CGAL 6.2 - 2D Straight Skeleton and Polygon Offsetting

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Functions
Offset Construction Functions
Reference Manual » Functions

##

The following functions are used to construct the inward or outward offsets of a polygon.

# Kernel Choices

Up to three different kernels can be used in the offset construction process:

- Input Kernel (`InK`): This is the kernel type of the input polygons.
- Skeleton Kernel (`SsK`): This is the kernel used for the construction of the (partial) straight skeleton. It might differ from the Input Kernel for example if the input is based on exact number types, but you wish to speed up the construction of the straight skeleton, using a lighter kernel such as `CGAL::Exact_predicates_inexact_constructions_kernel`.
- Offset (Output) Kernel (`OfK`): This is the kernel used for the construction of the offset polygons and is thus the kernel type for the output.

When the kernel differs, `CGAL::Cartesian_converter` is used to convert from one kernel to the other.

By default, `SsK` and `OfK` are chosen as `CGAL::Exact_predicates_inexact_constructions_kernel` as a compromise between speed and robustness. See also Section Create Offset Polygons from a Polygon (With or Without Holes).

# Polygon Return Type

There are two classes of offset construction functions: those that simply return all the offset polygons, and those that also create an arrangement of the polygons to create a polygon with holes. The latter contain _with_holes in the name.

The return type of the offset construction function is thus dediced by whether or not it is a simple polygon or a polygon with holes, and on the dictated output kernel (see Section Kernel Choices). Thus, for an input polygon type `InKPolygon` in an offset construction that does not arrange polygons (e.g. `create_interior_skeleton_and_offset_polygons_2()`):

- If the kernel type `InK` of the input polygon type is different from the output kernel type `OfK`, then the output polygon type is `CGAL::Polygon_2<OfK>`.
- If the kernel type `InK` is equal to the output kernel type `OfK`, but the input polygon is a polygon with holes (a model of `GeneralPolygonWithHoles_2`), then the return type is `InKPolygon::General_polygon_2`.
- If the kernel type `InK` of the input polygon type is equal to the output kernel type `OfK`, and the input polygon is a polygon without holes (a model of `SequenceContainer` with `InK::Point_2` value type), then the return type is `InKPolygon`.

Similarly, if the return type is a polygon with holes (e.g. `create_interior_skeleton_and_offset_polygons_with_holes_2()`, a similar dispatch exists:

- If the kernel type `InK` of the input polygon type is different from the output kernel type `OfK`, then the output polygon type is `CGAL::Polygon_with_holes_2<OfK>`.
- If the kernel type `InK` is equal to the output kernel type `OfK`, but the input polygon is a polygon with holes (a model of `GeneralPolygonWithHoles_2`), then the return type is `InKPolygon`.
- If the kernel type `InK` of the input polygon type is equal to the output kernel type `OfK`, and the input polygon is a polygon without holes (a model of `SequenceContainer` with `InK::Point_2` value type), then the return type is `CGAL::Polygon_with_holes_2<OfK>`.

