors Ron Wein, Alon Baram, Efi Fogel, Eyal Flato, Michael Hemmer, and Sebastian Morr

\section mink_secintro Introduction

Given two sets \f$ A,B \in \mathbb{R}^d\f$, their <I>Minkowski sum</I>,
denoted by \f$ A \oplus B\f$, is their point-wise sum, namely the set
\f$ \left\{ a + b ~|~ a \in A, b \in B \right\}\f$. Minkowski sums are
used in many applications, such as motion planning and computer-aided
design and manufacturing. This package contains functions that compute
the planar Minkowski sums of two polygons. (Here, \f$ A\f$ and \f$ B\f$
are two closed polygons in \f$ \mathbb{R}^2\f$, which may have holes; see
Chapter \ref Chapter_2D_Regularized_Boolean_Set-Operations
"2D Regularized Boolean Set-Operations" for the precise definition of valid
polygons), and the planar Minkowski sum of a simple polygon and a
disc---an operation also referred to as <I>offsetting</I> or <I>dilating</I>
a polygon.\cgalFootnote{The family of valid types of summands is slightly
broader for certain operations, e.g., a degenerate polygon consisting of
line segments is a valid operand for the approximate-offsetting operation.}
This package, like the \ref Chapter_2D_Regularized_Boolean_Set-Operations
"2D Regularized Boolean Set-Operations" package, is implemented on top of
the arrangement infrastructure provided
by the \ref chapterArrangement_on_surface_2 "2D Arrangements" package.
The two packages are integrated well to
allow mixed operations. For example, it is possible to apply Boolean set
operations on objects that are the result of Minkowski sum
computations.\cgalFootnote{The operands of the Minkowski sum operations
supported by this package must be (linear) polygons, as opposed to the
operands of the Boolean set operations supported by the
\ref Chapter_2D_Regularized_Boolean_Set-Operations
"2D Regularized Boolean Set-Operations" package.
The latter belong to the broader family of general polygons.}

\section mink_secsum_poly Computing the Minkowski Sum of Two Polygons

\cgalFigureBegin{mink_figconvex,ms_convex.png}
\cgalFigureEnd

Computing the Minkowski sum of two convex polygons \f$ P\f$ and \f$ Q\f$ with
\f$ m\f$ and \f$ n\f$ vertices, respectively, is rather easy. Observe that
\f$ P \oplus Q\f$ is a convex polygon bounded by copies of the \f$ m + n\f$
edges ordered according to the angle they form with the \f$ x\f$-axis. As the
two input polygons are convex, their edges are already sorted by the angle
they form with the \f$ x\f$-axis; see the figure above.
The Minkowski sum can therefore be computed using an operation similar to the
merge step of the merge-sort algorithm\cgalFootnote{See, for example,
<a href="https://en.wikipedia.org/wiki/Merge_sort">
https://en.wikipedia.org/wiki/Merge_sort</a>.} in \cgalBigO{m + n} time,
starting from the two bottommost vertices in \f$ P\f$ and in \f$ Q\f$ and
merging the ordered list of edges.

\cgalFigureBegin{mink_figonecyc,ms_convex_polygon.png,ms_concave_polygon.png,ms_convolution.png}
The convolution of a convex polygon and a non-convex polygon. The convolution
consists of a single self-intersecting cycle, drawn as a sequence of directed
line segments. Each face of the arrangement induced by the segments forming
the cycle contains its winding number. The Minkowski sum of the two polygons
is shaded. Dotted edges are not part of the reduced convolution.
\cgalFigureEnd

If the polygons are not convex, you can utilize either the
<I>Decomposition</I> or the <I>Convolution</I> approaches described
below.  Applications of some of the operations in this package are
restricted to polygons that do not contain holes. (Resulting sums may
contain holes though.)

<DL>
<DT><B>Decomposition:</B><DD>
We decompose \f$ P\f$ and \f$ Q\f$ into convex sub-polygons. Namely, we
obtain two sets of convex polygons
\f$ P_1, \ldots, P_k\f$ and \f$ Q_1, \ldots, Q_\ell\f$, such that
\f$ \bigcup_{i = 1}^{k}{P_i} = P\f$ and \f$ \bigcup_{i = j}^{\ell}{Q_j} = Q\f$.
We then calculate the pairwise sums \f$ S_{ij} = P_i \oplus Q_j\f$ using the
simple procedure described above, and finally compute the union
\f$ P \oplus Q = \bigcup_{ij}{S_{ij}}\f$;
see \ref ref_bso_union "Union Functions".

This approach relies on a successful decomposition of the input
polygons into convex pieces, and its performance depends on the
quality and performance of the decomposition. Some of the supplied
decomposition methods do not handle polygons that contain holes.

