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Trisectrix
In geometry, a trisectrix is a curve which can be used to trisect an arbitrary angle with ruler and compass and this curve as an additional tool. Such a method falls outside those allowed by compass and straightedge constructions, so they do not contradict the well known theorem which states that an arbitrary angle cannot be trisected with that type of construction. There is a variety of such curves and the methods used to construct an angle trisector differ according to the curve. Examples include: * Limaçon trisectrix (some sources refer to this curve as simply the trisectrix.) * Trisectrix of Maclaurin * Equilateral trefoil (a.k.a. Longchamps' Trisectrix) * Tschirnhausen cubic (a.k.a. Catalan's trisectrix and L'Hôpital's cubic) * Durer's folium * Cubic parabola * Hyperbola with eccentricity 2 * Rose curve specified by a sinusoid with angular frequency of one-third. * Parabola A related concept is a sectrix, which is a curve which can be used to divide an arbitrary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limaçon Trisectrix
In geometry, a limaçon trisectrix is the name for the quartic plane curve that is a trisectrix that is specified as a limaçon. The shape of the limaçon trisectrix can be specified by other curves particularly as a rose (mathematics), rose, conchoid (mathematics), conchoid or epitrochoid. The curve is one among a number of plane curve trisectrixes that includes the Conchoid of Nicomedes, the Tommaso Ceva#The Cycloid of Ceva, Cycloid of Ceva, Quadratrix of Hippias, Trisectrix of Maclaurin, and Tschirnhausen cubic. The limaçon trisectrix a special case of a sectrix of Maclaurin. Specification and loop structure The limaçon trisectrix specified as a polar equation is :r= a(1+2\cos\theta). The constant a may be positive or negative. The two curves with constants a and -a are reflection symmetry, reflections of each other across the line \theta=\pi/2. The period of r= a(1+2\cos\theta) is 2\pi given the period of the sine wave, sinusoid \cos\theta. The limaçon trisectrix is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trisectrix Of Maclaurin
In algebraic geometry, the trisectrix of Maclaurin is a cubic plane curve notable for its trisectrix property, meaning it can be used to trisect an angle. It can be defined as locus of the point of intersection of two lines, each rotating at a uniform rate about separate points, so that the ratio of the rates of rotation is 1:3 and the lines initially coincide with the line between the two points. A generalization of this construction is called a sectrix of Maclaurin. The curve is named after Colin Maclaurin who investigated the curve in 1742. Equations Let two lines rotate about the points P = (0,0) and P_1 = (a, 0) so that when the line rotating about P has angle \theta with the ''x'' axis, the rotating about P_1 has angle 3\theta. Let Q be the point of intersection, then the angle formed by the lines at Q is 2\theta. By the law of sines, : = \! so the equation in polar coordinates is (up to translation and rotation) :r= a \frac = \frac = (4 \cos \theta - \sec \theta).\! Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sectrix Of Delanges
In geometry, a trisectrix is a curve which can be used to trisect an arbitrary angle with ruler and compass and this curve as an additional tool. Such a method falls outside those allowed by compass and straightedge constructions, so they do not contradict the well known theorem which states that an arbitrary angle cannot be trisected with that type of construction. There is a variety of such curves and the methods used to construct an angle trisector differ according to the curve. Examples include: * Limaçon trisectrix (some sources refer to this curve as simply the trisectrix.) * Trisectrix of Maclaurin * Equilateral trefoil (a.k.a. Longchamps' Trisectrix) * Tschirnhausen cubic (a.k.a. Catalan's trisectrix and L'Hôpital's cubic) * Durer's folium * Cubic parabola * Hyperbola with eccentricity 2 * Rose curve specified by a sinusoid with angular frequency of one-third. * Parabola A related concept is a sectrix, which is a curve which can be used to divide an arbitrary angl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angle Trisection
Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a Compass (drawing tool), compass. In 1837, Pierre Wantzel proved that the problem, as stated, is Proof of impossibility, impossible to solve for arbitrary angles. However, some special angles can be trisected: for example, it is trivial to trisect a right angle. It is possible to trisect an arbitrary angle by using tools other than straightedge and compass. For example, neusis construction, also known to ancient Greeks, involves simultaneous sliding and rotation of a marked straightedge, which cannot be achieved with the original tools. Other techniques were developed by mathematicians over the centuries. Because it is defined in simple terms, but complex to prove unsolvable, the problem of angle trisection is a frequent subject of pse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sectrix Of Maclaurin
