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Tait–Kneser Theorem
In differential geometry, the Tait–Kneser theorem states that, if a smooth plane curve has monotonic curvature, then the osculating circles of the curve are disjoint and nested within each other. The logarithmic spiral or the pictured Archimedean spiral provide examples of curves whose curvature is monotonic for the entire curve. This monotonicity cannot happen for a simple closed curve (by the four-vertex theorem, there are at least four vertices where the curvature reaches an extreme point) but for such curves the theorem can be applied to the arcs of the curves between its vertices. The theorem is named after Peter Tait, who published it in 1896, and Adolf Kneser, who rediscovered it and published it in 1912. Tait's proof follows simply from the properties of the evolute, the curve traced out by the centers of osculating circles. For curves with monotone curvature, the arc length along the evolute between two centers equals the difference in radii of the corresponding circle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Osculating Circles Of The Archimedean Spiral of biological classification attempted to group creatures into circles which could touch or overlap with adjacent circles, a phenomenon called 'osculation'.
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In mathematics, osculate, meaning to touch (from the Latin ''osculum'' meaning kiss), may refer to: * osculant, an invariant of hypersurfaces *osculating circle *osculating curve *osculating plane *osculating orbit The obsolete Quinarian system The quinarian system was a method of zoological classification which was popular in the mid 19th century, especially among British naturalists. It was largely developed by the entomologist William Sharp Macleay in 1819. The system was further pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...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] |
Osculating Circle
In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ''p'' on the curve has been traditionally defined as the circle passing through ''p'' and a pair of additional points on the curve infinitesimally close to ''p''. Its center lies on the inner normal line, and its curvature defines the curvature of the given curve at that point. This circle, which is the one among all tangent circles at the given point that approaches the curve most tightly, was named ''circulus osculans'' (Latin for "kissing circle") by Leibniz. The center and radius of the osculating circle at a given point are called center of curvature and radius of curvature of the curve at that point. A geometric construction was described by Isaac Newton in his '' Principia'': Nontechnical description Imagine a car moving along a curved road on a vast flat plane. Suddenly, at one point along the road, the steering wheel locks in its present position. Thereaf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Archimedean Spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates it can be described by the equation r = a + b\cdot\theta with real numbers and . Changing the parameter moves the centerpoint of the spiral outward from the origin (positive toward and negative toward ) essentially through a rotation of the spiral, while controls the distance between loops. From the above equation, it can thus be stated: position of particle from point of start is proportional to angle as time elapses. Archimedes described such a spiral in his book '' On Spirals''. Conon of Samos was a friend of his and Pappus states that this spiral was discovered by Conon. Derivation of general equation of spiral A p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Closed Curve
In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior points. Every continuous path connecting a point of one region to a point of the other intersects with the curve somewhere. While the theorem seems intuitively obvious, it takes some ingenuity to prove it by elementary means. ''"Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it."'' (). More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces. The Jordan curve theorem is named after the mathematician Camille Jordan (1838–1922), who found its first proof. For decades, mathematicians generally thought that this proof was flawed and that the first rigo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Four-vertex Theorem
The four-vertex theorem of geometry states that the curvature along a simple, closed, smooth plane curve has at least four local extrema (specifically, at least two local maxima and at least two local minima). The name of the theorem derives from the convention of calling an extreme point of the curvature function a vertex. This theorem has many generalizations, including a version for space curves where a vertex is defined as a point of vanishing torsion. Definition and examples The curvature at any point of a smooth curve in the plane can be defined as the reciprocal of the radius of an osculating circle at that point, or as the norm of the second derivative of a parametric representation of the curve, parameterized consistently with the length along the curve. For the vertices of a curve to be well-defined, the curvature itself should vary continuously, as happens for curves of smoothness C^2. A vertex is then a local maximum or local minimum of curvature. If the curvature ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (curve)
In the geometry of plane curves, a vertex is a point of where the first derivative of curvature is zero. This is typically a local maximum or minimum of curvature, and some authors define a vertex to be more specifically a local extremum of curvature. However, other special cases may occur, for instance when the second derivative is also zero, or when the curvature is constant. For space curves, on the other hand, a vertex is a point where the torsion vanishes. Examples A hyperbola has two vertices, one on each branch; they are the closest of any two points lying on opposite branches of the hyperbola, and they lie on the principal axis. On a parabola, the sole vertex lies on the axis of symmetry and in a quadratic of the form: :ax^2 + bx + c\,\! it can be found by completing the square or by differentiation., p. 127. On an ellipse, two of the four vertices lie on the major axis and two lie on the minor axis. For a circle, which has constant curvature, every point is a verte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Log Spiral With Osculating Circles
Log most often refers to: * Trunk (botany), the stem and main wooden axis of a tree, called logs when cut ** Logging, cutting down trees for logs ** Firewood, logs used for fuel ** Lumber or timber, converted from wood logs * Logarithm, in mathematics Log, LOG or LoG may also refer to: Arts, entertainment and media * ''Log'' (magazine), an architectural magazine * ''The Log'', a boating and fishing newspaper published by the Duncan McIntosh Company * Lamb of God (band) or LoG, an American metal band * The Log, an electric guitar by Les Paul * Log, a fictional product in ''The Ren & Stimpy Show'' * The League of Gentlemen or LoG, a British comedy show. Places * Log, Russia, the name of several places * Log, Slovenia, the name of several places Science and mathematics *Logarithm, a mathematical function * Log file, a computer file in which events are recorded * Laplacian of Gaussian or LoG, an algorithm used in digital image processing Other uses * Logbook, or log, a record ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Tait (physicist)
Peter Guthrie Tait FRSE (28 April 1831 – 4 July 1901) was a Scottish mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook '' Treatise on Natural Philosophy'', which he co-wrote with Lord Kelvin, and his early investigations into knot theory. His work on knot theory contributed to the eventual formation of topology as a mathematical discipline. His name is known in graph theory mainly for Tait's conjecture. He is also one of the namesakes of the Tait–Kneser theorem on osculating circles. Early life Tait was born in Dalkeith on 28 April 1831 the only son of Mary Ronaldson and John Tait, secretary to the 5th Duke of Buccleuch. He was educated at Dalkeith Grammar School then Edinburgh Academy. He studied Mathematics and Physics at the University of Edinburgh, and then went to Peterhouse, Cambridge, graduating as senior wrangler and first Smith's prizeman in 1852. As a fellow and lecturer of his college he remain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adolf Kneser
Adolf Kneser (19 March 1862 – 24 January 1930) was a German mathematician. He was born in Grüssow, Mecklenburg, Germany and died in Breslau, Germany (now Wrocław, Poland). He is the father of the mathematician Hellmuth Kneser and the grandfather of the mathematician Martin Kneser. Kneser is known for the first proof of the four-vertex theorem that applied in general to non-convex curves. Kneser's theorem on differential equations is named after him, and provides criteria to decide whether a differential equation is oscillating. He is also one of the namesakes of the Tait–Kneser theorem on osculating circle In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ''p'' on the curve has been traditionally defined as the circle passing through ''p'' and a pair of additional points on the curve i ...s. Selected publications * *; *; * * References External links * * 1862 births 1930 deaths 19th-century ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
