|
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) w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemniscate Of Bernoulli
In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points and , known as foci, at distance from each other as the locus of points so that . The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from , which is Latin for "decorated with hanging ribbons". It is a special case of the Cassini oval and is a rational algebraic curve of degree 4. This lemniscate was first described in 1694 by Jakob Bernoulli as a modification of an ellipse, which is the locus of points for which the sum of the distances to each of two fixed ''focal points'' is a constant. A Cassini oval, by contrast, is the locus of points for which the ''product'' of these distances is constant. In the case where the curve passes through the point midway between the foci, the oval is a lemniscate of Bernoulli. This curve can be obtained as the inverse transform of a hyperbola, with the inversion circle centered at the center of the hyperbola (bisector o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Lemniscate
In mathematics, a polynomial lemniscate or ''polynomial level curve'' is a plane algebraic curve of degree 2n, constructed from a polynomial ''p'' with complex coefficients of degree ''n''. For any such polynomial ''p'' and positive real number ''c'', we may define a set of complex numbers by , p(z), = c. This set of numbers may be equated to points in the real Cartesian plane, leading to an algebraic curve ''ƒ''(''x'', ''y'') = ''c''2 of degree 2''n'', which results from expanding out p(z) \bar p(\bar z) in terms of ''z'' = ''x'' + ''iy''. When ''p'' is a polynomial of degree 1 then the resulting curve is simply a circle whose center is the zero of ''p''. When ''p'' is a polynomial of degree 2 then the curve is a Cassini oval. Erdős lemniscate A conjecture of Erdős which has attracted considerable interest concerns the maximum length of a polynomial lemniscate ''ƒ''(''x'', ''y'') = 1 of degree 2''n'' when ''p'' is m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Force
In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force. : \vec = \mathbf(\mathbf) = \left\vert F( \mathbf ) \right\vert \hat where \vec F is the force, F is a vector valued force function, ''F'' is a scalar valued force function, r is the position vector, , , r, , is its length, and \hat = \mathbf r / \, \mathbf r\, is the corresponding unit vector. Not all central force fields are conservative or spherically symmetric. However, a central force is conservative if and only if it is spherically symmetric or rotationally invariant. Properties Central forces that are conservative can always be expressed as the negative gradient of a potential energy:- : \mathbf(\mathbf) = - \mathbf V(\mathbf)\textV(\mathbf) = \int_^ F(r)\,\mathrmr (the upper bound of integration is arbitrary, as the potential is defined up to an additive constant). In a conservative field, the total mechanical energy (kinetic and p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pedal Curve
A pedal (from the Latin '' pes'' ''pedis'', "foot") is a lever designed to be operated by foot and may refer to: Computers and other equipment * Footmouse, a foot-operated computer mouse * In medical transcription, a pedal is used to control playback of voice dictations Geometry * Pedal curve, a curve derived by construction from a given curve * Pedal triangle, a triangle obtained by projecting a point onto the sides of a triangle Music Albums * ''Pedals'' (Rival Schools album) * ''Pedals'' (Speak album) Other music * Bass drum pedal, a pedal used to play a bass drum while leaving the drummer's hands free to play other drums with drum sticks, hands, etc. * Effects pedal, a pedal used commonly for electric guitars * Pedal keyboard, a musical keyboard operated by the player's feet * Pedal harp, a modern orchestral harp with pedals used to change the tuning of its strings * Pedal point, a type of nonchord tone, usually in the bass * Pedal tone, a fundamental tone played ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isoptic
In the geometry of curves, an orthoptic is the set of points for which two tangents of a given curve meet at a right angle. Examples: # The orthoptic of a parabola is its directrix (proof: see below), # The orthoptic of an ellipse \tfrac + \tfrac = 1 is the director circle x^2 + y^2 = a^2 + b^2 (see below), # The orthoptic of a hyperbola \tfrac - \tfrac = 1,\ a > b is the director circle x^2 + y^2 = a^2 - b^2 (in case of there are no orthogonal tangents, see below), # The orthoptic of an astroid x^ + y^ = 1 is a quadrifolium with the polar equation r=\tfrac\cos(2\varphi), \ 0\le \varphi < 2\pi (see below). Generalizations: # An isoptic is the set of points for which two tangents of a given curve meet at a ''fixed angle'' (see [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Curve
In inversive geometry, an inverse curve of a given curve is the result of applying an inverse operation to . Specifically, with respect to a fixed circle with center and radius the inverse of a point is the point for which lies on the ray and . The inverse of the curve is then the locus of as runs over . The point in this construction is called the center of inversion, the circle the circle of inversion, and the radius of inversion. An inversion applied twice is the identity transformation, so the inverse of an inverse curve with respect to the same circle is the original curve. Points on the circle of inversion are fixed by the inversion, so its inverse is itself. Equations The inverse of the point with respect to the unit circle is where :X = \frac,\qquad Y=\frac, or equivalently :x = \frac,\qquad y=\frac. So the inverse of the curve determined by with respect to the unit circle is :f\left(\frac, \frac\right)=0. It is clear from this that inverting an algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. 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 quantity, that is, it is expressed by a single real number. For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangential Angle
In geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent line to the curve at the given point and the -axis. (Some authors define the angle as the deviation from the direction of the curve at some fixed starting point. This is equivalent to the definition given here by the addition of a constant to the angle or by rotating the curve.) Equations If a curve is given parametrically by , then the tangential angle at is defined (up to a multiple of ) by : \frac = (\cos \varphi,\ \sin \varphi). Here, the prime symbol denotes the derivative with respect to . Thus, the tangential angle specifies the direction of the velocity vector , while the speed specifies its magnitude. The vector :\frac is called the unit tangent vector, so an equivalent definition is that the tangential angle at is the angle such that is the unit tangent vector at . If the curve is parametrized by arc length , so , then the definition simp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Colin Maclaurin
Colin Maclaurin (; gd, Cailean MacLabhruinn; February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra. He is also known for being a child prodigy and holding the record for being the youngest professor. The Maclaurin series, a special case of the Taylor series, is named after him. Owing to changes in orthography since that time (his name was originally rendered as M'Laurine), his surname is alternatively written MacLaurin. Early life Maclaurin was born in Kilmodan, Argyll. His father, John Maclaurin, minister of Glendaruel, died when Maclaurin was in infancy, and his mother died before he reached nine years of age. He was then educated under the care of his uncle, Daniel Maclaurin, minister of Kilfinan. A child prodigy, he entered university at age 11. Academic career At eleven, Maclaurin, a child prodigy at the time, entered the University of Glasgow. He graduated Master of Arts three years later by defending ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. Usually, the radius is required to be a positive number. A circle with r=0 (a single point) is a degenerate case. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. Specifically, a circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a '' disc''. A circle may also be defined as a special ki ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian coordinate system) is called the ''pole'', and the ray from the pole in the reference direction is the ''polar axis''. 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''. Angles in polar notation are generally expressed in either degrees or radians (2 rad being equal to 360°). Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-17th century, though the actual term "polar coordinates" has been attributed to Gregorio Fontana in the 18th century. The initial motivation for the introduction of the polar system was the study of circula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.png "Click for more on -> Tangential Angle")
.jpg "Click for more on -> Colin Maclaurin")
