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Astroid
In mathematics, an astroid is a particular type of roulette curve: a hypocycloid with four cusp (singularity), cusps. Specifically, it is the Locus (mathematics), locus of a point on a circle as it Rolling, rolls inside a fixed circle with four times the radius. By double generation, it is also the locus of a point on a circle as it rolls inside a fixed circle with 4/3 times the radius. It can also be defined as the Envelope (mathematics), envelope of a line segment of fixed length that moves while keeping an end point on each of the axes. It is therefore the Envelope (mathematics), envelope of the moving bar in the Trammel of Archimedes. Its modern name comes from the Greek word for "star". It was proposed, originally in the form of "Astrois", by Joseph Johann von Littrow in 1838. The curve had a variety of names, including tetracuspid (still used), cubocycloid, and paracycle. It is nearly identical in form to the evolute of an ellipse. Equations If the radius of the fixed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Astroid Created With Elipses With A Plus B Const
In mathematics, an astroid is a particular type of roulette curve: a hypocycloid with four cusps. Specifically, it is the locus of a point on a circle as it rolls inside a fixed circle with four times the radius. By double generation, it is also the locus of a point on a circle as it rolls inside a fixed circle with 4/3 times the radius. It can also be defined as the envelope of a line segment of fixed length that moves while keeping an end point on each of the axes. It is therefore the envelope of the moving bar in the Trammel of Archimedes. Its modern name comes from the Greek word for "star". It was proposed, originally in the form of "Astrois", by Joseph Johann von Littrow in 1838. The curve had a variety of names, including tetracuspid (still used), cubocycloid, and paracycle. It is nearly identical in form to the evolute of an ellipse. Equations If the radius of the fixed circle is ''a'' then the equation is given by x^ + y^ = a^. This implies that an astroid is also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypocycloid
In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. As the radius of the larger circle is increased, the hypocycloid becomes more like the cycloid created by rolling a circle on a line. History The 2-cusped hypocycloid called Tusi couple was first described by the 13th-century Persian people, Persian Islamic astronomy, astronomer and Islamic mathematics, mathematician Nasir al-Din al-Tusi in ''Tahrir al-Majisti (Commentary on the Almagest)''. German painter and German Renaissance theorist Albrecht Dürer described epitrochoids in 1525, and later Roemer and Bernoulli concentrated on some specific hypocycloids, like the astroid, in 1674 and 1691, respectively. Properties If the rolling circle has radius , and the fixed circle has radius , then the parametric equations for the curve can be given by either: \begin & x (\theta) = (R - r) \cos \theta + r \cos \left(\frac \theta \right) \\ & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Evolute
In the differential geometry of curves, the evolute of a curve is the locus (mathematics), locus of all its Center of curvature, centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that curve. The evolute of a circle is therefore a single point at its center. Equivalently, an evolute is the envelope (mathematics), envelope of the perpendicular, normals to a curve. The evolute of a curve, a surface, or more generally a submanifold, is the caustic (mathematics), caustic of the normal map. Let be a smooth, regular submanifold in . For each point in and each vector , based at and normal to , we associate the point . This defines a Lagrangian map, called the normal map. The caustic of the normal map is the evolute of . Evolutes are closely connected to involutes: A curve is the evolute of any of its involutes. History Apollonius of Perga, Apollonius ( 200 BC) discussed evolut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hedgehog (geometry)
In differential geometry, a hedgehog or plane hedgehog is a type of plane curve, the envelope of a family of lines determined by a support function. More intuitively, sufficiently well-behaved hedgehogs are plane curves with one tangent line in each oriented direction. A projective hedgehog is a restricted type of hedgehog, defined from an anti-symmetric support function, and (again when sufficiently well-behaved) forms a curve with one tangent line in each direction, regardless of orientation. Every closed strictly convex curve is the envelope of its supporting lines. The astroid forms a non-convex hedgehog, and the deltoid curve forms a projective hedgehog. Hedgehogs can also be defined from support functions of hyperplanes in higher dimensions. Definitions Formally, a planar support function can be defined as a continuously differentiable function h from the unit circle in the plane to real numbers, or equivalently as a function f(\theta)=h\bigl((\cos\theta,\sin\theta)\big ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Envelope (mathematics)
