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Lecture

Graph Surfaces and Sphere in Spherical Coordinates

Description

This lecture covers graph surfaces with an example of a surface defined by a parametric equation, followed by a detailed description of a sphere in spherical coordinates.

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Related concepts (32)

Sphere

A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. That given point is the centre of the sphere, and r is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry.

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. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.

Graph embedding

In topological graph theory, an embedding (also spelled imbedding) of a graph on a surface is a representation of on in which points of are associated with vertices and simple arcs (homeomorphic images of ) are associated with edges in such a way that: the endpoints of the arc associated with an edge are the points associated with the end vertices of no arcs include points associated with other vertices, two arcs never intersect at a point which is interior to either of the arcs. Here a surface is a compact, connected -manifold.

Graph minor

In graph theory, an undirected graph H is called a minor of the graph G if H can be formed from G by deleting edges, vertices and by contracting edges. The theory of graph minors began with Wagner's theorem that a graph is planar if and only if its minors include neither the complete graph K5 nor the complete bipartite graph K3,3. The Robertson–Seymour theorem implies that an analogous forbidden minor characterization exists for every property of graphs that is preserved by deletions and edge contractions.

Longitude

Longitude (ˈlɒndʒᵻtjuːd, ˈlɒŋɡᵻ-) is a geographic coordinate that specifies the east–west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter lambda (λ). Meridians are imaginary semicircular lines running from pole to pole that connect points with the same longitude. The prime meridian defines 0° longitude; by convention the International Reference Meridian for the Earth passes near the Royal Observatory in Greenwich, south-east London on the island of Great Britain.

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