BROWSE ALPHABETICALLY LEVEL:    Elementary    Advanced    Both INCLUDE TOPICS:    Basic Math    Algebra    Analysis    Biography    Calculus    Comp Sci    Discrete    Economics    Foundations    Geometry    Graph Thry    History    Number Thry    Phys Sci    Statistics    Topology    Trigonometry Dandelin’s Spheres – ellipse Dandelin’s Spheres    ARTICLE   A proof by the 17th century French mathematician Germinal Dandelin of the equivalence of the plane-geometry and conic-section definitions of the ellipse, parabola, and hyperbola. See the article for a full exposition. degenerate conic   A conic section in which the intersecting plane passes through the vertex. Related article: Conics degree   Geometry: A unit of measure of angles, denoted by “ ° ”. A complete circle contains 360°, a straight line 180°, and a right angle 90°. Compare radians.Algebra: The degree of a polynomial is the highest sum of exponents appearing in any term with a non-zero coefficient.Graph Theory: The degree of a vertex is the number of edges incident on that vertex, counting loops twice. The degree of an edge is an unordered pair of the degrees of its two ends. diameter   Geometry: A diameter of a circle (or sphere) is a line containing the center and with endpoints on the perimeter (resp. surface).Analysis: Given a set X in a metric space, the diameter of X is the supremum of the distances between all pairs of points of X.Graph Theory: The diameter of a given graph G is the maximum, over all pairs of vertices u, v of G that are in the same connected component of G, of the distance between u and v. In other words, it is the greatest distance between two vertices on the graph. difference   The difference of two numbers m and n, with n > m, is the number which when added to m yields n. For example, the difference of 3 and 5 is 2.Set Theory: The difference of two sets A and B, denoted either as AB or as A - B, is the set of elements of A that are not in B. discrete   General: So-called “Discrete Mathematics” consists of those branches of mathematics which are concerned with the relations among fixed rather than continuously varying quantities, e.g., combinatorics and probability.Topology: A topology on a set X is discrete if every subset of X is open, or equivalently if every one-point set of X is open. disjoint   Two sets are disjoint if they have empty intersection. disjoint union   A union of sets which are disjoint. disk   A set of points consisting of a circle together with its interior points. The set consisting only of the interior points of a circle is called an open disk.Cf. neighborhood. distance   The distance between two points in a space is given by the length of the geodesic joining those two points. In Euclidean space, the geodesic is given by a straight line, and the distance between two points is the length of this line. The distance between two points a and b on a real number line is the absolute value of their difference, i.e., d(a, b) = |a - b|. In two (or more) dimensions, the distance is given by the (generalized) Pythagorean theorem, i.e., in a Cartesian coordinate system of n dimensions, where a = (a1, ... ,an) and b = (b1, ... ,bn), the distance d(a, b) is given byThe concept of distance may be generalized to more abstract spaces – such a distance concept is referred to as a metric.Graph Theory: The length of the shortest path between two vertices of a graph. If there is no path between two vertices, their distance is defined to be infinite. The distance between two vertices v and u is denoted by d(v, u). In a connected graph, distance is a metric. distributive   distributive property   An algebraic property of numbers which states that for all numbers a, b, and c, a(b + c) = ab + ac. Cf. commutative, associative. divide   To divide a number a by another number b is to find a third number c such that the product of b and c is a, that is, b × c = a. The number a is called the dividend, the number b is called the divisor, and the number c is called the quotient. The operation of dividing may be denoted by a horizontal or diagonal slash separating the dividend and divisor (with the dividend on top), or by a horizontal dash with a dot above and below it placed between the dividend and divisor.In the case of whole numbers a and b there may not be a whole number quotient; however, there are always unique whole numbers q and r such that a = b × q + r, with r < b. In this case q is called the quotient and r is called the remainder. If in a particular case r = 0, we say that b divides a, and this is often denoted by b|a. dividend   A number that is being divided. divisor   A number that is dividing another. dodecahedron   A polyhedron having twelve faces.The faces of a regular dodecahedron are regular pentagons.Cf. Platonic solid. domain   General: A universe of discourse, that is, the class of objects under consideration. Functions and relations: The domain of a function (relation) is the set of elements which the function (relation) maps to its range set. e   See Euler number. edge   General: A line formed by the intersection of two planes. In a 3-dimensional figure (such as a polyhedron), the line or curve where two faces or surfaces meet.Graph Theory: One of two kinds of entities in a graph. Restricted to being incident on exactly two vertices. edge set   The set of edges of some graph. For a graph G, the edge set of G is denoted by E(G), or, if there is no ambiguity as to the graph in question, simply by E. ellipse   The locus of points in the plane, the sum of whose distances from two fixed points, called the foci, remains constant.Like the hyperbola and parabola, the ellipse is a conic section. Related article: Conics Dandelin’s Spheres – ellipse
 HOME | ABOUT | CONTACT | AD INFO | PRIVACYCopyright © 1997-2013, Math Academy Online™ / Platonic Realms™. Except where otherwise prohibited, material on this site may be printed for personal classroom use without permission by students and instructors for non-profit, educational purposes only. All other reproduction in whole or in part, including electronic reproduction or redistribution, for any purpose, except by express written agreement is strictly prohibited. Please send comments, corrections, and enquiries using our contact page.