## Functions
template<class K , class InputPolygonPtrIterator , class OutputPolygonWithHolesPtrIterator >
bool | CGAL::arrange_offset_polygons_2 (InputPolygonPtrIterator begin, InputPolygonPtrIterator end, OutputPolygonWithHolesPtrIterator out, const K &k)
| The function `arrange_offset_polygons_2()` arranges the sequence of 2D polygons obtained by `create_offset_polygons_2()` into 2D polygons with holes by determining geometric parent-hole relationships using a simple algorithm based on the particular characteristics of offset polygons.
template<class InputIterator , class Traits >
std::optional< typename Traits::FT > | CGAL::compute_outer_frame_margin (InputIterator first, InputIterator beyond, typename Traits::FT offset, const Traits &traits=Default_traits)
| computes the separation required between a polygon and the outer frame used to obtain an exterior skeleton suitable for the computation of outer offset polygons at a given distance.
template<typename OfKPolygon , typename FT , typename StraightSkeleton , typename OfK >
std::vector< std::shared_ptr< OfKPolygon > > | CGAL::create_offset_polygons_2 (FT offset, const StraightSkeleton &ss, OfK k=Exact_predicates_inexact_constructions_kernel())
| returns a container with the offset polygons at distance `offset` obtained from the straight skeleton `ss`.
template<typename OfKPolygon , typename FT , typename InKPolygon , typename HoleIterator , typename OfK , typename SsK >
std::vector< std::shared_ptr< OfKPolygon > > | CGAL::create_interior_skeleton_and_offset_polygons_2 (FT offset, const InKPolygon &outer_boundary, HoleIterator holes_begin, HoleIterator holes_end, OfK ofk=CGAL::Exact_predicates_inexact_constructions_kernel, SsK ssk=CGAL::Exact_predicates_inexact_constructions_kernel())
| returns a container with all the inner offset polygons at distance `offset` of a 2D polygon with holes.
template<typename OfKPolygon , typename FT , typename InKPolygon , typename OfK , typename SsK >
std::vector< std::shared_ptr< OfKPolygon > > | CGAL::create_interior_skeleton_and_offset_polygons_2 (FT offset, const InKPolygon &poly, OfK ofk=CGAL::Exact_predicates_inexact_constructions_kernel, SsK ssk=CGAL::Exact_predicates_inexact_constructions_kernel())
| returns a container with all the inner offset polygons at distance `offset` of the 2D polygon `poly`.
template<typename OfKPolygon , typename FT , typename InKPolygon , typename OfK , typename SsK >
std::vector< std::shared_ptr< OfKPolygon > > | CGAL::create_exterior_skeleton_and_offset_polygons_2 (FT offset, const InKPolygon &poly, OfK ofk=Exact_predicates_inexact_constructions_kernel(), SsK ssk=Exact_predicates_inexact_constructions_kernel())
| returns a container with all the outer offset polygons at distance `offset` of the 2D polygon `poly`.
template<class OfKPolygon , class FT , class InKPolygon , class OfK , class SsK >
std::vector< std::shared_ptr< OfKPolygon > > | CGAL::create_interior_skeleton_and_offset_polygons_with_holes_2 (FT offset, const InKPolygon &poly_with_holes, OfK ofk=CGAL::Exact_predicates_inexact_constructions_kernel, SsK ssk=CGAL::Exact_predicates_inexact_constructions_kernel)
| returns a container with all the inner offset polygons with holes at distance `offset` of the 2D polygon with holes `poly_with_holes`.
template<class OfKPolygon , class FT , class InKPolygon , class OfK , class SsK >
std::vector< std::shared_ptr< OfKPolygon > > | CGAL::create_exterior_skeleton_and_offset_polygons_with_holes_2 (FT offset, const InKPolygon &poly_with_holes, OfK ofk=Exact_predicates_inexact_constructions_kernel(), SsK ssk=Exact_predicates_inexact_constructions_kernel())
| returns a container with all the outer offset polygons with holes at distance `offset` of the 2D polygon `poly_with_holes`.
template<class Target_skeleton , class Source_skeleton , class Items_converter >
std::shared_ptr< Target_skeleton > | CGAL::convert_straight_skeleton_2 (Source_skeleton const &s, Items_converted const &ic=Items_converter())
| returns a new straight skeleton data structure with the same combinatorial and geometric data as `s` using the items converter `ic` to convert the geometric embedding to the types of the target skeleton.

## Function Documentation

## ◆ arrange_offset_polygons_2()

template<class K , class InputPolygonPtrIterator , class OutputPolygonWithHolesPtrIterator >

bool CGAL::arrange_offset_polygons_2 | ( | InputPolygonPtrIterator | begin,
|  | InputPolygonPtrIterator | end,
|  | OutputPolygonWithHolesPtrIterator | out,
|  | const K & | k
| ) |  |

`#include <CGAL/arrange_offset_polygons_2.h>`

The function `arrange_offset_polygons_2()` arranges the sequence of 2D polygons obtained by `create_offset_polygons_2()` into 2D polygons with holes by determining geometric parent-hole relationships using a simple algorithm based on the particular characteristics of offset polygons.

The function determines parent-hole relationships among the polygons given by `[begin,end]` creating `std::shared_ptr< GeneralPolygonWithHoles_2 >` objects added to the output sequence given `out`. A `CLOCKWISE` oriented polygon `H` is a hole of a `COUNTERCLOCKWISE` polygon `P`, iff at least one vertex of `H` is `ON_BOUNDED_SIDE` of `P`.

This function should not be used to arrange arbitrary polygons into polygons with holes unless they meet the requirements specified below.

Precondition
  The input polygons must be simple.
  The set of input polygons are unique and interior disjoint. That is, given distinct polygons `P` and `Q`, there are vertices of `P` which are not on the boundary of `Q` and are all on the bounded or unbounded side of `Q` (but not both).

