A Multi-Triangulation (MT) is a tool which provides representations of a surface through triangle meshes at variable resolution, i.e., meshes which can be more or less refined according to user-defined requirements. The user queries an MT by specifying two parameters:
The MT answers the query by returning a triangle mesh which represents the surface with the required accuracy inside the focus area.
A large number of relevant spatial operations can be performed at variable resolution on a surface by querying an MT built for such a surface. Examples of such queries are:
A variable resolution means that the resolution (density) of the triangle mesh can be locally adapted to the needs of the application. For instance:
In the following example we can see a triangle mesh by two different
viewpoints. The mesh is extracted considering a conical region with
resolution decreasing with distance from the vertex of the cone.
An MT is made of a collection of elementary triangle meshes, each
representing a portion of the surface at a certain resolution. Many
alternative representations of the same piece of surface exist in an
MT. A mesh representing the whole surface at variable resolution can be
obtained by assembling the elementary triangle meshes. The triangle
meshes forming an MT are arranged in a directed acyclic graph, which
provides all the ways in which they can be combined to provide
consistent surface descriptions.
The construction of an MT is based on iterative methods for the generation of triangle meshes representing a surface. Such methods are of two kinds:
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