<DT><B>Convolution:</B><DD>
Let \f$ P = \left( p_0, \ldots, p_{m-1} \right)\f$ and
\f$ Q = \left(q_0, \ldots, q_{n-1} \right)\f$ denote the vertices of
the input polygons. We assume that both \f$ P\f$ and \f$ Q\f$ have
positive orientations (i.e., their boundaries wind in a
counterclockwise order around their interiors). The
<I>convolution</I> of these two polygons \cgalCite{grs-kfcg-83},
denoted \f$ P * Q\f$, is a collection of line segments of the form
\f$ [p_i + q_j, p_{i+1} + q_j]\f$,\cgalFootnote{Throughout this
chapter, we increment or decrement an index of a vertex modulo the
size of the polygon.} where the vector \f$ {\mathbf{p_i p_{i+1}}}\f$
lies between \f$ {\mathbf{q_{j-1} q_j}}\f$ and
\f$ {\mathbf{q_j q_{j+1}}}\f$,\cgalFootnote{We say that a vector
\f$ {\mathbf v}\f$ lies between two vectors \f$ {\mathbf u}\f$ and
\f$ {\mathbf w}\f$ if we reach \f$ {\mathbf v}\f$ strictly before
reaching \f$ {\mathbf w}\f$ if we move all three vectors to the
origin and rotate \f$ {\mathbf u}\f$ counterclockwise. Note that this
also covers the case where \f$ {\mathbf u}\f$ has the same direction
as \f$ {\mathbf v}\f$.} and, symmetrically, of segments of the form
\f$ [p_i + q_j, p_i + q_{j+1}]\f$, where the vector \f$ {\mathbf{q_j
q_{j+1}}}\f$ lies between \f$ {\mathbf{p_{i-1} p_i}}\f$ and \f$
{\mathbf{p_i p_{i+1}}}\f$.

The segments of the convolution form a number of closed (not
necessarily simple) polygonal curves called <I>convolution cycles</I>.
The Minkowski sum \f$ P \oplus Q\f$ is the set of points
having a non-zero winding number with respect to the cycles of
\f$ P * Q\f$.\cgalFootnote{Informally speaking, the winding number
of a point \f$ p \in\mathbb{R}^2\f$ with respect to some planar curve
\f$ \gamma\f$ is an integer number counting how many times does
\f$ \gamma\f$ wind in a counterclockwise direction around \f$ p\f$.}
See \cgalFigureRef{mink_figonecyc} for an illustration.

We construct the arrangement induced by the convolution cycles of
\f$P \f$ and \f$Q \f$, then compute the winding numbers of the cells
of the arrangement. Finally, we extract the Minkowski sum from the
arrangement. This variant is referred to as the full-convolution method.

A segment \f$[p_i + q_j, p_{i+1} + q_j] \f$ (resp.
\f$[p_i + q_j, p_i + q_{j+1}] \f$) cannot possibly contribute to the
boundary of the Minkowski sum if \f$q_j \f$ (resp. \f$p_i \f$) is a
reflex vertex (see dotted edges in \cgalFigureRef{mink_figonecyc}).
The remaining subset of convolution segments is called the
<I>reduced convolution</I> \cgalCite{cgal:bl-frmsurc-11}. This subset
is still a superset of the Minkowski sum boundary, but the winding
number property does not apply any longer as there are no closed
cycles anymore. We apply two different filters, which identify holes in
the Minkowski sum:
<OL>
<LI>A loop that is on the Minkowski sum boundary has to be orientable;
  that is, all normal directions of its edges have to point either
  inward or outward.</LI>
<LI>For any point \f$x\f$ inside of a hole of the Minkowski sum, the
  following condition holds: \f$(-P + x) \cap Q = \emptyset\f$. If, on
  the other hand, the inversed version of \f$P\f$, translated by
  \f$x\f$, overlaps \f$Q\f$, the loop is a <I>false</I> hole and is in
  the interior of the Minkowski sum.</LI>
</OL>

After applying these two filters, only those segments which constitute
the Minkowski sum boundary remain. This variant is referred to as the
reduced-convolution method.
</DL>

The number of segments in the convolution of two polygons is usually
smaller than the number of segments that constitute the boundaries of
the sub-sums \f$ S_{ij}\f$ when using the decomposition approach. As
both approaches construct the arrangement of these segments and
extract the sum from this arrangement, computing Minkowski sum using
the convolution approach usually generates a smaller intermediate
arrangement; hence it is faster and consumes less space. In most cases,
the reduced convolution method is faster than the full convolution
method, as the respective induced arrangement is usually much smaller.
However, in degenerate cases with many holes in the Minkowski sum, the
full convolution method can be preferable, as it avoids costly
intersection tests.