In geometry, an envelope of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope. Classically, a point on the envelope can be thought of as the intersection of two " infinitesimally adjacent" curves, meaning the limit of intersections of nearby curves. This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions. To have an envelope, it is necessary that the individual members of the family of curves are differentiable curves as the concept of tangency does not apply otherwise, and there has to be a smooth transition proceeding through the members. But these conditions are not sufficient – a given family may fail to have an envelope. A simple example of this is given by a family of concentric circles of expanding radius. Envelope of a family of curves Let each curve ''C''''t'' in the family be given as the solution of an equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pedal Equation
In Euclidean geometry, for a plane curve and a given fixed point , the pedal equation of the curve is a relation between and where is the distance from to a point on and is the perpendicular distance from to the tangent line to at the point. The point is called the pedal point and the values and are sometimes called the pedal coordinates of a point relative to the curve and the pedal point. It is also useful to measure the distance of to the normal (the contrapedal coordinate) even though it is not an independent quantity and it relates to as p_c:=\sqrt. Some curves have particularly simple pedal equations and knowing the pedal equation of a curve may simplify the calculation of certain of its properties such as curvature. These coordinates are also well suited for solving certain type of force problems in classical mechanics and celestial mechanics. Equations Cartesian coordinates For ''C'' given in rectangular coordinates by ''f''(''x'', ''y'') = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superellipse
A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but defined by an equation that allows for various shapes between a rectangle and an ellipse. In two dimensional Cartesian coordinate system, a superellipse is defined as the set of all points (x,y) on the curve that satisfy the equation\left, \frac\^n\!\! + \left, \frac\^n\! = 1,where a and b are positive numbers referred to as Semidiameter, semi-diameters or Semi-major and semi-minor axes, semi-axes of the superellipse, and n is a positive parameter that defines the shape. When n=2, the superellipse is an ordinary ellipse. For n>2, the shape is more rectangular with rounded corners, and for 00, where m either equals to or differs from ''n''. If m=n, it is the Lamé's superellipses. If m\neq n, the curve possesses more flexibility of behavior, and is better possible fit to des ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cusp (singularity)
In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of singular point of a curve. For a plane curve defined by an analytic, parametric equation :\begin x &= f(t)\\ y &= g(t), \end a cusp is a point where both derivatives of and are zero, and the directional derivative, in the direction of the tangent, changes sign (the direction of the tangent is the direction of the slope \lim (g'(t)/f'(t))). Cusps are ''local singularities'' in the sense that they involve only one value of the parameter , in contrast to self-intersection points that involve more than one value. In some contexts, the condition on the directional derivative may be omitted, although, in this case, the singularity may look like a regular point. For a curve defined by an implicit equation :F(x,y) = 0, which is smooth, cusps are points where the terms of lowest degre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Coordinate System
In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two 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 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 intuitive to model using polar coordinates. The polar coordinate system i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deltoid Curve
In geometry, a deltoid curve, also known as a tricuspoid curve or Steiner curve, is a hypocycloid of three cusps. In other words, it is the roulette created by a point on the circumference of a circle as it rolls without slipping along the inside of a circle with three or one-and-a-half times its radius. It is named after the capital Greek letter delta (Δ) which it resembles. More broadly, a ''deltoid'' can refer to any closed figure with three vertices connected by curves that are concave to the exterior, making the interior points a non-convex set. Equations A hypocycloid can be represented (up to rotation and translation) by the following parametric equations :x=(b-a)\cos(t)+a\cos\left(\fracat\right) \, :y=(b-a)\sin(t)-a\sin\left(\fracat\right) \, , where ''a'' is the radius of the rolling circle, ''b'' is the radius of the circle within which the aforementioned circle is rolling and ''t'' ranges from zero to 6. (In the illustration above ''b = 3a'' tracing the deltoid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nephroid
In geometry, a nephroid () is a specific plane curve. It is a type of epicycloid in which the smaller circle's radius differs from the larger one by a factor of one-half. Name Although the term ''nephroid'' was used to describe other curves, it was applied to the curve in this article by Richard A. Proctor in 1878. Strict definition A nephroid is * an algebraic curve of Degree of a polynomial, degree 6. * an epicycloid with two Cusp (singularity), cusps * a plane simple closed curve = a Jordan curve Equations Parametric If the small circle has radius a, the fixed circle has midpoint (0,0) and radius 2a, the rolling angle of the small circle is 2\varphi and point (2a,0) the starting point (see diagram) then one gets the Parametric equation, parametric representation: :x(\varphi) = 3a\cos\varphi- a\cos3\varphi=6a\cos\varphi-4a \cos^3\varphi \ , :y(\varphi) = 3a \sin\varphi - a\sin3\varphi =4a\sin^3\varphi\ , \qquad 0\le \varphi < 2\pi The complex map |
