mirror of https://github.com/CGAL/cgal
178 lines
6.5 KiB
TeX
178 lines
6.5 KiB
TeX
% +------------------------------------------------------------------------+
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% | Reference manual page: BisectionGeometricTraits_3.tex
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% +------------------------------------------------------------------------+
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% | 23.02.2009 Author
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% | Package: Package
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% |
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\RCSdef{\RCSBisectionGeometricTraitsRev}{$Id: header.tex 40270 2007-09-07 15:29:10Z lsaboret $}
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\RCSdefDate{\RCSBisectionGeometricTraitsDate}{$Date: 2007-09-07 17:29:10 +0200 (Ven, 07 sep 2007) $}
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% |
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\ccRefPageBegin
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%%RefPage: end of header, begin of main body
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% +------------------------------------------------------------------------+
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\begin{ccRefConcept}{BisectionGeometricTraits_3}
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%% \ccHtmlCrossLink{} %% add further rules for cross referencing links
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%% \ccHtmlIndexC[concept]{} %% add further index entries
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\ccDefinition
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The concept \ccRefName\ describes a geometric traits class
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that provides the basic types and operations
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to implement a model of \ccc{MeshDomain_3}
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based solely on intersection detections.
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Points in the non-empty intersections are herein computed
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by bisection.
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Such traits class is relevant when intersection detections
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can be performed efficiently. For instance, when bounding surfaces
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are implicitly described by a function (such as an isosurface of a 3D
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function from $\mathbb{R}^3$ to $\mathbb{R}$), the do-intersect predicate with a segment
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is computed by evaluations of the function values at both end points
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of the segment.
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\ccTypes
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\ccNestedType{FT}{Numerical type. Must be a model of
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\ccc{SqrtFieldNumberType} and constructible from a \ccc{double}.}
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\ccGlue
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\ccNestedType{Point_3}{The point type. Must have a
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constructor \ccc{Point_3(FT, FT, FT)}.}
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\ccGlue
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\ccNestedType{Segment_3}{Segment type.}
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\ccGlue
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\ccNestedType{Ray_3}{Ray type.}
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\ccGlue
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\ccNestedType{Line_3}{Line type.}
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\ccGlue
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\ccNestedType{Vector_3}{Vector type.}
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\ccGlue
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\ccNestedType{Sphere_3}{Sphere type.}
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\ccTwo{Construct_translated_point_3}{}
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\ccNestedType{Compute_scalar_product_3}
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{Function object providing the operator\\
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\ccc{FT operator()(Vector_3 v, Vector_3 w)} which returns the scalar
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(inner) product of the two vectors \ccc{v} and \ccc{w}.}
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\ccNestedType{Compute_squared_distance_3}
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{Function object providing the operator\\
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\ccc{FT operator()(Point_3, Point_3)} which returns the squared distance
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between two points.}
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\ccNestedType{Compute_squared_radius_3}
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{Function object providing the operator\\
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\ccc{FT operator()(Sphere_3 s)} which returns the squared radius
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of~\ccc{s}.}
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\ccNestedType{Construct_center_3}
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{Function object providing the operator\\
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\ccc{Point_3 operator()(Sphere_3 s)} which returns the center of
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the sphere~\ccc{s}.}
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\ccNestedType{Construct_midpoint_3}
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{Function object providing the operator\\
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\ccc{Point_3 operator()(Point_3 p, Point_3 q)} which computes
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the midpoint of the segment \ccc{pq}.}
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\ccNestedType{Construct_point_on_3}
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{Function object providing the following operators:\\
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\ccc{Point_3 operator()(Line_3 l,int i);} which returns an
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arbitrary point on \ccc{l}. It holds \ccStyle{point(i) == point(j)}, iff
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\ccStyle{i==j}. Furthermore, is directed from \ccStyle{point(i)} to
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\ccStyle{point(j)}, for all \ccStyle{i} $<$ \ccStyle{j}.\\
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\ccc{Point_3 operator()(Ray_3 r,int i);} which returns a point on
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\ccc{r}. \ccStyle{point(0)} is the source, \ccStyle{point(i)}, with
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$i>0$, is different from the source. \ccPrecond $i \geq 0$.\\
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\ccc{Point_3 operator()(Segment_3 s,int i);} which returns either source
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or target of \ccc{s}: \ccStyle{point(0)} returns the source of \ccc{s},
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\ccStyle{point(1)} returns the target of \ccc{s}. Parameter
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\ccStyle{i} is taken modulo 2, which gives easy access to the other end
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point.
