Euler operator
Encyclopedia
In mathematics
, Euler operators are a small set of functions to create polygon mesh
es. They are closed and sufficient on the set of meshes, and they are invertible
.
, with vertices
, and with edges that connect these vertices. In addition to a graph, a mesh has also faces
: Let the graph be drawn ("embedded") in a two-dimensional plane
, in such a way that the edges do not cross (which is possible only if the graph is a planar graph
). Then the contiguous 2D regions on either side of each edge are the faces of the mesh.
The Euler operators are functions
to manipulate meshes. They are very straightforward: Create a new vertex (in some face), connect vertices, split a face by inserting a diagonal, subdivide an edge by inserting a vertex. It is immediately clear that these operations are invertible.
Further Euler operators exist to create higher-genus
shapes, for instance to connect the ends of a bent tube to create a torus
.
relationship, i.e., which face is bounded by which face, which vertex is connected to which other vertex, and so on. They are not concerned with the geometric properties: The length of an edge, the position of a vertex, and whether a face is curved or planar, are just geometric "attributes".
Note: In topology
, objects can arbitrarily deform. So a valid mesh can, e.g., collapse to a single point if all of its vertices happen to be at the same position in space.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Euler operators are a small set of functions to create polygon mesh
Polygon mesh
A polygon mesh or unstructured grid is a collection of vertices, edges and faces that defines the shape of a polyhedral object in 3D computer graphics and solid modeling...
es. They are closed and sufficient on the set of meshes, and they are invertible
Inverse function
In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...
.
Purpose
A "polygon mesh" can be thought of as a graphGraph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
, with vertices
Vertex (graph theory)
In graph theory, a vertex or node is the fundamental unit out of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges , while a directed graph consists of a set of vertices and a set of arcs...
, and with edges that connect these vertices. In addition to a graph, a mesh has also faces
Face (geometry)
In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
: Let the graph be drawn ("embedded") in a two-dimensional plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
, in such a way that the edges do not cross (which is possible only if the graph is a planar graph
Planar graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints...
). Then the contiguous 2D regions on either side of each edge are the faces of the mesh.
The Euler operators are functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
to manipulate meshes. They are very straightforward: Create a new vertex (in some face), connect vertices, split a face by inserting a diagonal, subdivide an edge by inserting a vertex. It is immediately clear that these operations are invertible.
Further Euler operators exist to create higher-genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
shapes, for instance to connect the ends of a bent tube to create a torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
.
Properties
Euler operators are topological operators: They modify only the incidenceIncidence (geometry)
In geometry, the relations of incidence are those such as 'lies on' between points and lines , and 'intersects' . That is, they are the binary relations describing how subsets meet...
relationship, i.e., which face is bounded by which face, which vertex is connected to which other vertex, and so on. They are not concerned with the geometric properties: The length of an edge, the position of a vertex, and whether a face is curved or planar, are just geometric "attributes".
Note: In topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, objects can arbitrarily deform. So a valid mesh can, e.g., collapse to a single point if all of its vertices happen to be at the same position in space.