Template Parameters

  K | must be a model of `Kernel`.
  InputPolygonPtrIterator | must be a model of `InputIterator` whose `value_type` is a smart pointer (such as `std::shared_ptr`) whose `element_type` is a model of `SequenceContainer` with value type `K::Point_2`.
  OutputPolygonWithHolesPtrIterator | must be a model of `OutputIterator` whose `value_type` is a smart pointer (such as `std::shared_ptr`) whose `element_type` is a model of `GeneralPolygonWithHoles_2`.

Returns
  `true` if no error was encountered, and `false` otherwise.

See also
  `CGAL::create_exterior_straight_skeleton_2()`
  `CGAL::Straight_skeleton_builder_2`

## ◆ compute_outer_frame_margin()

template<class InputIterator , class Traits >

std::optional< typename Traits::FT > CGAL::compute_outer_frame_margin | ( | InputIterator | first,
|  | InputIterator | beyond,
|  | typename Traits::FT | offset,
|  | const Traits & | traits = `Default_traits`
| ) |  |

`#include <CGAL/compute_outer_frame_margin.h>`

computes the separation required between a polygon and the outer frame used to obtain an exterior skeleton suitable for the computation of outer offset polygons at a given distance.

Given a non-degenerate strictly-simple 2D polygon whose vertices are passed in the range [`first`,`beyond`), calculates the largest Euclidean distance `d` between each input vertex and its corresponding offset vertex at a distance `offset`.

If such a distance can be approximately computed, returns an `optional<FT>` with the value `d + (offset * 1.05)`. If the distance cannot be computed, not even approximately, due to overflow for instance, returns an empty `optional<FT>` (an absent result).

This result is the required separation between the input polygon and the rectangular frame used to construct an exterior offset contour at distance `offset` (which is done by placing the polygon as a hole of that frame).

Such a separation must be computed in this way because if the frame is too close to the polygon, the inward offset contour from the frame could collide with the outward offset contour of the polygon, resulting in a merged contour offset instead of two contour offsets, one of them corresponding to the frame.

Simply using `2*offset` as the separation is incorrect since `offset` is the distance between an offset line and its original, not between an offset vertex and its original. The latter, which is calculated by this function and needed to place the frame sufficiently away from the polygon, can be thousands of times larger than `offset`.

If the result is absent, any attempt to construct an exterior offset polygon at distance `offset` will fail. This will occur whenever the polygon has a vertex with an internal angle approaching `0` (because the offset vertex of a vertex whose internal angle equals 0 is at infinity).

The default traits class `Default_traits` is an instance of the class `Polygon_offset_builder_traits_2<Kernel>` parameterized on the kernel in which the type `InputIterator::value_type` is defined.

Template Parameters

  Traits | must be a model for `PolygonOffsetBuilderTraits_2`.
  InputIterator | must be a model of `InputIterator` with `value_type` equivalent to `Traits::Point_2`.

Precondition
  `offset > 0`.
  The range [`first`,`beyond`) contains the vertices of a non-degenerate strictly-simple 2D polygon.

See also
  `PolygonOffsetBuilderTraits_2`
  `CGAL::Polygon_offset_builder_traits_2<K>`

Examples
  Straight_skeleton_2/Low_level_API.cpp.

## ◆ convert_straight_skeleton_2()

template<class Target_skeleton , class Source_skeleton , class Items_converter >

std::shared_ptr< Target_skeleton > CGAL::convert_straight_skeleton_2 | ( | Source_skeleton const & | s,
|  | Items_converted const & | ic = `Items_converter()`
| ) |  |

`#include <CGAL/Straight_skeleton_converter_2.h>`

returns a new straight skeleton data structure with the same combinatorial and geometric data as `s` using the items converter `ic` to convert the geometric embedding to the types of the target skeleton.

Template Parameters

  Target_skeleton | must be a model of `StraightSkeleton_2`
  Source_skeleton | must be a model of `StraightSkeleton_2`
  Items_converter | must be a model of `StraightSkeletonItemsConverter_2`

See also
  `CGAL::Straight_skeleton_items_converter_2<SrcSs,TgtSs,NTCV>`
  `CGAL::Straight_skeleton_converter_2<SrcSs,TgtSs,ItemsCV>`

## ◆ create_exterior_skeleton_and_offset_polygons_2()

template<typename OfKPolygon , typename FT , typename InKPolygon , typename OfK , typename SsK >

std::vector< std::shared_ptr< OfKPolygon > > CGAL::create_exterior_skeleton_and_offset_polygons_2 | ( | FT | offset,
|  | const InKPolygon & | poly,
|  | OfK | ofk = `Exact_predicates_inexact_constructions_kernel()`,
|  | SsK | ssk = `Exact_predicates_inexact_constructions_kernel()`
| ) |  |

`#include <CGAL/create_offset_polygons_2.h>`

returns a container with all the outer offset polygons at distance `offset` of the 2D polygon `poly`.

