mirror of https://github.com/CGAL/cgal
59 lines
2.0 KiB
TeX
59 lines
2.0 KiB
TeX
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This chapter describes algorithms which for a given point set compute
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the ``best'' inscribed object from a specific
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class. We provide algorithms for
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computing maximal inscribed $k$-gons (triangles, quadrilaterals,
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\dots) of a planar point set $P$. Maximal $k$-gons are convex, and it
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is known that their vertices can be chosen to be vertices of the
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convex hull of $P$. Hence, the functions
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\ccc{CGAL::maximum_area_inscribed_k_gon_2} and
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\ccc{CGAL::maximum_perimeter_inscribed_k_gon_2} operate on convex polygons
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only. The example below shows that the largest area triangle (green)
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and the largest perimeter triangle (orange, containing the top point)
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of a point set are different in general.
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\begin{ccHtmlOnly}
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<center>
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<img border="0" src="max_triangle.gif" align="center">
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</center>
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\end{ccHtmlOnly}
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\begin{ccTexOnly}
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\begin{center}
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\includegraphics[width=5cm]{Inscribed_areas/max_triangle}
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\end{center}
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\end{ccTexOnly}
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We further provide an algorithm for computing the maximal area
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inscribed axis parallel rectangle
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Given a set of points in the plane, the class \ccc{CGAL::Largest_empty_iso_rectangle_2<T>}
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is a data structure that maintains an iso-rectangle with the largest area among
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all iso-rectangles that are inside a given iso-rectangles, and
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that do not contain any point of the point set.
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\begin{ccHtmlOnly}
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<center>
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<img border="0" src="largestEmptyRect.gif" align="center">
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</center>
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\end{ccHtmlOnly}
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\begin{ccTexOnly}
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\begin{center}
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\includegraphics[width=5cm]{Inscribed_areas/largestEmptyRect}
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\end{center}
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\end{ccTexOnly}
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Inscribed volumes are also frequently applied to extract
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geometric properties of objects. The largest area triangle is for example used in
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heuristics for matching archaeological aerial photographs. Largest
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perimeter triangles are used in scoring cross country soaring flights,
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where the goal is basically to fly as far as possible, but still
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return to the departure airfield. To score simply based on the total
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distance flown is not a good measure, since circling in thermals
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allows to increase it easily.
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