\subsection mink_ssec_hole_filter Filtering Out Holes

If a hole in one polygon is relatively small compared to the other
polygon, the hole is irrelevant for the computation of
\f$P\oplus Q \f$ \cgalCite{bfhhm-epsph-15}; It implies that the hole
can be removed (that is, filled up) before the main computation starts.
Theoretically, we can always fill up all the holes of at least one
polygon, transforming it into a simple polygon, and still obtain
exactly the same Minkowski sum. Practically, we remove all holes in one
polygon whose bounding boxes are, in \f$x \f$- or \f$y \f$-direction,
smaller than, or as large as, the bounding box of the other polygon.
Obliterating holes in the input summands speeds up the computation of
Minkowski sums, regardless of the approach used to compute the
Minkowski sum.

\subsection mink_ssec_conv Computing Minkowski Sum using Convolutions

The function template \link minkowski_sum_2()
`minkowski_sum_2(P, Q)`\endlink accepts two polygons
\f$ P\f$ and \f$ Q\f$ and computes their Minkowski sum
\f$ S = P \oplus Q\f$ using the convolution approach.
The call \link minkowski_sum_2() `minkowski_sum_2(P, Q)`\endlink
defaults to the call \link minkowski_sum_by_reduced_convolution_2()
`minkowski_sum_by_reduced_convolution_2(P, Q)`\endlink, which applies
the reduced convolution aforementioned method. Explicitly call
\link minkowski_sum_by_full_convolution_2()
`minkowski_sum_by_full_convolution_2(P, Q)`\endlink to apply
the full convolution method. The types of the operands accepted by
the function \link minkowski_sum_by_full_convolution_2()
`minkowski_sum_by_full_convolution_2(P, Q)`\endlink are instances of
the \link Polygon_2 `Polygon_2`\endlink class template. The types of
operands accepted by the function \link
minkowski_sum_by_reduced_convolution_2()
`minkowski_sum_by_reduced_convolution_2(P, Q)`\endlink (and by the
function \link minkowski_sum_2() `minkowski_sum_2(P, Q)`\endlink)
are instances of either the \link Polygon_2 `Polygon_2`\endlink or
\link Polygon_with_holes_2 `Polygon_with_holes_2`\endlink class templates.
Even when the input polygons are restricted to be simple polygons, they
still may not be convex; thus, their Minkowski sum may not be simply
connected and may contain polygonal holes; see for example
\cgalFigureRef{mink_figonecyc}. The type of the returned object \f$ S \f$
is therefore an instance of the
\link Polygon_with_holes_2 `Polygon_with_holes_2`\endlink class template
in all cases. Recall that the outer boundary of \f$S \f$  is a polygon
that can be accessed using `S.outer_boundary()`, and its polygonal
holes are given by the range [`S.holes_begin()`, `S.holes_end()`) (where
\f$ S \f$ contains `S.number_of_holes()` holes in its interior).

\cgalFigureBegin{mink_figsum_tri_sqr,ms_sum_triangle_square.png}
The Minkowski sum of a triangle and a square, as computed by the example
program \ref Minkowski_sum_2/sum_triangle_square.cpp.
\cgalFigureEnd

The example program below constructs the Minkowski sum of a triangle and
a square, as depicted in \cgalFigureRef{mink_figsum_tri_sqr}. The result in
this case is a convex hexagon. This program, like other example programs
in this package, includes the header file `bops_linear.h`, which defines
the polygon types.

\cgalExample{Minkowski_sum_2/sum_triangle_square.cpp}

\cgalFigureBegin{mink_figsum_holes,ms_tight.png}
The Minkowski sum of two non-convex polygons \f$ P\f$ and \f$ Q\f$, as
computed by the example programs \ref Minkowski_sum_2/sum_with_holes.cpp
and \ref Minkowski_sum_2/sum_by_decomposition.cpp.
\cgalFigureEnd

The program below computes the Minkowski sum of two polygons that are
read from an input file. In this case the sum may contain holes. The
Minkowski sum, for example, of the polygons described in the default
input file `room_star.dat` is not simple and contains four holes,
as illustrated in \cgalFigureRef{mink_figsum_holes}.

\cgalExample{Minkowski_sum_2/sum_with_holes.cpp}

While in general the convolution approach to computing Minkowski sums
runs faster, we observed that when the proportion of reflex vertices
in both summands is large, the decomposition approach runs faster. In
the following we describe how to employ the decomposition-based
Minkowski sum procedure.