The exterior skeleton is limited by an outer rectangular frame placed at a margin sufficiently large to allow the offset polygons to be constructed.

Note
  A temporary straight skeleton is constructed in the limited exterior of the input polygon to obtain the offsets. The construction of this skeleton is the most expensive operation, therefore, to construct offsets at more than one single distance, use the separate functions `create_exterior_straight_skeleton_2()` and `create_offset_polygons_2()` instead.

Template Parameters

  OfKPolygon | is a polygon without holes type determined from `OfK` and `InKPolygon`, see Section Polygon Return Type.
  FT | must be a model of `FieldNumberType` convertible to `OfK::FT` and `SsK::FT`.
  InKPolygon | must be a model of `SequenceContainer` with value type `InK::Point_2` (e.g. `Polygon_2<InK>`) or a model of `GeneralPolygonWithHoles_2` (e.g. `Polygon_with_holes_2<InK>`).
  OfK | must be a model of `Kernel`. It is used to instantiate `Polygon_offset_builder_traits_2<OfK>` for constructing the offset polygons.
  SsK | must be a model of `Kernel`. It is used to instantiate `Straight_skeleton_builder_traits_2<SsK>` for constructing the straight skeleton.

Note
  If `SsK != OfK` the constructed straight skeleton is converted to `CGAL::Straight_skeleton_2<OfK>`.

Precondition
  `offset` is positive
  `poly` is weakly simple, counterclockwise polygon.

See also
  `CGAL::create_interior_skeleton_and_offset_polygons_2()`
  `CGAL::create_exterior_skeleton_and_offset_polygons_with_holes_2()`
  `Polygon_offset_builder_2`

Examples
  Straight_skeleton_2/Create_skeleton_and_offset_polygons_2.cpp.

## ◆ create_exterior_skeleton_and_offset_polygons_with_holes_2()

template<class OfKPolygon , class FT , class InKPolygon , class OfK , class SsK >

std::vector< std::shared_ptr< OfKPolygon > > CGAL::create_exterior_skeleton_and_offset_polygons_with_holes_2 | ( | FT | offset,
|  | const InKPolygon & | poly_with_holes,
|  | OfK | ofk = `Exact_predicates_inexact_constructions_kernel()`,
|  | SsK | ssk = `Exact_predicates_inexact_constructions_kernel()`
| ) |  |

`#include <CGAL/create_offset_polygons_from_polygon_with_holes_2.h>`

returns a container with all the outer offset polygons with holes at distance `offset` of the 2D polygon `poly_with_holes`.

Note that the offset of the outer frame is ignored.

This is equivalent to a call to `CGAL::arrange_offset_polygons_2()` on the output of `create_exterior_skeleton_and_offset_polygons_2(offset, poly_with_holes, ofk, ssk)` after having filtered out the polygon corresponding to the offset of the outer frame and having reversed the orientation of all other polygons.

Template Parameters

  OfKPolygon | is a polygon without holes type determined by `OfK` and `InKPolygon`, see Section Polygon Return Type.
  FT | must be a model of `FieldNumberType` convertible to `OfK::FT` and `SsK::FT`.
  InKPolygon | must be a model of `SequenceContainer` with value type `InK::Point_2` (e.g. `Polygon_2<InK>`) or a model of `GeneralPolygonWithHoles_2` (e.g. `Polygon_with_holes_2<InK>`).
  OfK | must be a model of `Kernel`. It is used to instantiate `Polygon_offset_builder_traits_2<OfK>` for constructing the offset polygons.
  SsK | must be a model of `Kernel`. It is used to instantiate `Straight_skeleton_builder_traits_2<SsK>` for constructing the straight skeleton.

Note
  If `SsK != OfK` the constructed straight skeleton is converted to `CGAL::Straight_skeleton_2<OfK>`.

