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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 ang ... [...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, conchoid or epitrochoid. The curve is one among a number of plane curve trisectrixes that includes the Conchoid of Nicomedes, the 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 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 sinusoid \cos\theta. The limaçon trisectrix is composed of two loops. * The outer loop is defined when 1+2\cos\theta\ge0 on the polar angle interval -2\ ... [...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)\!. The ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sectrix Of Ceva
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] |
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. Pierre Wantzel proved in 1837 that the problem, as stated, is impossible to solve for arbitrary angles. However, although there is no way to trisect an angle ''in general'' with just a compass and a straightedge, some special angles can be trisected. For example, it is relatively straightforward to trisect a right angle (that is, to construct an angle of measure 30 degrees). 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. Becaus ... [...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: grc, νεύσεις, neuseis, label=none) 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'' ( el, διάστημα, lit=distance). 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 ... [...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 geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both 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 parallel to another plane that is tangential to the conical surface. The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the "vertex" and is the point where the parabola is most sharply curved. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doubling The Cube
Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the 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 tools. The Egyptians, Indians, and particularly the Greeks were aware of the problem and made many futile attempts at solving what they saw as an obstinate but soluble problem. However, 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 that volu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratrix Of Hippias
The quadratrix or trisectrix of Hippias (also quadratrix of Dinostratus) is a curve which is created by a uniform motion. It is one of the oldest examples for a kinematic curve (a curve created through motion). Its discovery is attributed to the Greek sophist Hippias of Elis, who used it around 420 BC in an attempt to solve the angle trisection problem (hence trisectrix). Later around 350 BC Dinostratus used it in an attempt to solve the problem of squaring the circle (hence quadratrix). Definition Consider a square ABCD, and an inscribed quarter circle arc centered at A with radius equal to the side of the square. Let E be a point that travels with a constant angular velocity along the arc from D to B, and let F be a point that travels simultaneously with a constant velocity from D to ABCD along line segment \overline, so that E and F start at the same time at D and arrive at the same time at B and A. Then the quadratrix is defined as the locus of the intersection of line seg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedean Spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the 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 coordinates it can be described by the equation r = a + b\cdot\theta with real numbers and . Changing the parameter moves the centerpoint of the spiral outward from the origin (positive toward and negative toward ) essentially through a rotation of the spiral, while controls the distance between loops. From the above equation, it can thus be stated: position of 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 friend of his and Pappus states that this spiral was discovered by Conon. Derivation of general equation of spiral A p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
