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A vertex of an angle is the endpoint where two lines or rays come together. In geometry, a vertex (pl.: vertices or vertexes) is a point where two or more curves, lines, or edges meet or intersect. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. [1] [2] [3]
The pencil of conic sections with the x axis as axis of symmetry, one vertex at the origin (0, 0) and the same semi-latus rectum can be represented by the equation = + (),, with the eccentricity. For e = 0 {\displaystyle e=0} the conic is a circle (osculating circle of the pencil),
A peripheral vertex in a graph of diameter d is one whose eccentricity is d —that is, a vertex whose distance from its furthest vertex is equal to the diameter. Formally, v is peripheral if ϵ(v) = d. A pseudo-peripheral vertex v has the property that, for any vertex u, if u is as far away from v as possible, then v is as far away from u as
A vertex with degree 1 is called a leaf vertex or end vertex or a pendant vertex, and the edge incident with that vertex is called a pendant edge. In the graph on the right, {3,5} is a pendant edge. This terminology is common in the study of trees in graph theory and especially trees as data structures .
In the geometry of plane curves, a vertex is a point of where the first derivative of curvature is zero. [1] This is typically a local maximum or minimum of curvature, [ 2 ] and some authors define a vertex to be more specifically a local extremum of curvature. [ 3 ]
The quadratic formula can equivalently be written using various alternative expressions, for instance = (), which can be derived by first dividing a quadratic equation by , resulting in + + = , then substituting the new coefficients into the standard quadratic formula.
The formula for calculating it can be derived and expressed in several ways. Knowing the shortest distance from a point to a line can be useful in various situations—for example, finding the shortest distance to reach a road, quantifying the scatter on a graph, etc.
The formula can be proved by using mathematical induction: starting with a triangle, for which the angle sum is 180°, then replacing one side with two sides connected at another vertex, and so on. The sum of the external angles of any simple polygon, if only one of the two external angles is assumed at each vertex, is 2π radians (360°).