\subsection mink_ssec_decomp_strategies Decomposition Strategies

In order to compute Minkowski sums of two polygon \f$ P \f$ and
\f$ Q \f$ using the decomposition method, issue the call
\link minkowski_sum_2()
`minkowski_sum_2(P, Q, decompP, decompQ)`\endlink, where each of \f$P \f$
and \f$Q \f$ is either a simple polygon or a polygon with holes.
If \f$P \f$ is a simple polygon, `decompP` must be an object of
a type that models the concept `PolygonConvexDecomposition_2`.
If \f$P \f$ is a polygon with holes, them `decompP` is an object
of a type that models the concept
`PolygonWithHolesConvexDecomposition_2`, which refines the concept
`PolygonConvexDecomposition_2`. The same holds for \f$Q \f$.
The two concepts `PolygonConvexDecomposition_2` and
`PolygonWithHolesConvexDecomposition_2` refine a `Functor` concept
variant. Namely, they both require the provision of a function
operator (`operator()`). The function operator of the model of the
concept `PolygonConvexDecomposition_2` accepts a planar simple
polygon, while the function operator of the model of the concept
`PolygonWithHolesConvexDecomposition_2` accepts a planar polygon
with holes. Both return a range of convex polygons that represents
the convex decomposition of the input polygon. If the decomposition
strategy that decomposes \f$P \f$ is the same as the strategy that
decomposes \f$Q \f$, you can omit the forth argument, and issue the call
\link minkowski_sum_2() `minkowski_sum_2(P, Q, decomp)`\endlink,
where `decomp` is an object that represents the common strategy.
The class template `Polygon_nop_decomposition_2`, which models the
concept `PolygonConvexDecomposition_2`, is a trivial convex
decomposition strategy referred to as the <I>nop</I> strategy; it merely
passes the input polygon to the next stage intact; use it in cases you
know that the corresponding input polygon is convex to start with. If
both \f$P \f$ and \f$Q \f$ are known to be convex, you can issue the call
\link minkowski_sum_2() `minkowski_sum_2(P, Q, nop)`\endlink, where `nop`
is an object that represents the nop strategy.

The Minkowski-sum package includes four models of the concept
`PolygonConvexDecomposition_2` (besides the trivial model
`Polygon_nop_decomposition_2`) and two models of the refined concept
`PolygonWithHolesConvexDecomposition_2` as described below. The first
three are class templates that wrap the corresponding decomposition
functions included in the
\ref Chapter_2D_Polygon "Planar Polygon Partitioning" package.

<UL>
<LI>The `Optimal_convex_decomposition_2<Kernel>` class template uses
the dynamic-programming algorithm of Greene \cgalCite{g-dpcp-83} for
computing an optimal decomposition of a polygon into a minimal number
of convex sub-polygons. While this algorithm results in a small number
of convex polygons, it consumes rather many resources, as it runs in
\cgalBigO{n^4} time and \cgalBigO{n^3} space in the worst case, where
\f$ n \f$ is the number of vertices in the input polygon.

<LI>The `Hertel_Mehlhorn_convex_decomposition_2<Kernel>` class
template implements the approximation algorithm suggested by Hertel and
Mehlhorn \cgalCite{hm-ftsp-83}, which triangulates the input polygon
and then discards unnecessary triangulation edges. After triangulation
(carried out by the constrained-triangulation procedure of CGAL) the
algorithm runs in \cgalBigO{n} time and space, and guarantees that the
number of sub-polygons it generates is not more than four times the
optimum.

<LI>The `Greene_convex_decomposition_2<Kernel>` class template is an
implementation of Greene's approximation algorithm
\cgalCite{g-dpcp-83}, which computes a convex decomposition of the
polygon based on its partitioning into \f$ y\f$-monotone polygons.
This algorithm runs in \cgalBigO{n \log n} time and \cgalBigO{n} space,
and has the same guarantee on the quality of approximation as Hertel
and Mehlhorn's algorithm.

<LI>The `Small_side_angle_bisector_decomposition_2<Kernel>` class
template is an implementation of a decomposition algorithm introduced
in \cgalCite{cgal:afh-pdecm-02}. It is based on the angle-bisector
decomposition method suggested by Chazelle and Dobkin
\cgalCite{cd-ocd-85}, which runs in \cgalBigO{n^2} time. In addition,
it applies a heuristic by Flato that reduces the number of output
polygons in many common cases.  The convex decompositions that it
produces usually yield efficient running times for Minkowski sum
computations. It starts by examining each pair of reflex vertices in
the input polygon, such that the entire interior of the diagonal
connecting these vertices is contained in the polygon. Out of all
available pairs, the vertices \f$ p_i\f$ and \f$ p_j\f$ are selected,
such that the number of reflex vertices encountered when traversing
the boundary of the polygon from \f$ p_i\f$ to \f$ p_j\f$ in clockwise
order is minimal. The polygon is split by the diagonal \f$ p_i
p_j\f$. This process is repeated recursively on both resulting
sub-polygons. In case it is not possible to eliminate two reflex
vertices at once any more, each reflex vertex is eliminated by a
diagonal that is closest to the angle bisector emanating from this
vertex and having rational-coordinate endpoints on both sides.
</UL>

The following are two models of the refined concept
`PolygonWithHolesConvexDecomposition_2`. An instance of any one these
two types can be used to decompose a polygon with holes.
<UL>
<LI>The `Polygon_vertical_decomposition_2<Kernel>` class template
uses vertical decomposition to decompose the underlying arrangement;
see \ref chapterArrangement_on_surface_2 "2D Arrangements" package.

