CurvatureIn 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.
Gothic architectureGothic architecture is an architectural style that was prevalent in Europe from the late 12th to the 16th century, during the High and Late Middle Ages, surviving into the 17th and 18th centuries in some areas. It evolved from Romanesque architecture and was succeeded by Renaissance architecture. It originated in the Île-de-France and Picardy regions of northern France. The style at the time was sometimes known as opus Francigenum (French work); the term Gothic was first applied contemptuously during the later Renaissance, by those ambitious to revive the architecture of classical antiquity.
English Gothic architectureEnglish Gothic is an architectural style that flourished from the late 12th until the mid-17th century. The style was most prominently used in the construction of cathedrals and churches. Gothic architecture's defining features are pointed arches, rib vaults, buttresses, and extensive use of stained glass. Combined, these features allowed the creation of buildings of unprecedented height and grandeur, filled with light from large stained glass windows. Important examples include Westminster Abbey, Canterbury Cathedral and Salisbury Cathedral.
Gaussian curvatureIn differential geometry, the Gaussian curvature or Gauss curvature Κ of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, κ1 and κ2, at the given point: The Gaussian radius of curvature is the reciprocal of Κ. For example, a sphere of radius r has Gaussian curvature 1/r2 everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.
Gothic Revival architectureGothic Revival (also referred to as Victorian Gothic or neo-Gothic) is an architectural movement that began in the late 1840s in England. The movement gained momentum and expanded in the first half of the 19th century, as increasingly serious and learned admirers of the neo-Gothic styles sought to revive medieval Gothic architecture, intending to complement or even supersede the neoclassical styles prevalent at the time. Gothic Revival draws upon features of medieval examples, including decorative patterns, finials, lancet windows, and hood moulds.
Vault (architecture)In architecture, a vault (French voûte, from Italian volta) is a self-supporting arched form, usually of stone or brick, serving to cover a space with a ceiling or roof. As in building an arch, a temporary support is needed while rings of voussoirs are constructed and the rings placed in position. Until the topmost voussoir, the keystone, is positioned, the vault is not self-supporting. Where timber is easily obtained, this temporary support is provided by centering consisting of a framed truss with a semicircular or segmental head, which supports the voussoirs until the ring of the whole arch is completed.
Scalar curvatureIn the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls.
SpiralIn mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. It is a subtype of whorled patterns, a broad group that also includes concentric objects. Two major definitions of "spiral" in the American Heritage Dictionary are: a curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point. a three-dimensional curve that turns around an axis at a constant or continuously varying distance while moving parallel to the axis; a helix.
French Gothic architectureFrench Gothic architecture is an architectural style which emerged in France in 1140, and was dominant until the mid-16th century. The most notable examples are the great Gothic cathedrals of France, including Notre-Dame Cathedral, Reims Cathedral, Chartres Cathedral, and Amiens Cathedral. Its main characteristics are verticality, or height, and the innovative use of the rib vault and flying buttresses and other architectural innovations to distribute the weight of the stone structures to supports on the outside, allowing unprecedented height and volume.
Principal curvatureIn differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by different amounts in different directions at that point. At each point p of a differentiable surface in 3-dimensional Euclidean space one may choose a unit normal vector. A normal plane at p is one that contains the normal vector, and will therefore also contain a unique direction tangent to the surface and cut the surface in a plane curve, called normal section.
CurveIn mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's Elements: "The [curved] line is [...] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which [...
Algebraic curveIn mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation h(x, y, t) = 0 can be restricted to the affine algebraic plane curve of equation h(x, y, 1) = 0.
Analytic geometryIn mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry. Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions.
Golden spiralIn geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes. There are several comparable spirals that approximate, but do not exactly equal, a golden spiral. For example, a golden spiral can be approximated by first starting with a rectangle for which the ratio between its length and width is the golden ratio.
Moduli of algebraic curvesIn algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. One also distinguishes between fine and coarse moduli spaces for the same moduli problem.
Differentiable curveDifferential 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.
Barrel vaultA barrel vault, also known as a tunnel vault, wagon vault or wagonhead vault, is an architectural element formed by the extrusion of a single curve (or pair of curves, in the case of a pointed barrel vault) along a given distance. The curves are typically circular in shape, lending a semi-cylindrical appearance to the total design. The barrel vault is the simplest form of a vault: effectively a series of arches placed side by side (i.e., one after another). It is a form of barrel roof.
GeometryGeometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.
Archimedean spiralThe 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 (r, θ) it can be described by the equation with real numbers a and b.
Moduli spaceIn mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space.
