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In inversive geometry, an inverse curve of a given curve C is the result of applying an inverse operation to C. Specifically, with respect to a fixed circle with center O and radius k the inverse of a point Q is the point P for which P lies on the ray OQ and OP·OQ = k 2. The inverse of the curve C is then the locus of P as Q runs over C.
A curve may have equivalent parametrizations when there is a continuous increasing monotonic function relating the parameter of one curve to the parameter of the other. When there is a decreasing continuous function relating the parameters, then the parametric representations are opposite and the orientation of the curve is reversed. [1] [2]
Apex graphs are closed under the operation of taking minors: contracting any edge, or removing any edge or vertex, leads to another apex graph.For, if G is an apex graph with apex v, then any contraction or removal that does not involve v preserves the planarity of the remaining graph, as does any edge removal of an edge incident to v.
Several graph-theoretic concepts are related to each other via complementation: The complement of an edgeless graph is a complete graph and vice versa. Any induced subgraph of the complement graph of a graph G is the complement of the corresponding induced subgraph in G. An independent set in a graph is a clique in the complement graph and vice ...
The term apex may used in different contexts: In an isosceles triangle, the apex is the vertex where the two sides of equal length meet, opposite the unequal third side. [1] Here the point A is the apex. In a pyramid or cone, the apex is the vertex at the "top" (opposite the base). In a pyramid, the vertex is the point that is part of all the ...
A universal graph is a graph that contains as subgraphs all graphs in a given family of graphs, or all graphs of a given size or order within a given family of graphs. 2. A universal vertex (also called an apex or dominating vertex) is a vertex that is adjacent to every other vertex in the graph.
The concept of duality can be extended to graph embeddings on two-dimensional manifolds other than the plane. The definition is the same: there is a dual vertex for each connected component of the complement of the graph in the manifold, and a dual edge for each graph edge connecting the two dual vertices on either side of the edge. In most ...
The following example shows that in some cases the envelope of a family of curves may be seen as the topologic boundary of a union of sets, whose boundaries are the curves of the envelope. For s > 0 {\displaystyle s>0} and t > 0 {\displaystyle t>0} consider the (open) right triangle in a Cartesian plane with vertices ( 0 , 0 ) {\displaystyle (0 ...