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{
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{
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    class Manifold;
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    class Manifold;
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    template<typename ITEM>
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    template<typename ITEM>
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    class VertexAttributeVector;
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    class VertexAttributeVector;
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    double voronoi_area(const Manifold& m,
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    double mixed_area(const Manifold& m,
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                        VertexID v);
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                        VertexID v);
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    double barycentric_area(const Manifold& m,
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    double barycentric_area(const Manifold& m,
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                            VertexID v);
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                            VertexID v);
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