<LI>The `Polygon_triangulation_decomposition_2<Kernel>` class template
uses constrained triangulation to decompose the input polygons, which
may have holes, into triangles.
</UL>

The example below demonstrates the computation of the Minkowski sum of
the same input polygons used in
\ref Minkowski_sum_2/sum_with_holes.cpp (depicted in Figure
\cgalFigureRef{mink_figsum_holes}), using the small-side
angle-bisector decomposition strategy.

\cgalExample{Minkowski_sum_2/sum_by_decomposition.cpp}

\subsection mink_ssec_optimal_decomp Optimal Decomposition

Decomposition methods that handle polygons with holes are typically
more costly than decomposition methods that handle only simple
polygons. The hole filtration (see \ref mink_ssec_hole_filter) is
applied before the actual construction starts (be it convolution
based or decomposition based). The filtration may result with a
polygon that does not have holes, or even a convex polygon, but this
is unknown at the time of the call. To this end, we introduce the
overloaded function template \link minkowski_sum_by_decomposition_2()
`minkowski_sum_by_decomposition_2(P, Q, no_holes_decomp, with_holes_decomp)`\endlink,
where `no_holes_decomp` and `with_holes_decomp` are objects that model
the concepts `PolygonConvexDecomposition_2` and
`PolygonWithHolesConvexDecomposition_2`, respectively. If after the
application of the hole filtration \f$P\f$ remains a polygon with holes,
then the strategy represented by the object `with_holes_decomp` is
applied to it. If, however, \f$P\f$ turns into a polygon without holes,
then the strategy represented by the object `no_holes_decomp` is applied
to it, unless the result is a convex polygon, in which case the nop
strategy is applied. If \f$P\f$ is a polygon without holes to start with,
then only convexity is checked. (Checking whether the result is convex
inccurs a small overhead though.) The same holds for \f$Q\f$.

\section mink_secoffset Offsetting a Polygon

The operation of computing the Minkowski sum \f$ P \oplus B_r\f$ of a
polygon \f$ P\f$ with a disc \f$ b_r\f$ of radius \f$ r\f$ centered at
the origin is widely known as <I>offsetting</I> the polygon \f$ P\f$ by
a radius \f$ r\f$.

\cgalFigureAnchor{mink_figpgn_offset}
<center>
<table border=0>
<tr>
<td>
\image html ms_offset_convex.png
\image latex ms_offset_convex.png
</td>
<td>
\image html ms_offset_nonconvex.png
\image latex ms_offset_nonconvex.png
</td>
<td>
\image html ms_offset_convolution.png
\image latex ms_offset_convolution.png
</td>
<tr align="center"><td>(a)</td><td>(b)</td><td>(c)</td></tr>
</table>
</center>
\cgalFigureCaptionBegin{mink_figpgn_offset}
(a) The offset of a convex polygon.
(b) The offset of a non-convex polygon as computed by decomposing it into
    two convex sub-polygons.
(c) The offset of a non-convex polygon as computed using the convolution
    approach. The convolution cycle induces an arrangement with three faces,
    whose winding numbers are indicated.
\cgalFigureCaptionEnd

Let \f$ P \f$ be a simple polygon, and let \f$ p_0, \ldots, p_{n - 1} \f$
be the vertices of \f$ P \f$ oriented counterclockwise around the interior
of \f$ P \f$. If \f$ P\f$ is a convex polygon the offset is easily computed
by shifting each polygon edge by \f$ r\f$ away from the polygon, namely to
the right side of the edge. As a result we obtain a collection of
\f$ n\f$ disconnected <I>offset edges</I>. Each pair of adjacent offset
edges, induced by \f$ p_{i-1} p_i\f$ and \f$ p_i p_{i+1}\f$, are connected
by a circular arc of radius \f$ r\f$, whose supporting circle is centered
at \f$ p_i\f$. The angle that defines such a circular arc equals
\f$ 180^{\circ} - \angle (p_{i-1}, p_i, p_{i+1})\f$; see
\cgalFigureRef{mink_figpgn_offset} (a) for an illustration. The running
time of this simple process is naturally linear with respect to the size of
the polygon.