In geometry, a sectrix of Maclaurin is defined as the curve swept out by the point of intersection of two lines which are each revolving at constant rates about different points called poles. Equivalently, a sectrix of Maclaurin can be defined as a curve whose equation in biangular coordinates is linear. The name is derived from the trisectrix of Maclaurin (named for Colin Maclaurin), which is a prominent member of the family, and their sectrix property, which means they can be used to divide an angle into a given number of equal parts. There are special cases known as arachnida or araneidans because of their spider-like shape, and Plateau curves after Joseph Plateau who studied them. Equations in polar coordinates We are given two lines rotating about two poles P and P_1. By translation and rotation we may assume P = (0,0) and P_1 = (a, 0). At time t, the line rotating about P has angle \theta = \kappa t + \alpha and the line rotating about P_1 has angle \theta_1 = \kappa_1 t + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neusis Construction
In geometry, the neusis (; ; plural: ) is a geometric construction method that was used in antiquity by Greek mathematicians. Geometric construction The neusis construction consists of fitting a line element of given length () in between two given lines ( and ), in such a way that the line element, or its extension, passes through a given point . That is, one end of the line element has to lie on , the other end on , while the line element is "inclined" towards . Point is called the pole of the neusis, line the directrix, or guiding line, and line the catch line. Length is called the ''diastema'' (). A neusis construction might be performed by means of a marked ruler that is rotatable around the point (this may be done by putting a pin into the point and then pressing the ruler against the pin). In the figure one end of the ruler is marked with a yellow eye with crosshairs: this is the origin of the scale division on the ruler. A second marking on the ruler (the blue ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a Point (geometry), point (the Focus (geometry), focus) and a Line (geometry), line (the Directrix (conic section), directrix). The focus does not lie on the directrix. The parabola is the locus (mathematics), locus of points in that plane that are equidistant from the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane (geometry), plane Parallel (geometry), parallel to another plane that is tangential to the conical surface. The graph of a function, graph of a quadratic function y=ax^2+bx+ c (with a\neq 0 ) is a parabola with its axis parallel to the -axis. Conversely, every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doubling The Cube
Doubling the cube, also known as the Delian problem, is an ancient geometry, geometric problem. Given the Edge (geometry), edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related problems of squaring the circle and trisecting the angle, doubling the cube is now known to be impossible to construct by using only a compass and straightedge, but even in ancient times solutions were known that employed other methods. According to Eutocius, Archytas was the first to solve the problem of doubling the cube (the so-called Delian problem) with an ingenious geometric construction. The nonexistence of a compass-and-straightedge solution was finally proven by Pierre Wantzel in 1837. In algebraic terms, doubling a unit cube requires the construction of a line segment of length , where ; in other words, , the cube root of two. This is because a cube of side length 1 has a volume of , and a cube of twice tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratrix Of Hippias
The quadratrix or trisectrix of Hippias (also called the quadratrix of Dinostratus) is a curve which is created by a uniform motion. It is traced out by the crossing point of two Line (geometry), lines, one moving by translation (geometry), translation at a uniform speed, and the other moving by rotation (geometry), rotation around one of its points at a uniform speed. An alternative definition as a parametric curve leads to an equivalence between the quadratrix, the image of the Lambert W function, and the graph of the function y=x\cot x. The discovery of this curve is attributed to the Greek sophist Hippias, Hippias of Elis, who used it around 420 BC in an attempt to solve the Angle trisection, angle trisection problem, hence its name as a trisectrix. Later around 350 BC Dinostratus used it in an attempt to solve the problem of squaring the circle, hence its name as a quadratrix. Dinostratus's theorem, used in this attempt, relates an endpoint of the curve to the value of . Bot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedean Spiral
The Archimedean spiral (also known as Archimedes' spiral, the arithmetic spiral) is a spiral named after the 3rd-century BC Ancient Greece, Greek mathematician Archimedes. The term ''Archimedean spiral'' is sometimes used to refer to the more general class of spirals of this type (see below), in contrast to ''Archimedes' spiral'' (the specific arithmetic spiral of Archimedes). It is the locus (mathematics), locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in Polar coordinate system, polar coordinates it can be described by the equation r = b\cdot\theta with real number . Changing the parameter controls the distance between loops. From the above equation, it can thus be stated: position of the particle from point of start is proportional to angle as time elapses. Archimedes described such a spiral in his book ''On Spirals''. Conon of Samos was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
