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Limaçon Trisectrix Outer Loop
In geometry, a limaçon or limacon , also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette (curve), roulette curve formed by the path of a Point (geometry), point fixed to a circle when that circle Rolling, rolls around the outside of a circle of equal radius. It can also be defined as the roulette formed when a circle rolls around a circle with half its radius so that the smaller circle is inside the larger circle. Thus, they belong to the family of curves called centered trochoids; more specifically, they are epitrochoids. The cardioid is the special case in which the point generating the roulette lies on the rolling circle; the resulting curve has a cusp (singularity), cusp. Depending on the position of the point generating the curve, it may have inner and outer loops (giving the family its name), it may be heart (symbol), heart-shaped, or it may be oval. A limaçon is a circular algebraic curve, bicircular rational plane algebraic curve of Degree of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Germany
Germany, officially the Federal Republic of Germany, is a country in Central Europe. It lies between the Baltic Sea and the North Sea to the north and the Alps to the south. Its sixteen States of Germany, constituent states have a total population of over 84 million in an area of , making it the most populous member state of the European Union. It borders Denmark to the north, Poland and the Czech Republic to the east, Austria and Switzerland to the south, and France, Luxembourg, Belgium, and the Netherlands to the west. The Capital of Germany, nation's capital and List of cities in Germany by population, most populous city is Berlin and its main financial centre is Frankfurt; the largest urban area is the Ruhr. Settlement in the territory of modern Germany began in the Lower Paleolithic, with various tribes inhabiting it from the Neolithic onward, chiefly the Celts. Various Germanic peoples, Germanic tribes have inhabited the northern parts of modern Germany since classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or surface is contained in a larger space, curvature can be defined ''extrinsically'' relative to the ambient space. Curvature of Riemannian manifolds of dimension at least two can be defined ''intrinsically'' without reference to a larger space. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature ''at a point'' of a differentiable curve is the curvature of its osculating circle — that is, the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inflection Point
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa. For the graph of a function of differentiability class (its first derivative , and its second derivative , exist and are continuous), the condition can also be used to find an inflection point since a point of must be passed to change from a positive value (concave upward) to a negative value (concave downward) or vice versa as is continuous; an inflection point of the curve is where and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or und ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acnode
An acnode is an isolated point in the solution set of a polynomial equation in two real variables. Equivalent terms are isolated point and hermit point. For example the equation :f(x,y)=y^2+x^2-x^3=0 has an acnode at the origin, because it is equivalent to :y^2 = x^2 (x-1) and x^2(x-1) is non-negative only when x ≥ 1 or x = 0. Thus, over the ''real'' numbers the equation has no solutions for x < 1 except for (0, 0). In contrast, over the complex numbers the origin is not isolated since square roots of negative real numbers exist. In fact, the complex solution set of a polynomial equation in two complex variables can never have an isolated point. An acnode is a critical point, or singularity, of the defining polynomial function, in the sense that both partial derivatives and vanish. Furth ... [...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
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] |
Rose (mathematics)
In mathematics, a rose or rhodonea curve is a sinusoid specified by either the cosine or sine functions with no phase angle that is plotted in polar coordinates. Rose curves or "rhodonea" were named by the Italian mathematician who studied them, Guido Grandi, between the years 1723 and 1728. General overview Specification A rose is the set of points in polar coordinates specified by the polar equation :r=a\cos(k\theta) or in Cartesian coordinates using the parametric equations :\begin x &= r\cos(\theta) = a\cos(k\theta)\cos(\theta) \\ y &= r\sin(\theta) = a\cos(k\theta)\sin(\theta) \end Roses can also be specified using the sine function. Since :\sin(k \theta) = \cos\left( k \theta - \frac \right) = \cos\left( k \left( \theta-\frac \right) \right). Thus, the rose specified by is identical to that specified by rotated counter-clockwise by radians, which is one-quarter the period of either sinusoid. Since they are specified using the cosine or sine function, rose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sinusoidal Spiral
In algebraic geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates :r^n = a^n \cos(n \theta)\, where is a nonzero constant and is a rational number other than 0. With a rotation about the origin, this can also be written :r^n = a^n \sin(n \theta).\, The term "spiral" is a misnomer, because they are not actually spirals, and often have a flower-like shape. Many well known curves are sinusoidal spirals including: * Rectangular hyperbola () * Line () * Parabola () * Tschirnhausen cubic () * Cayley's sextet () * Cardioid () * Circle () * Lemniscate of Bernoulli () The curves were first studied by Colin Maclaurin. Equations Differentiating :r^n = a^n \cos(n \theta)\, and eliminating ''a'' produces a differential equation for ''r'' and θ: :\frac\cos n\theta + r\sin n\theta =0. Then :\left(\frac,\ r\frac\right)\cos n\theta \frac = \left(-r\sin n\theta ,\ r \cos n\theta \right) = r\left(-\sin n\theta ,\ \cos n\theta \right) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows for a geometric interpretation of complex numbers. Under addition, they add like vector (geometry), vectors. The multiplication of two complex numbers can be expressed more easily in polar coordinates: the magnitude or ' of the product is the product of the two absolute values, or moduli, and the angle or ' of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation. The complex plane is sometimes called the Argand plane or Gauss plane. Notational conventions Complex numbers In complex analysis, the complex numbers are customarily represented by the symbol , which can be sepa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Coordinate
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular oriented lines, called '' coordinate lines'', ''coordinate axes'' or just ''axes'' (plural of ''axis'') of the system. The point where the axes meet is called the '' origin'' and has as coordinates. The axes directions represent an orthogonal basis. The combination of origin and basis forms a coordinate frame called the Cartesian frame. Similarly, the position of any point in three-dimensional space can be specified by three ''Cartesian coordinates'', which are the signed distances from the point to three mutually perpendicular planes. More generally, Cartesian coordinates specify the point in an -dimensional Euclidean space for any dimension . These coordinates are the signed distances from the point to mutually perpendicular fixed h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Coordinates
In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference point called the ''pole'', and *the point's direction from the pole relative to the direction of the ''polar axis'', a ray (geometry), ray drawn from the pole. The distance from the pole is called the ''radial coordinate'', ''radial distance'' or simply ''radius'', and the angle is called the ''angular coordinate'', ''polar angle'', or ''azimuth''. The pole is analogous to the origin in a Cartesian coordinate system. Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as spirals. Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
