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Pedal Equation
For a plane curve ''C'' and a given fixed point ''O'', the pedal equation of the curve is a relation between ''r'' and ''p'' where ''r'' is the distance from ''O'' to a point on ''C'' and ''p'' is the perpendicular distance from ''O'' to the tangent line to ''C'' at the point. The point ''O'' is called the ''pedal point'' and the values ''r'' and ''p'' 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 ''O'' to the normal p_c (the ''contrapedal coordinate'') even though it is not an independent quantity and it relates to (r,p) 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane Curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. Plane curves also include the Jordan curves (curves that enclose a region of the plane but need not be smooth) and the graphs of continuous functions. Symbolic representation A plane curve can often be represented in Cartesian coordinates by an implicit equation of the form f(x,y)=0 for some specific function ''f''. If this equation can be solved explicitly for ''y'' or ''x'' – that is, rewritten as y=g(x) or x=h(y) for specific function ''g'' or ''h'' – then this provides an alternative, explicit, form of the representation. A plane curve can also often be represented in Cartesian coordinates by a parametric equation of the form (x,y)=(x(t), y(t)) for specific functions x(t) and y(t). Plane curves can sometimes also be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectangular Hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola. Hyperbolas arise in many ways: * as the curve representing the reciprocal function y(x) = 1/x in the Cartesian plane, * as the path followed by the shadow of the tip of a sundial, * as the shape of an open orbit (as distinct from a closed elliptical orbit), such as the orbit of a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cassini Oval
In geometry, a Cassini oval is a quartic plane curve defined as the locus (mathematics), locus of points in the plane (geometry), plane such that the Product_(mathematics), product of the distances to two fixed points (Focus (geometry), foci) is constant. This may be contrasted with an ellipse, for which the ''sum'' of the distances is constant, rather than the product. Cassini ovals are the special case of polynomial lemniscates when the polynomial used has degree of a polynomial, degree 2. Cassini ovals are named after the astronomer Giovanni Domenico Cassini who studied them in the late 17th century. Cassini believed that the Sun traveled around the Earth on one of these ovals, with the Earth at one focus of the oval. Other names include Cassinian ovals, Cassinian curves and ovals of Cassini. Formal definition A Cassini oval is a set of points, such that for any point P of the set, the ''product'' of the distances , PP_1, ,\, , PP_2, to two fixed points P_1, P_2 is a consta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Oval
In geometry, a Cartesian oval is a plane curve consisting of points that have the same linear combination of distances from two fixed points (foci). These curves are named after French mathematician René Descartes, who used them in optics. Definition Let and be fixed points in the plane, and let and denote the Euclidean distances from these points to a third variable point . Let and be arbitrary real numbers. Then the Cartesian oval is the locus of points ''S'' satisfying . The two ovals formed by the four equations and are closely related; together they form a quartic plane curve called the ovals of Descartes. Special cases In the equation , when and the resulting shape is an ellipse. In the limiting case in which ''P'' and ''Q'' coincide, the ellipse becomes a circle. When m = a/\!\operatorname(P, Q) it is a limaçon of Pascal. If m = -1 and 0 < a < \operatorname(P, Q) the equation gives a branch of a |
Logarithmic Spiral
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). More than a century later, the curve was discussed by Descartes (1638), and later extensively investigated by Jacob Bernoulli, who called it ''Spira mirabilis'', "the marvelous spiral". The logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant. Definition In polar coordinates (r, \varphi) the logarithmic spiral can be written as r = ae^,\quad \varphi \in \R, or \varphi = \frac \ln \frac, with e being the base of natural logarithms, and a > 0, k\ne 0 being real constants. In Cartesian coordinates The logarithmic spiral with the polar equation r = a e^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Involute
In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from or wrapped around the curve. It is a class of curves coming under the roulette family of curves. The evolute of an involute is the original curve. The notions of the involute and evolute of a curve were introduced by Christiaan Huygens in his work titled '' Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae'' (1673). Involute of a parameterized curve Let \vec c(t),\; t\in _1,t_2 be a regular curve in the plane with its curvature nowhere 0 and a\in (t_1,t_2), then the curve with the parametric representation \vec C_a(t)=\vec c(t) -\frac\; \int_a^t, \vec c'(w), \; dw is an ''involute'' of the given curve. Adding an arbitrary but fixed number l_0 to the integral \Bigl(\int_a^t, \ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Astroid
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 al ... [...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 hypocycoid 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. (In the illustration above ''b = 3a'' tracing the deltoid.) In complex coordinates this bec ... [...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 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 6. * an epicycloid with two 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 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 maps the unit circle to a nephroid [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lituus (mathematics)
300px, Branch for positive In mathematics, a lituus is a spiral with polar equation :r^2\theta = k where is any non-zero constant. Thus, the angle is inversely proportional to the square of the radius . This spiral, which has two branches depending on the sign of , is asymptotic to the -axis. Its points of inflexion are at :(\theta, r) = \left(\tfrac12, \pm\sqrt\right). The curve was named for the ancient Roman lituus by Roger Cotes Roger Cotes (10 July 1682 – 5 June 1716) was an English mathematician, known for working closely with Isaac Newton by proofreading the second edition of his famous book, the '' Principia'', before publication. He also invented the quadratur ... in a collection of papers entitled ''Harmonia Mensurarum'' (1722), which was published six years after his death. External links * * Interactive example using JSXGraph* * https://hsm.stackexchange.com/a/3181 on the history of the lituus curve. Spirals Plane curves {{geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Spiral
A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance from the spiral center, contrasting with the Archimedean spiral (for which this distance is invariant) and the logarithmic spiral (for which the distance between turns is proportional to the distance from the center). Fermat spirals are named after Pierre de Fermat.Anastasios M. Lekkas, Andreas R. Dahl, Morten Breivik, Thor I. Fossen"Continuous-Curvature Path Generation Using Fermat's Spiral" In: ''Modeling, Identification and Control''. Vol. 34, No. 4, 2013, pp. 183–198, . Their applications include curvature continuous blending of curves, modeling plant growth and the shapes of certain spiral galaxies, and the design of variable capacitors, solar power reflector arrays, and cyclotrons. Coordinate representation Polar The representatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Spiral
A hyperbolic spiral is a plane curve, which can be described in polar coordinates by the equation :r=\frac of a hyperbola. Because it can be generated by a circle inversion of an Archimedean spiral, it is called Reciprocal spiral, too.. Pierre Varignon first studied the curve in 1704. Later Johann Bernoulli and Roger Cotes worked on the curve as well. The hyperbolic spiral has a pitch angle that increases with distance from its center, unlike the logarithmic spiral (in which the angle is constant) or Archimedean spiral (in which it decreases with distance). For this reason, it has been used to model the shapes of spiral galaxies, which in some cases similarly have an increasing pitch angle. However, this model does not provide a good fit to the shapes of all spiral galaxies. In cartesian coordinates the hyperbolic spiral with the polar equation :r=\frac a \varphi ,\quad \varphi \ne 0 can be represented in Cartesian coordinates by :x = a \frac \varphi, \qquad y = a \frac \var ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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