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}
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\ccNestedType{Construct_segment_3}
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{Function object providing the operator\\
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\ccc{Segment_3 operator()(Point_3 p, Point_3 q);} which
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returns a segment with source $p$ and target $q$, directed from the
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source to the target.}
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\ccNestedType{Construct_scaled_vector_3}
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{Function object providing the operator\\
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\ccc{Vector_3 operator()(Vector_3 v, FT scale)} which returns
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the vector \ccc{v} scaled by a factor \ccc{scale}.}
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\ccNestedType{Construct_translated_point_3}
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{Function object providing the operator\\
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\ccc{Point_3 operator()(Point_3 p, Vector_3 v)} which returns
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the point obtained by translating \ccc{p} by the vector~\ccc{v}.}
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\ccNestedType{Construct_vector_3}
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{Function object providing the operator\\
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\ccc{Vector_3 operator()(Point_3 a, Point_3 b)} which returns
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the vector \ccc{b-a}.}
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\ccNestedType{Has_on_bounded_side_3}
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{Function object providing the operator\\
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\ccc{bool operator()(Sphere_3 s, Point_3 p);} which
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returns true iff $p$ lies on the bounded side of $s$.}
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\ccOperations
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The following functions give access to the predicate and construction
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objects:
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\ccThree{onstruct_translated_point_3xxx}{construct_translated_point_3_object()x}{}
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\ccCreationVariable{traits}
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\ccMethod{Compute_scalar_product_3 compute_scalar_product_3_object();}{}
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\ccMethod{Compute_squared_distance_3 compute_squared_distance_3_object();}{}
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\ccMethod{Compute_squared_radius_3 compute_squared_radius_3_object();}{}
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\ccMethod{Construct_center_3 construct_center_3_object();}{}
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\ccMethod{Construct_midpoint_3 construct_midpoint_3_object();}{}
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\ccMethod{Construct_point_on_3 construct_point_on_3_object();}{}
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\ccMethod{Construct_scaled_vector_3 construct_scaled_vector_3_object();}{}
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\ccMethod{Construct_segment_3 construct_segment_3_object();}{}
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\ccMethod{Construct_translated_point_3 construct_translated_point_3_object();}{}
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\ccMethod{Construct_vector_3 construct_vector_3_object();}{}
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\ccMethod{Has_on_bounded_side_3 has_on_bounded_side_3_object();}{}
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\ccHasModels
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Any \cgal{} Kernel.
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\ccSeeAlso
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\ccRefConceptPage{ImplicitSurfaceTraits_3} \\
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\ccRefConceptPage{IntersectionGeometricTraits_3} \\
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\ccRefIdfierPage{CGAL::Implicit_mesh_domain_3<Function,BGT>} \\
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\ccRefIdfierPage{CGAL::Labeled_image_mesh_domain_3<Image,BGT>}
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% \ccExample
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% A short example program.
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% Instead of a short program fragment, a full running program can be
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% included using the
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% \verb|\ccIncludeExampleCode{Package/BisectionGeometricTraits_3.C}|
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% macro. The program example would be part of the source code distribution and
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% also part of the automatic test suite.
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% \begin{ccExampleCode}
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% void your_example_code() {
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% }
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% \end{ccExampleCode}
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%% \ccIncludeExampleCode{Package/BisectionGeometricTraits_3.C}
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\end{ccRefConcept}
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% +------------------------------------------------------------------------+
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%%RefPage: end of main body, begin of footer
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\ccRefPageEnd
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% EOF
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% +------------------------------------------------------------------------+
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