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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 u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trilinear Coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices and respectively; the ratio is the ratio of the perpendicular distances from the point to the sidelines opposite vertices and respectively; and likewise for and vertices and . In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (, , ), or equivalently in ratio form, for any positive constant . If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0. If an exterior point is on the opposite side of a sideline from the interior of the triangle, its trilinear coordinate associated with that sideline is negative. It is impossible ... [...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 Greek geometer. Definition The Apollonian circles are defined in two different ways by a line segment denoted ''CD''. Each circle in the first family (the blue circles in the figure) is associated with a positive real number ''r'', and is defined as the locus of points ''X'' such that the ratio of distances from ''X'' to ''C'' and to ''D'' equals ''r'', :\left\. For values of ''r'' close to zero, the corresponding circle is close to ''C'', while for values of ''r'' close to ∞, the corresponding circle is close to ''D''; for the intermediate value ''r'' = 1, the circle degenerates to a line, the perpendicular bisector of ''CD''. The equation defining these circles as a lo ... [...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 lines that pass through a given point in a 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 set is a ''range'' of points. Penci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipolar Coordinates
Bipolar coordinates are a two-dimensional 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 Confusingly, the same term is also sometimes used for two-center bipolar coordinates. 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 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 di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Line
A field line is a graphical visual aid for visualizing vector fields. It consists of an imaginary directed line which is tangent to the field 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. Definition and description A vector field defines a direction and magnitude at each point in space. A field line for that vector field 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 to the field vector at each point. A field line is usually shown as a directed line segment, with an arrow indicatin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of atoms and molecules. Electromagnetism can be thought of as a combination of electricity and magnetism, two distinct but closely intertwined phenomena. In essence, electric forces occur between any two charged particles, causing an attraction between particles with opposite charges and repulsion between particles with the same charge, while magnetism is an interaction that occurs exclusively between ''moving'' charged particles. These two effects combine to create electromagnetic fields in the vicinity of charge particles, which can exert influence on other particles via the Lorentz force. At high energy, the weak force and electromagnetic force are unified as a single electroweak force. The electromagnetic force is responsible for m ... [...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 variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' − ''bc'' ≠ 0. Geometrically, a Möbius transformation can be obtained by first performing stereographic projection from the plane to the unit two-sphere, rotating and moving the sphere to a new location and orientation in space, and then performing stereographic projection (from the new position of the sphere) 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 called the Möbius group, which is the projective linear group PGL(2,C). Together with its subgroups, it has numerous applications in mathematics and physics. Möbius tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Inversion
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. Usually, the radius is required to be a positive number. A circle with r=0 (a single point) is a degenerate case. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. Specifically, a circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a '' disc''. A circle may also be defined as a special kin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