If \f$ P\f$ is not convex, its offset can be obtained by decomposing it
into convex sub-polygons \f$ P_1, \ldots P_m\f$ such that
\f$ \bigcup_{i=1}^{m}{P_i} = P\f$, computing the offset of each
sub-polygon, and finally calculating the union of these offsets
sub-polygons; see \cgalFigureRef{mink_figpgn_offset} (b). However, as
with the case of the Minkowski sum of a pair of polygons, it is also more
efficient to compute the <I>convolution cycle</I> of the polygon and the
disc \f$ B_r\f$,\cgalFootnote{As the disc is convex, it is guaranteed
that the convolution curve comprises a single cycle.} which can be
constructed by applying the process described in the previous
paragraph for convex polygons: The only difference is that a circular arc
induced by a reflex vertex \f$ p_i\f$ is defined by an angle
\f$ 180^{\circ} + \measuredangle p_{i-1} p_i p_{i+1}\f$; see
\cgalFigureRef{mink_figpgn_offset} (c) for an illustration. Recall that
the last step consists of computing the winding numbers of the faces
of the arrangement induced by the convolution cycle and discarding the
faces with zero winding numbers.

\subsection mink_ssec_approx_offset Approximating the Offset with a Guaranteed Error Bound

Let \f$ P \f$ be a counterclockwise-oriented simple polygon all
vertices of which  \f$ p_0, \ldots, p_{n-1} \f$ have rational coordinates,
i.e., for each vertex \f$ p_i = (x_i, y_i)\f$ we have
\f$ x_i, y_i \in {\mathbb Q}\f$. Consider the Minkowski sum of \f$ P \f$
with a disc of radius \f$ r\f$, where \f$ r\f$ is also a rational number.
The boundary of this sum is comprised line segments and circular arcs, where:
<UL>
<LI>Each circular arc is supported by a circle of radius \f$ r\f$ centered
at a polygon vertex \f$ p_i \f$. The equation of this circle,
\f$ (x - x_i)^2 + (y - y_i)^2 = r^2\f$, has only rational coefficients.
<LI>Each line segment is supported by a line parallel to a polygon
edge \f$ p_i p_{i+1}\f$ at distance \f$ r\f$ from this edge. Let
\f$ A, B, C \in {\mathbb Q}\f$ denote the coefficients of the equation
\f$ p_i p_{i+1}\f$ by \f$ Ax + By + C = 0 \f$ of the supporting line of
\f$ p_i p_{i+1} \f$.

The locus of all points that lie at distance \f$r \f$ from the line
\f$ Ax + By + C = 0 \f$ is given by:
\f[ \frac{(Ax + By + C)^2}{A^2 + B^2} = r^2 \ .\f]
Thus, the linear offset edges are segments of an algebraic
curve of degree \f$ 2 \f$ (a conic curve) with rational coefficients.
This curve is actually a pair of the parallel lines
\f$ Ax + By + (C \pm r \cdot \sqrt{A/B+1}) = 0 \f$. The offset edge
is supported by the line \f$ Ax + By + C' = 0 \f$, where
\f$ C' = C + r \cdot \sqrt{A/B+1} \f$ is in general <I>not</I> a
rational number. Therefore, the line segments that compose the offset
boundaries cannot be represented as segments of lines with rational
coefficients.
In Section \ref mink_ssec_exact_offset we use the line-pair representation
to construct the offset polygon in an exact manner using the traits class
for conic arcs.
</UL>

\cgalFigureBegin{mink_figapprox_offset,ms_approx_offset.png}
Approximating the offset edge \f$ o_1 o_2\f$ induced by the
polygon edge \f$ p_1 p_2\f$ by two line segments \f$ o'_1 q'\f$ and
\f$ q' o'_2\f$.
\cgalFigureEnd

The class-template `Gps_circle_segment_traits_2<Kernel>`, included in
the Boolean Set-Operations package, is used for representing generalized
polygons the edges of which are circular arcs or line segments, and for
applying Boolean set operations (e.g., intersection and union) on such
generalized polygons. When it is instantiated, the template parameter
'Kernel' must be substituted with a geometric kernel that employs exact
rational arithmetic. The curves that compose the polygon edges should
are then arcs of circles with rational coefficients or segments of lines
with rational coefficients. However, the line segments that result from
the (exact) offsetting operation may be irrational. As we still wish to
use the polygons defined by the `Gps_circle_segment_traits_2` class
template to compute Boolean set operations more efficiently, we apply a
simple approximation scheme such that each irrational line segment is
approximated by two rational segments.

