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[note 3] Mesh analysis can only be applied if it is possible to map the graph onto a plane or a sphere without any of the branches crossing over. Such graphs are called planar graphs. Ability to map onto a plane or a sphere are equivalent conditions. Any finite graph mapped onto a plane can be shrunk until it will map onto a small region of a ...
It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group . The fundamental group is a homotopy invariant —topological spaces that are homotopy equivalent (or the stronger case of homeomorphic ) have isomorphic fundamental groups.
In graph drawing and geometric graph theory, a Tutte embedding or barycentric embedding of a simple, 3-vertex-connected, planar graph is a crossing-free straight-line embedding with the properties that the outer face is a convex polygon and that each interior vertex is at the average (or barycenter) of its neighbors' positions.
A contour line (also isoline, isopleth, isoquant or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. [ 1 ] [ 2 ] It is a plane section of the three-dimensional graph of the function f ( x , y ) {\displaystyle f(x,y)} parallel to the ( x , y ...
Tutte's spring theorem states that every 3-connected planar graph can be drawn on a plane without crossings so that its edges are straight line segments and an outside face is a convex polygon (Tutte 1963). It is so called because such an embedding can be found as the equilibrium position for a system of springs representing the edges of the graph.
Geometric graph theory in the broader sense is a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means. In a stricter sense, geometric graph theory studies combinatorial and geometric properties of geometric graphs, meaning graphs drawn in the Euclidean plane with possibly intersecting straight-line edges, and topological graphs, where the edges are ...
Footage from the wreckage of Azerbaijan Airlines flight JS-8243 shows the plane’s tail peppered with holes. Thirty-eight passengers and crew died when the plane, travelling from the airline’s ...
Conversely, it is easily shown that if a, b, c and d are constants and a, b, and c are not all zero, then the graph of the equation + + + =, is a plane having the vector = (,,) as a normal. [ citation needed ] This familiar equation for a plane is called the general form of the equation of the plane.