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added example and benchmarks

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Michael Hemmer 2013-11-07 17:31:57 +01:00
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Visibility_2/doc/Visibility_2/visibility_2.txt
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@ -84,13 +84,44 @@ Class | Function | Prepro
Where \f$ n \f$ is the number of vertices of \f$ f \f$ and \f$ h \f$ is the number of holes+1.
\section benchmarks Runtime in Practice
\cgalFigureBegin{cathedral-fig, cathedral_2.png}
Example environment representing a cathedral.
\cgalFigureEnd
The left hand side of Figure \cgalFigureRef{definition-fig} depicts the outer boundary of a cathedral,
which is a simple polygon with 565 vertices. The right hand side shows the cathedral also with its inner
pillars, which is a polygon with 1153 vertices. The following table shows the total runtime to compute
all visibility polygons for all vertices of the boundary of the cathedral.
Boundary Cathedral | total preprocessing time | total query time |
-------------------------------|-----------------------------------------------------|-------------------------------|
`Simple_polygon_visibility_2` | - | 0.388021 |
`Rotational_sweep_visibility_2` | - | 1.01606 |
`Triangular_expansion_visibility_2` | 0 | 0.064005 |
`Preprocessed_rotational_sweep_visibility_2` | - | - |
The second table shows the same for the complete cathedral. The table does not report the time for `Simple_polygon_visibility_2`
as this algorithm can only handle simple polygons.
Complete Cathedral | total preprocessing time | total query time |
-------------------------------|-----------------------------------------------------|-------------------------------|
`Rotational_sweep_visibility_2` | - | 1.91211 |
`Triangular_expansion_visibility_2` | 0.004001 | 0.044001 |
`Preprocessed_rotational_sweep_visibility_2` | - | - |
Thus, in general we recommend to use `Triangular_expansion_visibility_2` even if the polygon is simple. The main advantage
of the algorithm is its locality. After the triangle that contains the query point is located in the triangulation,
the algorithm explores neighboring triangles, but only those that are actually seen. In this sense the algorithm can be
considered as output sensitive. However, if the simple polygon is rather convex (i.e., nearly all boundary is seen) or
if only one (or very little) queries are required, using one of the algorithms without preprocessing is advantageous.
\section simple_polygon_visibility_example Example of Visibility in a Simple Polygon
The following example shows how to obtain the regularized and non-regularized visibility regions.
\cgalExample{Visibility_2/simple_polygon_visibility_2.cpp}
\cgalFigureBegin{simple_example, simple_example.png}
Two different visibility regions in the example above.
\cgalFigureEnd
\section general_polygon_example Example of Visibility in a Polygon with Holes
The following example shows how to obtain the regularized visibility region by model `Triangular_expansion_visibility_2`. See \cgalFigureRef{general_polygon}. The arrangement has six bounded faces and an unbounded face. The query point \f$ q \f$ is on a vertex. The red arrow denotes the halfedge \f$ \overrightarrow{pq} \f$, which helps to identify the face in which the visibility region is computed.