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Quadratrix No Anim
In geometry, a quadratrix () is a curve having ordinates which are a measure of the area (or quadrature) of another curve. The two most famous curves of this class are those of Dinostratus and Ehrenfried Walther von Tschirnhaus, E. W. Tschirnhaus, which are both related to the circle. Quadratrix of Dinostratus The quadratrix of Dinostratus (also called the ''quadratrix of Hippias'') was well known to the ancient Greek geometers, and is mentioned by Proclus, who ascribes the invention of the curve to a contemporary of Socrates, probably Hippias of Elis. Dinostratus, a Greek geometer and disciple of Plato, discussed the curve, and showed how it effected a mechanical solution of squaring the circle. Pappus of Alexandria, Pappus, in his ''Collections'', treats its history, and gives two methods by which it can be generated. # Let a helix be drawn on a right circular cylinder (geometry), cylinder; a screw surface is then obtained by drawing line (geometry), lines from every point of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can be defined between two lines (or two line segments), between a line and a plane, and between two planes. Perpendicularity is one particular instance of the more general mathematical concept of '' orthogonality''; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its '' normal vector''. Definitions A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one sid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeros And Poles
In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. In some sense, it is the simplest type of singularity. Technically, a point is a pole of a function if it is a zero of the function and is holomorphic in some neighbourhood of (that is, complex differentiable in a neighbourhood of ). A function is meromorphic in an open set if for every point of there is a neighborhood of in which either or is holomorphic. If is meromorphic in , then a zero of is a pole of , and a pole of is a zero of . This induces a duality between ''zeros'' and ''poles'', that is fundamental for the study of meromorphic functions. For example, if a function is meromorphic on the whole complex plane plus the point at infinity, then the sum of the multiplicities of its poles equals the sum of the multiplicities of its zeros. Definitions A function of a complex variable is holomorphic in an open d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cotangent
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis. The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions. The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend the sine a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Coordinate System
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference coordinate line is called a ''coordinate axis'' or just ''axis'' (plural ''axes'') of the system, and the point where they meet is its ''origin'', at ordered pair . The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, ''n'' Cartesian coordinates (an element of real ''n''-space) specify the point in an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locus (mathematics)
In geometry, a locus (plural: ''loci'') (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.. In other words, the set of the points that satisfy some property is often called the ''locus of a point'' satisfying this property. The use of the singular in this formulation is a witness that, until the end of the 19th century, mathematicians did not consider infinite sets. Instead of viewing lines and curves as sets of points, they viewed them as places where a point may be ''located'' or may move. History and philosophy Until the beginning of the 20th century, a geometrical shape (for example a curve) was not considered as an infinite set of points; rather, it was considered as an entity on which a point may be located or on which it moves. Thus a circle in the Euclidean plane was defined as the ''locus'' of a point that is at a given d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel (geometry)
In geometry, parallel lines are coplanar straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. ''Parallel curves'' are curves that do not touch each other or intersect and keep a fixed minimum distance. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called '' skew lines''. Parallel lines are the subject of Euclid's parallel postulate. Parallelism is primarily a property of affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines can have analogous properties that are referred to as parallelism. Symbol The parallel symbol is \parallel. For example, AB \parallel CD indicates that line ''AB'' is parallel to line ''CD''. In the Unicode character set, the "parallel" and "not parallel" signs have co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circular Sector
A circular sector, also known as circle sector or disk sector (symbol: ⌔), is the portion of a disk (a closed region bounded by a circle) enclosed by two radii and an arc, where the smaller area is known as the ''minor sector'' and the larger being the ''major sector''. In the diagram, is the central angle, r the radius of the circle, and L is the arc length of the minor sector. The angle formed by connecting the endpoints of the arc to any point on the circumference that is not in the sector is equal to half the central angle. Types A sector with the central angle of 180° is called a '' half-disk'' and is bounded by a diameter and a semicircle. Sectors with other central angles are sometimes given special names, such as quadrants (90°), sextants (60°), and octants (45°), which come from the sector being one 4th, 6th or 8th part of a full circle, respectively. Confusingly, the arc of a quadrant (a circular arc) can also be termed a quadrant. Compass Traditionally ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratrix No Anim
In geometry, a quadratrix () is a curve having ordinates which are a measure of the area (or quadrature) of another curve. The two most famous curves of this class are those of Dinostratus and Ehrenfried Walther von Tschirnhaus, E. W. Tschirnhaus, which are both related to the circle. Quadratrix of Dinostratus The quadratrix of Dinostratus (also called the ''quadratrix of Hippias'') was well known to the ancient Greek geometers, and is mentioned by Proclus, who ascribes the invention of the curve to a contemporary of Socrates, probably Hippias of Elis. Dinostratus, a Greek geometer and disciple of Plato, discussed the curve, and showed how it effected a mechanical solution of squaring the circle. Pappus of Alexandria, Pappus, in his ''Collections'', treats its history, and gives two methods by which it can be generated. # Let a helix be drawn on a right circular cylinder (geometry), cylinder; a screw surface is then obtained by drawing line (geometry), lines from every point of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conical Surface
In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the ''apex'' or ''vertex'' — and any point of some fixed space curve — the ''directrix'' — that does not contain the apex. Each of those lines is called a ''generatrix'' of the surface. Every conic surface is ruled and developable. In general, a conical surface consists of two congruent unbounded halves joined by the apex. Each half is called a nappe, and is the union of all the rays that start at the apex and pass through a point of some fixed space curve. (In some cases, however, the two nappes may intersect, or even coincide with the full surface.) Sometimes the term "conical surface" is used to mean just one nappe. If the directrix is a circle C, and the apex is located on the circle's ''axis'' (the line that contains the center of C and is perpendicular to its plane), one obtains the ''right circul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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