Consider the example depicted in \cgalFigureRef{mink_figapprox_offset},
where the exact offset edge \f$ o_1 o_2\f$ is obtained by shifting the
polygon edge \f$ p_1 p_2\f$ by a vector of length \f$ r\f$ that forms an
angle \f$ \phi\f$ with the \f$ x\f$-axis. We select two points
\f$ o'_1\f$ and \f$ o'_2\f$ with rational coordinates that lie on the two
circles of radius \f$ r \f$ centered at \f$ p_1\f$ and \f$ p_2\f$,
respectively. These points are selected such that
\f$ \phi'_1 < \phi < \phi'_2 \f$, where \f$ \phi'_j, j = 1,2 \f$ is the
angle that the vector \f$ \overrightarrow{p_j o_j} \f$ forms with the
$x$-axis. Then, we construct two tangents to the two circles at \f$ o'_1\f$
and \f$ o'_2\f$, respectively. The tangent lines have rational coefficients.
Finally, we compute the intersection point of the two tangents, denoted by
\f$ q'\f$. The two line segments \f$ o'_1 q'\f$ and \f$ q' o'_2\f$
approximate the original offset edge \f$ o_1 o_2\f$.

The \link approximated_offset_2() `approximated_offset_2(P, r, epsilon)`\endlink
function template accepts a polygon \f$ P\f$ that is either simple or
degenerate (consisting of two points only), an offset radius \f$ r\f$,
and (a floating-point number) \f$ \epsilon > 0 \f$. It constructs an
approximation of the offset of \f$ P\f$ by the radius \f$ r\f$ using the
procedure described above. Furthermore, it is guaranteed that the
approximation error, namely the distance of the point \f$ q'\f$ from
\f$ o_1 o_2\f$, is bounded by \f$ \epsilon \f$. Using this function, it is
possible to use the `Gps_circle_segment_traits_2<Kernel>` class template,
which considerably speeds up the (approximate) construction of the offset
polygon and the application of Boolean set operations on such polygons;
see Section
\ref arr_sssectr_circ_seg "A Traits Class for Circular Arcs and Line Segments"
for the exploitation of the efficient square root extension number type.
The function returns an object of the nested type
`Gps_circle_segment_traits_2::Polygon_with_holes_2` representing
the approximated offset polygon. Recall that if \f$ P\f$ is not convex, its
offset may not be simple and may contain holes, the boundaries of which are
also formed by line segments and circular arcs.

\cgalFigureBegin{mink_figex_offset,ms_exact_offset.png}
The offset computation performed by the example programs
\ref Minkowski_sum_2/approx_offset.cpp and \ref Minkowski_sum_2/exact_offset.cpp.
The input polygon is shaded and the boundary of its
offset is drawn in a thick black line.
\cgalFigureEnd

The example below demonstrates the construction of an approximated
offset of a non-convex polygon, as depicted in Figure
\cgalFigureRef{mink_figex_offset}. The program uses types defined in
the header file `bops_circular.h`, which defines the polygon types.

\cgalExample{Minkowski_sum_2/approx_offset.cpp}

\subsection mink_ssec_exact_offset Computing the Exact Offset

As mentioned in the previous section, it is possible to represent
offset polygons in an exact manner if the edges of the polygons are
represented as arcs of conic curves with rational coefficients. The
\link offset_polygon_2() `offset_polygon_2(P, r, traits)`\endlink
function template computes the offset of a given polygon \f$ P\f$ by a
rational radius \f$ r\f$ in an exact manner. The input polygon \f$
P\f$ must be either simple or degenerate consisting of two vertices
(representing a line segment). The `traits` argument must model the
concept `AosTraits_2` and it should be capable of handling
conic arcs in an exact manner---using an instance of the
`Arr_conic_traits_2` class template with the number types provided by
the \core library is the preferred option; see \ref arr_sssectr_conic
"A Traits Class for Conic Arcs" for more details. The function
template returns an object of the nested type
`Gps_traits_2::Polygons_with_holes_2` (see \ref
bso_ssecgeneral_polygon_concept "General Polygon Set Traits Adapter"
for more details on the traits-class adaptor `Gps_traits_2`), which
represents the exact offset polygon.

The following example demonstrates the construction of the offset of
the same polygon that serves as an input for the example program
\ref Minkowski_sum_2/approx_offset.cpp, presented in the previous
subsection (see also \cgalFigureRef{mink_figex_offset}).
Note that the resulting polygon is smaller than the one generated by the approximated-offset function
(recall that each irrational line segment in this case is approximated
by two rational line segments), but the offset computation is
considerably slower:

\cgalExample{Minkowski_sum_2/exact_offset.cpp}

\cgalAdvancedBegin
Both functions templates `approximated_offset_2()` and `offset_polygon_2()`
also have overloaded versions that accept a decomposition strategy
and use the polygon-decomposition approach to compute (or approximate)
the offset. These functions are typically considerably slower than their
counterparts that employ the convolution approach. However, similar to the
functions that compute the general Minkowski sum, they are able to compute
the offset of polygons with holes, given a decomposition strategy that
handles polygons with holes, such as the
`Polygon_vertical_decomposition_2<Kernel>` class template.
\cgalAdvancedEnd

