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Whewell Equation
The Whewell equation of a plane curve is an equation that relates the tangential angle () with arclength (), where the tangential angle is the angle between the tangent to the curve and the -axis, and the arc length is the distance along the curve from a fixed point. These quantities do not depend on the coordinate system used except for the choice of the direction of the -axis, so this is an intrinsic equation of the curve, or, less precisely, ''the'' intrinsic equation. If a curve is obtained from another by translation then their Whewell equations will be the same. When the relation is a function, so that tangential angle is given as a function of arclength, certain properties become easy to manipulate. In particular, the derivative of the tangential angle with respect to arclength is equal to the curvature. Thus, taking the derivative of the Whewell equation yields a Cesàro equation for the same curve. The concept is named after William Whewell, who introduced it in 1849, in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intrinsic Coordinates (Whewell Equation)
In geometry, an intrinsic equation of a curve is an equation that defines the curve using a relation between the curve's intrinsic properties, that is, properties that do not depend on the location and possibly the orientation of the curve. Therefore an intrinsic equation defines the shape of the curve without specifying its position relative to an arbitrarily defined coordinate system. The intrinsic quantities used most often are arc length s , tangential angle \theta , curvature \kappa or radius of curvature, and, for 3-dimensional curves, torsion \tau . Specifically: * The natural equation is the curve given by its curvature and torsion. * The Whewell equation is obtained as a relation between arc length and tangential angle. * The Cesàro equation In geometry, the Cesàro equation of a plane curve is an equation relating the curvature () at a point of the curve to the arc length () from the start of the curve to the given point. It may also be given as an equation relatin ... [...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] |
Equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in French an ''équation'' is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. ''Solving'' an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables. An equation is written as two expressions, connected by a ... [...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] |
Intrinsic Equation
In geometry, an intrinsic equation of a curve is an equation that defines the curve using a relation between the curve's intrinsic properties, that is, properties that do not depend on the location and possibly the orientation of the curve. Therefore an intrinsic equation defines the shape of the curve without specifying its position relative to an arbitrarily defined coordinate system. The intrinsic quantities used most often are arc length s , tangential angle \theta , curvature \kappa or radius of curvature, and, for 3-dimensional curves, torsion \tau . Specifically: * The natural equation is the curve given by its curvature and torsion. * The Whewell equation is obtained as a relation between arc length and tangential angle. * The Cesàro equation In geometry, the Cesàro equation of a plane curve is an equation relating the curvature () at a point of the curve to the arc length () from the start of the curve to the given point. It may also be given as an equation relatin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translation (geometry)
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry. As a function If \mathbf is a fixed vector, known as the ''translation vector'', and \mathbf is the initial position of some object, then the translation function T_ will work as T_(\mathbf)=\mathbf+\mathbf. If T is a translation, then the image of a subset A under the function T is the translate of A by T . The translate of A by T_ is often written A+\mathbf . Horizontal and vertical translations In geometry, a vertical translation (also known as vertical shift) is a translation of a geometric object in a direction parallel to the vertical axis of the Cartesian coordinate system. Often, vertical translations a ... [...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] |
Cesàro Equation
In geometry, the Cesàro equation of a plane curve is an equation relating the curvature () at a point of the curve to the arc length () from the start of the curve to the given point. It may also be given as an equation relating the radius of curvature () to arc length. (These are equivalent because .) Two congruent curves will have the same Cesàro equation. Cesàro equations are named after Ernesto Cesàro. Examples Some curves have a particularly simple representation by a Cesàro equation. Some examples are: * Line: \kappa = 0. * Circle: \kappa = \frac, where is the radius. * Logarithmic spiral: \kappa=\frac, where is a constant. * Circle involute: \kappa=\frac, where is a constant. * Cornu spiral: \kappa=Cs, where is a constant. * Catenary: \kappa=\frac. Related parameterizations The Cesàro equation of a curve is related to its Whewell equation The Whewell equation of a plane curve is an equation that relates the tangential angle () with arclength (), where the tan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Whewell
William Whewell ( ; 24 May 17946 March 1866) was an English polymath, scientist, Anglican priest, philosopher, theologian, and historian of science. He was Master of Trinity College, Cambridge. In his time as a student there, he achieved distinction in both poetry and mathematics. The breadth of Whewell's endeavours is his remarkable feature. In a time of increasing specialization, Whewell belonged in an earlier era when natural philosophers investigated widely. He published work in mechanics, physics, geology, astronomy, and economics, while also composing poetry, writing a Bridgewater Treatise, translating the works of Goethe, and writing sermons and theological tracts. In mathematics, Whewell introduced what is now called the Whewell equation, defining the shape of a curve without reference to an arbitrarily chosen coordinate system. He also organized thousands of volunteers internationally to study ocean tides, in what is now considered one of the first citizen scienc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge Philosophical Society
The Cambridge Philosophical Society (CPS) is a scientific society at the University of Cambridge. It was founded in 1819. The name derives from the medieval use of the word philosophy to denote any research undertaken outside the fields of law, theology and medicine. The society was granted a royal charter by King William IV in 1832. The society is governed by an elected council of senior academics, which is chaired by the Society's President, according to a set of statutes. The society has published several scientific journals, including ''Biological Reviews'' (established 1926) and ''Mathematical Proceedings of the Cambridge Philosophical Society'' (formerly entitled ''Proceedings of the Cambridge Philosophical Society'', published since 1843). ''Transactions of the Cambridge Philosophical Society'' was published between 1821–1928, but was then discontinued. History The society was founded in 1819 by Edward Clarke, Adam Sedgwick and John Stevens Henslow, and is Cambri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line (mathematics)
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segment in everyday life, which has two points to denote its ends. Lines can be referred by two points that lay on it (e.g., \overleftrightarrow) or by a single letter (e.g., \ell). Euclid described a line as "breadthless length" which "lies evenly with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of the 19th century (such as non-Euclidean, projective and affine geometry). In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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