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Frenet–Serret Formulas
In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space \mathbb^, or the geometric properties of the curve itself irrespective of any motion. More specifically, the formulas describe the derivatives of the so-called tangent, normal, and binormal unit vectors in terms of each other. The formulas are named after the two French mathematicians who independently discovered them: Jean Frédéric Frenet, in his thesis of 1847, and Joseph Alfred Serret, in 1851. Vector notation and linear algebra currently used to write these formulas were not yet available at the time of their discovery. The tangent, normal, and binormal unit vectors, often called T, N, and B, or collectively the Frenet–Serret frame or TNB frame, together form an orthonormal basis spanning \mathbb^ and are defined as follows: * T is the unit vector tangent to the curve, pointing in the direction o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalar (mathematics)
A scalar is an element of a field which is used to define a ''vector space''. In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector. Generally speaking, a vector space may be defined by using any field instead of real numbers (such as complex numbers). Then scalars of that vector space will be elements of the associated field (such as complex numbers). A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, allowing two vectors to be multiplied in the defined way to produce a scalar. A vector space equipped with a scalar product is called an inner product space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Osculating Plane
{{Unreferenced, date=May 2019, bot=noref (GreenC bot) In mathematics, particularly in differential geometry, an osculating plane is a plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second order of contact at the point. The word ''osculate'' is from the Latin ''osculatus'' which is a past participle of ''osculari'', meaning ''to kiss''. An osculating plane is thus a plane which "kisses" a submanifold. The osculating plane in the geometry of Euclidean space curves can be described in terms of the Frenet-Serret formulas as the linear span of the tangent and normal vectors. See also * Normal plane (geometry) * 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 ... * Differential geometry of curves#Specia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Camille Jordan
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''. Biography Jordan was born in Lyon and educated at the École polytechnique. He was an engineer by profession; later in life he taught at the École polytechnique and the Collège de France, where he had a reputation for eccentric choices of notation. He is remembered now by name in a number of results: * The Jordan curve theorem, a topological result required in complex analysis * The Jordan normal form and the Jordan matrix in linear algebra * In mathematical analysis, Jordan measure (or ''Jordan content'') is an area measure that predates measure theory * In group theory, the Jordan–Hölder theorem on composition series is a basic result. * Jordan's theorem on finite linear groups Jordan's work did much to bring Galois theory into the mainstream. He also investigated the Mathie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skew-symmetric Matrix
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if a_ denotes the entry in the i-th row and j-th column, then the skew-symmetric condition is equivalent to Example The matrix :A = \begin 0 & 2 & -45 \\ -2 & 0 & -4 \\ 45 & 4 & 0 \end is skew-symmetric because : -A = \begin 0 & -2 & 45 \\ 2 & 0 & 4 \\ -45 & -4 & 0 \end = A^\textsf . Properties Throughout, we assume that all matrix entries belong to a field \mathbb whose characteristic is not equal to 2. That is, we assume that , where 1 denotes the multiplicative identity and 0 the additive identity of the given field. If the characteristic of the field is 2, then a skew-symmetric matrix is the same thing as a symmetric matrix. * The sum of two skew-symmetric matrices is skew-symmetric. * A scala ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsion Of Curves
Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Biology and medicine * Torsion fracture or spiral fracture, a bone fracture when torque is applied * Organ torsion, twisting that interrupts the blood supply to that organ: ** Splenic torsion, causing splenic infarction ** Ovarian torsion ** Testicular torsion * Penile torsion, a congenital condition * Torsion of the digestive tract in some domestic animals: ** Torsion, a type of horse colic ** Gastric torsion, or gastric dilatation volvulus * Torsion (gastropod), a developmental feature of all gastropods Mathematics * Torsion of a curve * Torsion tensor, in differential geometry * Torsion (algebra), in ring theory * Torsion group, in group theory and arithmetic geometry * Tor functor, the derived functors of the tensor product of modules ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnitude (mathematics)
In mathematics, the magnitude or size of a mathematical object is a property which determines whether the object is larger or smaller than other objects of the same kind. More formally, an object's magnitude is the displayed result of an order theory, ordering (or ranking)—of the class (mathematics), class of objects to which it belongs. In physics, magnitude can be defined as quantity or distance. History The Greeks distinguished between several types of magnitude, including: *Positive fractions *Line segments (ordered by length) *Geometric shape, Plane figures (ordered by area) *Solid geometry, Solids (ordered by volume) *Angle, Angles (ordered by angular magnitude) They proved that the first two could not be the same, or even isomorphic systems of magnitude. They did not consider negative number, negative magnitudes to be meaningful, and ''magnitude'' is still primarily used in contexts in which zero is either the smallest size or less than all possible sizes. Numbers Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiable Curve
Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the synthetic approach. Differential geometry takes another path: curves are represented in a parametrized form, and their geometric properties and various quantities associated with them, such as the curvature and the arc length, are expressed via derivatives and integrals using vector calculus. One of the most important tools used to analyze a curve is the Frenet frame, a moving frame that provides a coordinate system at each point of the curve that is "best adapted" to the curve near that point. The theory of curves is much simpler and narrower in scope than the theory of surfaces and its higher-dimensional generalizations because a regular curve in a Euclidean space has no intrinsic geometry. Any regular curve may be parametrized by the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arc Length
ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * Airport Regions Conference, a European organization of major airports * Amalgamated Roadstone Corporation, a British stone quarrying company * American Record Company (1904–1908, re-activated 1979), one of two United States record labels by this name * American Record Corporation (1929–1938), a United States record label also known as American Record Company * ARC (American Recording Company) (1978-present), a vanity label for Earth, Wind & Fire * ARC Document Solutions, a company based in California, formerly American Reprographics Company * Amey Roadstone Construction, a former British construction company * Aqaba Railway Corporation, a freight railway in Jordan * ARC/Architectural Resources Cambridge, Inc., Cambridge, Massachusett ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the orientation of the ''net'' force acting on that object. The magnitude of an object's acceleration, as described by Newton's Second Law, is the combined effect of two causes: * the net balance of all external forces acting onto that object — magnitude is directly proportional to this net resulting force; * that object's mass, depending on the materials out of which it is made — magnitude is inversely proportional to the object's mass. The SI unit for acceleration is metre per second squared (, \mathrm). For example, when a vehicle starts from a standstill (zero velocity, in an inertial frame of reference) and travels in a straight line at increasing speeds, it is accelerating in the direction of travel. If the vehicle turns, an acc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
