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Pencil Of Parallel Lines
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. Penc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meridian (geography)
In geography and geodesy, a meridian is the locus connecting points of equal longitude, which is the angle (in degrees or other units) east or west of a given prime meridian (currently, the IERS Reference Meridian). In other words, it is a coordinate line for longitudes, a line of longitude. The position of a point along the meridian at a given longitude is given by its latitude, measured in angular degrees north or south of the Equator. On a Mercator projection or on a Gall-Peters projection, each meridian is perpendicular to all circles of latitude. Assuming a spherical Earth, a meridian is a great semicircle on Earth's surface. Adopting instead a spheroidal or ellipsoid model of Earth, the meridian is half of a north-south great ellipse. The length of a meridian is twice the length of an Earth quadrant, equal to on a modern ellipsoid ( WGS 84). Pre-Greenwich The first prime meridian was set by Eratosthenes in 200 BC. This prime meridian was used to provide mea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imaginary Number
An imaginary number is the product of a real number and the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square (algebra), square of an imaginary number is . For example, is an imaginary number, and its square is . The number 0, zero is considered to be both real and imaginary. Originally coined in the 17th century by René Descartes as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler (in the 18th century) and Augustin-Louis Cauchy and Carl Friedrich Gauss (in the early 19th century). An imaginary number can be added to a real number to form a complex number of the form , where the real numbers and are called, respectively, the ''real part'' and the ''imaginary part'' of the complex number. History Although the Greek mathematician and engineer Heron of Alexandria is noted as the first t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix. They are also used in fundamental elliptic curve cryptography algorithms. If homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. Since homogeneous coordinates are also given to points at infini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalar (mathematics)
A scalar is an element of a field which is used to define a ''vector space''. In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector. Generally speaking, a vector space may be defined by using any field instead of real numbers (such as complex numbers). Then scalars of that vector space will be elements of the associated field (such as complex numbers). A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, allowing two vectors to be multiplied in the defined way to produce a scalar. A vector space equipped with a scalar product is called an inner product space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Collinear
In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned objects, that is, things being "in a line" or "in a row". Points on a line In any geometry, the set of points on a line are said to be collinear. In Euclidean geometry this relation is intuitively visualized by points lying in a row on a "straight line". However, in most geometries (including Euclidean) a Line (geometry), line is typically a Primitive notion, primitive (undefined) object type, so such visualizations will not necessarily be appropriate. A Mathematical model, model for the geometry offers an interpretation of how the points, lines and other object types relate to one another and a notion such as collinearity must be interpreted within the context of that model. For instance, in spherical g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line At Infinity
In geometry and topology, the line at infinity is a projective line that is added to the affine plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at infinity is also called the ideal line. Geometric formulation In projective geometry, any pair of lines always intersects at some point, but parallel lines do not intersect in the real plane. The line at infinity is added to the real plane. This completes the plane, because now parallel lines intersect at a point which lies on the line at infinity. Also, if any pair of lines do not intersect at a point on the line, then the pair of lines are parallel. Every line intersects the line at infinity at some point. The point at which the parallel lines intersect depends only on the slope of the lines, not at all on their y-intercept. In the affine plane, a line extends in two opposite directions. In the projective plane, the two opposite dire ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Of A Point
In elementary plane geometry, the power of a point is a real number that reflects the relative distance of a given point from a given circle. It was introduced by Jakob Steiner in 1826. Specifically, the power \Pi(P) of a point P with respect to a circle c with center O and radius r is defined by : \Pi(P)=, PO, ^2 - r^2. If P is ''outside'' the circle, then \Pi(P)>0, if P is ''on'' the circle, then \Pi(P)=0 and if P is ''inside'' the circle, then \Pi(P)<0. Due to the Pythagorean theorem the number has the simple geometric meanings shown in the diagram: For a point outside the circle is the squared tangential distance of point to the circle . Points with equal power, isolines of , are circles concentric to circle . Steiner used the power of a point for proofs of several statements on circles, for example: * Determination of a circle, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Apollonian 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 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antipodal Points
In mathematics, two points of a sphere (or n-sphere, including a circle) are called antipodal or diametrically opposite if they are the endpoints of a diameter, a straight line segment between two points on a sphere and passing through its center (geometry), center. Given any point on a sphere, its antipodal point is the unique point at greatest distance, whether measured intrinsically (great-circle distance on the surface of the sphere) or extrinsically (Chord (geometry), chordal distance through the sphere's interior). Every great circle on a sphere passing through a point also passes through its antipodal point, and there are infinitely many great circles passing through a pair of antipodal points (unlike the situation for any non-antipodal pair of points, which have a unique great circle passing through both). Many results in spherical geometry depend on choosing non-antipodal points, and degeneracy (mathematics), degenerate if antipodal points are allowed; for example, a sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
