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Radical Axis
In Euclidean geometry, the radical axis of two non-concentric circles is the set of points whose Power of a point, power with respect to the circles are equal. For this reason the radical axis is also called the power line or power bisector of the two circles. In detail: For two circles with centers and radii the powers of a point with respect to the circles are :\Pi_1(P)=, PM_1, ^2 - r_1^2,\qquad \Pi_2(P)= , PM_2, ^2 - r_2^2. Point belongs to the radical axis, if : \Pi_1(P)=\Pi_2(P). If the circles have two points in common, the radical axis is the common secant line of the circles. If point is outside the circles, has equal tangential distance to both the circles. If the radii are equal, the radical axis is the line segment bisector of . In any case the radical axis is a line perpendicular to \overline. ;On notations The notation ''radical axis'' was used by the French mathematician Michel Chasles, M. Chasles as ''axe radical''. Jean-Victor Poncelet, J.V. Poncelet us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coaxal Circles
In geometry, Apollonian circles are two families (pencils) of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. They were discovered by Apollonius of Perga, a renowned ancient Greek geometer. Definition The Apollonian circles are defined in two different ways by a line segment denoted . Each circle in the first family (the blue circles in the figure) is associated with a positive real number , and is defined as the locus of points such that the ratio of distances from to and to equals , \left\. For values of close to zero, the corresponding circle is close to , while for values of close to , the corresponding circle is close to ; for the intermediate value , the circle degenerates to a line, the perpendicular bisector of . The equation defining these circles as a locus can be generalized to define the Fermat–Apollonius circles of larger se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pencil (mathematics)
In geometry, a pencil is a family of geometric objects with a common property, for example the set of Line (geometry), lines that pass through a given point in a plane (mathematics), plane, or the set of circles that pass through two given points in a plane. Although the definition of a pencil is rather vague, the common characteristic is that the pencil is completely determined by any two of its members. Analogously, a set of geometric objects that are determined by any three of its members is called a bundle. Thus, the set of all lines through a point in three-space is a bundle of lines, any two of which determine a pencil of lines. To emphasize the two-dimensional nature of such a pencil, it is sometimes referred to as a ''flat pencil''. Any geometric object can be used in a pencil. The common ones are lines, planes, circles, conics, spheres, and general curves. Even points can be used. A pencil of points is the set of all points on a given line. A more common term for this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipolar Coordinates
Bipolar coordinates are a two-dimensional orthogonal coordinates, orthogonal coordinate system based on the Apollonian circles.Eric W. Weisstein, Concise Encyclopedia of Mathematics CD-ROM, ''Bipolar Coordinates'', CD-ROM edition 1.0, May 20, 1999 There is also a third system, based on two poles (biangular coordinates). The term "bipolar" is further used on occasion to describe other curves having two singular points (foci), such as ellipses, hyperbolas, and Cassini ovals. However, the term ''bipolar coordinates'' is reserved for the coordinates described here, and never used for systems associated with those other curves, such as elliptic coordinates. Definition The system is based on two Focus (geometry), foci ''F''1 and ''F''2. Referring to the figure at right, the ''σ''-coordinate of a point ''P'' equals the angle ''F''1 ''P'' ''F''2, and the ''τ''-coordinate equals the natural logarithm of the ratio of the distances ''d''1 and ''d''2: : \tau = \ln \frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Line
A field line is a graphical Scientific visualization, visual aid for visualizing vector fields. It consists of an imaginary integral curve which is tangent to the field Euclidean vector, vector at each point along its length. A diagram showing a representative set of neighboring field lines is a common way of depicting a vector field in scientific and mathematical literature; this is called a field line diagram. They are used to show electric fields, magnetic fields, and gravitational fields among many other types. In fluid mechanics, field lines showing the velocity field of a fluid flow are called Streamlines, streaklines, and pathlines, streamlines. Definition and description A vector field defines a direction and magnitude at each point in space. A field line is an integral curve for that vector field and may be constructed by starting at a point and tracing a line through space that follows the direction of the vector field, by making the field line tangent line, tang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interactions of atoms and molecules. Electromagnetism can be thought of as a combination of electrostatics and magnetism, which are distinct but closely intertwined phenomena. Electromagnetic forces occur between any two charged particles. Electric forces cause an attraction between particles with opposite charges and repulsion between particles with the same charge, while magnetism is an interaction that occurs between charged particles in relative motion. These two forces are described in terms of electromagnetic fields. Macroscopic charged objects are described in terms of Coulomb's law for electricity and Ampère's force law for magnetism; the Lorentz force describes microscopic charged particles. The electromagnetic force is responsible for ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalised Circle
In geometry, a generalized circle, sometimes called a ''cline'' or ''circline'', is a straight line or a circle, the curves of constant curvature in the Euclidean plane. The natural setting for generalized circles is the extended plane, a plane along with one point at infinity through which every straight line is considered to pass. Given any three distinct points in the extended plane, there exists precisely one generalized circle passing through all three. Generalized circles sometimes appear in Euclidean geometry, which has a well-defined notion of distance between points, and where every circle has a center and radius: the point at infinity can be considered infinitely distant from any other point, and a line can be considered as a degenerate circle without a well-defined center and with infinite radius (zero curvature). A reflection across a line is a Euclidean isometry (distance-preserving transformation) which maps lines to lines and circles to circles; but an inversio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying . Geometrically, a Möbius transformation can be obtained by first applying the inverse stereographic projection from the plane to the unit sphere, moving and rotating the sphere to a new location and orientation in space, and then applying a stereographic projection to map from the sphere back to the plane. These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle. The Möbius transformations are the projective transformations of the complex projective line. They form a group (mathematics), group called the Möbius group, which is the projective linear group . Together with its subgroups, it has numerous applications in mathematics and physics. Möbius geometry, Möbius geometries and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Inversion
In geometry, inversive geometry is the study of ''inversion'', a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry become much more tractable when an inversion is applied. Inversion seems to have been discovered by a number of people contemporaneously, including Steiner (1824), Quetelet (1825), Bellavitis (1836), Stubbs and Ingram (1842–3) and Kelvin (1845). The concept of inversion can be generalized to higher-dimensional spaces. Inversion in a circle Inverse of a point To invert a number in arithmetic usually means to take its reciprocal. A closely related idea in geometry is that of "inverting" a point. In the plane, the inverse of a point ''P'' with respect to a ''reference circle (Ø)'' with center ''O'' and radius ''r'' is a point ''P'', lying on the ray from ''O'' through ''P'' such that :OP \cdot OP^ = r^2. This is call ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