See also
  `CGAL::create_exterior_skeleton_and_offset_polygons_2()`
  `CGAL::create_interior_skeleton_and_offset_polygons_with_holes_2()`
  `Polygon_offset_builder_2`

## ◆ create_interior_skeleton_and_offset_polygons_2() [1/2]

template<typename OfKPolygon , typename FT , typename InKPolygon , typename HoleIterator , typename OfK , typename SsK >

std::vector< std::shared_ptr< OfKPolygon > > CGAL::create_interior_skeleton_and_offset_polygons_2 | ( | FT | offset,
|  | const InKPolygon & | outer_boundary,
|  | HoleIterator | holes_begin,
|  | HoleIterator | holes_end,
|  | OfK | ofk = `CGAL::Exact_predicates_inexact_constructions_kernel`,
|  | SsK | ssk = `CGAL::Exact_predicates_inexact_constructions_kernel()`
| ) |  |

`#include <CGAL/create_offset_polygons_2.h>`

returns a container with all the inner offset polygons at distance `offset` of a 2D polygon with holes.

The outer boundary is `outer_boundary` and its holes are given by `[holes_begin,holes_end)`.

Note
  A temporary straight skeleton is constructed in the interior of the input polygon to obtain the offsets. The construction of this skeleton is the most expensive operation, therefore, to construct offsets at more than one single distance, it is advised to use `create_interior_straight_skeleton_2()` to create the skeleton only once, and then call `create_offset_polygons_2()` for each distance.

Template Parameters

  OfKPolygon | is a polygon without holes type determined from `OfK` and `InKPolygon`, see Section Polygon Return Type.
  FT | must be a model of `FieldNumberType` convertible to `OfK::FT` and `SsK::FT`.
  HoleIterator | must be a model of `InputIterator` with value type being a model of `ConstRange` with value type `SsK::Point_2`.
  InKPolygon | must be a model of `SequenceContainer` with value type `InK::Point_2` (e.g. `Polygon_2<InK>`).
  OfK | must be a model of `Kernel`. It is used to instantiate `Polygon_offset_builder_traits_2<OfK>` for constructing the offset polygons.
  SsK | must be a model of `Kernel`. It is used to instantiate `Straight_skeleton_builder_traits_2<SsK>` for constructing the straight skeleton.

Note
  If `SsK != OfK` the constructed straight skeleton is converted to `CGAL::Straight_skeleton_2<OfK>`.

Precondition
  `outer_boundary` describes a weakly simple, counterclockwise polygon.
  `[holes_begin,holes_end)` is a range of weakly simple, clockwise polygons.
  Holes do not intersect each other or the outer boundary.

See also
  `CGAL::create_exterior_skeleton_and_offset_polygons_2()`
  `CGAL::create_interior_skeleton_and_offset_polygons_with_holes_2()`
  `Polygon_offset_builder_2`

Examples
  Straight_skeleton_2/Create_saop_from_polygon_with_holes_2.cpp, and Straight_skeleton_2/Create_skeleton_and_offset_polygons_2.cpp.

## ◆ create_interior_skeleton_and_offset_polygons_2() [2/2]

template<typename OfKPolygon , typename FT , typename InKPolygon , typename OfK , typename SsK >

std::vector< std::shared_ptr< OfKPolygon > > CGAL::create_interior_skeleton_and_offset_polygons_2 | ( | FT | offset,
|  | const InKPolygon & | poly,
|  | OfK | ofk = `CGAL::Exact_predicates_inexact_constructions_kernel`,
|  | SsK | ssk = `CGAL::Exact_predicates_inexact_constructions_kernel()`
| ) |  |

`#include <CGAL/create_offset_polygons_2.h>`

returns a container with all the inner offset polygons at distance `offset` of the 2D polygon `poly`.

Note
  A temporary straight skeleton is constructed in the interior of the input polygon to obtain the offsets. The construction of this skeleton is the most expensive operation, therefore, to construct offsets at more than one single distance, it is advised to use `create_interior_straight_skeleton_2()` to create the skeleton only once, and then call `create_offset_polygons_2()` for each distance

Template Parameters

  OfKPolygon | is a polygon without holes type determined by `OfK` and `InKPolygon`, see Section Polygon Return Type.
  FT | must be a model of `FieldNumberType` convertible to `OfK::FT` and `SsK::FT`.
  InKPolygon | must be a model of `SequenceContainer` with value type `InK::Point_2` (e.g. `Polygon_2<InK>`) or a model of `GeneralPolygonWithHoles_2` (e.g. `Polygon_with_holes_2<InK>`).
  OfK | must be a model of `Kernel`. It is used to instantiate `Polygon_offset_builder_traits_2<OfK>` for constructing the offset polygons.
  SsK | must be a model of `Kernel`. It is used to instantiate `Straight_skeleton_builder_traits_2<SsK>` for constructing the straight skeleton.