\subsection mink_ssec_inner_offset Computing Inner Offsets

An operation closely related to the (outer) offset computation, is
computing the <I>inner offset</I> of a polygon, or <I>insetting</I> it
by a given radius. The inset of a polygon \f$ P\f$ with radius
\f$ r\f$ is the set of points inside \f$ P\f$ the distance of which
from the polygon boundary, denoted \f$ \partial P\f$, is at least \f$ r\f$,
namely: \f$ \{ p \in P \;|\; {\rm dist}(p, \partial P) \geq r \}\f$.
Note that the resulting point set may be disconnected when \f$ P\f$ is a
non-convex polygon that has some narrow components. In such a case the
resulting set is characterized by a (possibly empty) set of polygons the
edges of which are line segments and circular arcs of radius \f$ r\f$.

The inset can be computed using the convolution method traversing the
polygon in a <I>clockwise</I> order (as oppose to the
<I>counterclockwise</I> order applied in the case of offsetting a polygon).
As with the (outer) offset functions, the Minkowski-sum package contains two
functions for insetting a simple polygon, namely,
`approximated_inset_2(P, r, eps, oi)` and `inset_polygon_2(P, r, traits, oi)`.

The \link approximated_inset_2() `approximated_inset_2(P, r, eps, oi)`\endlink
function template accepts a polygon \f$ P\f$, an inset radius \f$ r\f$,
(a floating-point number) \f$ \epsilon > 0 \f$, and an output
iterator `oi`, whose value-type must be an instance of the class template
`Gps_circle_segment_traits_2::Polygon_2`. . It constructs an approximation
of the inset of \f$ P\f$ by the radius \f$ r\f$, where the approximation
error is bounded by \f$ \epsilon\f$. The function returns the polygons that
approximate the inset polygon through the output iterator `oi`.

\cgalExample{Minkowski_sum_2/approx_inset.cpp}

Similarly, the function template
\link inset_polygon_2() `inset_polygon_2(P, r, traits, oi)` \endlink
computes the exact inset of \f$ P\f$ with radius \f$ r\f$, and returns
its output through the given output iterator `oi`. The `traits` parameter
must model the concept `AosTraits_2` and it should be capable of
handling conic arcs in an exact manner, whereas the value-type of `oi`
must be an instance of `Gps_traits_2::Polygon_2`.

\cgalExample{Minkowski_sum_2/exact_inset.cpp}

Unlike the functions that compute the offsets, there are no overloaded
versions of the functions that compute the insets and use convex polygon
decomposition, as decomposition approach cannot be easily generalized for
inset computations.

The package also provides overloaded versions of the functions
\link approximated_offset_2() `approximated_offset_2(P, r, eps)`\endlink
 and
\link offset_polygon_2() `offset_polygon_2(P, r, traits)`\endlink
that accept a <I>polygon with holes</I>
\f$ P\f$ and compute its offset. This offset is obtain by computing the
outer offset of the outer boundary of \f$ P\f$'s, and computing the inner
offsets of the holes of \f$ P\f$. The former polygon defines the output
boundary of \f$ P \oplus B_r\f$, and the latter define the holes within
the result.

\section Minkoski_sum_2Design Design and Implementation History
Eyal flato set the ground for the this package around the end of the previous
millennium with his research work about Minkowski sum construction using the
decomposition approach. However, it was only a few years later when Ron Wein
developed the first version of this package, which was introduced with
release 3.3. The first version of the package contained functionality for
computing the Minkowski sum of two polygons, computing the offset of
a polygon by a disk, and approximating the offset of a polygon by a disk.
Release 3.4 contained an enhancement of the offset computation, which allowed
for polygon with holes as operands. Release 3.4 also featured new functionality
for computing the inner offset of a polygon and approximating the inner offset.
All this was also developed by Ron Wein. Release 3.7 contained an enhancement
of the offset approximation for degenerate polygons such as line segments. This
enhancement was developed by Efi Fogel, who also developed a new decomposition
strategy, which can handle polygons with holes, essentially enabling the
computation of Minkowski sum of two polygons with holes using the decomposition
approach. The latter was introduced with release 4.6. Ron Wein, Efi Fogel,
Ophir Setter, Andreas Fabri, and Laurent Rineau helped maintaining the package
applying bug fixes and other improvements. In particular, Andreas Fabri and
Laurent Rineau helped tracing and solving several bugs in the approximated
offset computation. They have also suggested a few algorithmic improvements
that made their way into version 3.4, yielding a faster approximation scheme.
During the <I>Google Summer of Code</I> 2014, Sebastian Morr, mentored by
Michael Hemmer, implemented the reduced convolution approach, based on Alon
Baram's master's thesis.

*/
} /* namespace CGAL */