Note
  If `SsK != OfK` the constructed straight skeleton is converted to `CGAL::Straight_skeleton_2<OfK>`.

Precondition
  `offset` is positive
  `poly` is weakly simple, counterclockwise polygon.

See also
  `CGAL::create_exterior_skeleton_and_offset_polygons_2()`
  `CGAL::create_interior_skeleton_and_offset_polygons_with_holes_2()`
  `Polygon_offset_builder_2`

## ◆ create_interior_skeleton_and_offset_polygons_with_holes_2()

template<class OfKPolygon , class FT , class InKPolygon , class OfK , class SsK >

std::vector< std::shared_ptr< OfKPolygon > > CGAL::create_interior_skeleton_and_offset_polygons_with_holes_2 | ( | FT | offset,
|  | const InKPolygon & | poly_with_holes,
|  | OfK | ofk = `CGAL::Exact_predicates_inexact_constructions_kernel`,
|  | SsK | ssk = `CGAL::Exact_predicates_inexact_constructions_kernel`
| ) |  |

`#include <CGAL/create_offset_polygons_from_polygon_with_holes_2.h>`

returns a container with all the inner offset polygons with holes at distance `offset` of the 2D polygon with holes `poly_with_holes`.

This is equivalent to `arrange_offset_polygons_2(create_interior_skeleton_and_offset_polygons_2(offset, poly_with_holes, ofk, ssk))`.

Template Parameters

  OfKPolygon | is a polygon without holes type determined by `OfK` and `InKPolygon`, see Section Polygon Return Type.
  FT | must be a model of `FieldNumberType` convertible to `OfK::FT` and `SsK::FT`.
  InKPolygon | must be a model of `SequenceContainer` with value type `InK::Point_2` (e.g. `Polygon_2<InK>`) or a model of `GeneralPolygonWithHoles_2` (e.g. `Polygon_with_holes_2<InK>`).
  OfK | must be a model of `Kernel`. It is used to instantiate `Polygon_offset_builder_traits_2<OfK>` for constructing the offset polygons.
  SsK | must be a model of `Kernel`. It is used to instantiate `Straight_skeleton_builder_traits_2<SsK>` for constructing the straight skeleton.

Note
  If `SsK != OfK` the constructed straight skeleton is converted to `CGAL::Straight_skeleton_2<OfK>`.

See also
  `CGAL::create_interior_skeleton_and_offset_polygons_2()`
  `CGAL::create_exterior_skeleton_and_offset_polygons_with_holes_2()`
  `Polygon_offset_builder_2`

Examples
  Straight_skeleton_2/Create_skeleton_and_offset_polygons_with_holes_2.cpp, and Straight_skeleton_2/Show_offset_polygon.cpp.

## ◆ create_offset_polygons_2()

template<typename OfKPolygon , typename FT , typename StraightSkeleton , typename OfK >

std::vector< std::shared_ptr< OfKPolygon > > CGAL::create_offset_polygons_2 | ( | FT | offset,
|  | const StraightSkeleton & | ss,
|  | OfK | k = `Exact_predicates_inexact_constructions_kernel()`
| ) |  |

`#include <CGAL/create_offset_polygons_2.h>`

returns a container with the offset polygons at distance `offset` obtained from the straight skeleton `ss`.

If `ss` is the interior skeleton of a polygon with holes, the offset polygons will be generated in its interior. If `ss` is the outer skeleton of a polygon with holes, the offset polygons will be generated in its exterior.

Template Parameters

  OfKPolygon | is a polygon without holes type determined from `OfK`, see Section Polygon Return Type.
  FT | must be a model of `FieldNumberType` convertible to `OfK::FT`.
  StraightSkeleton | is an object of type `CGAL::Straight_skeleton_2<SsK>`.
  OfK | must be a model of `Kernel`. It is used to instantiate `Polygon_offset_builder_traits_2<OfK>` for constructing the offset polygons.

Note
  If `SsK != OfK` the constructed straight skeleton is converted to `CGAL::Straight_skeleton_2<OfK>`.

Precondition
  `offset` is positive.
  `ss` is a valid straight skeleton.

See also
  `CGAL::create_interior_skeleton_and_offset_polygons_2()`
  `CGAL::create_exterior_skeleton_and_offset_polygons_2()`
  `Polygon_offset_builder_2`