Euclidean geometryEuclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into a logical system in which each result is proved from axioms and previously proved theorems.
Vector spaceIn mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space.
Boundary layerIn physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary condition (zero velocity at the wall). The flow velocity then monotonically increases above the surface until it returns to the bulk flow velocity. The thin layer consisting of fluid whose velocity has not yet returned to the bulk flow velocity is called the velocity boundary layer.
Normal (geometry)In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point. A normal vector may have length one (in which case it is a unit normal vector) or its length may represent the curvature of the object (a ); its algebraic sign may indicate sides (interior or exterior).
Line segmentIn geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using a line above the symbols for the two endpoints (such as ).
System of linear equationsIn mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by the ordered triple since it makes all three equations valid. The word "system" indicates that the equations should be considered collectively, rather than individually.
Boundary layer thicknessThis page describes some of the parameters used to characterize the thickness and shape of boundary layers formed by fluid flowing along a solid surface. The defining characteristic of boundary layer flow is that at the solid walls, the fluid's velocity is reduced to zero. The boundary layer refers to the thin transition layer between the wall and the bulk fluid flow. The boundary layer concept was originally developed by Ludwig Prandtl and is broadly classified into two types, bounded and unbounded.
Elliptic geometryElliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry.
Euclidean vectorIn mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a directed line segment, or graphically as an arrow connecting an initial point A with a terminal point B, and denoted by . A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier".
Directed graphIn mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. In formal terms, a directed graph is an ordered pair where V is a set whose elements are called vertices, nodes, or points; A is a set of ordered pairs of vertices, called arcs, directed edges (sometimes simply edges with the corresponding set named E instead of A), arrows, or directed lines.
Planetary boundary layerIn meteorology, the planetary boundary layer (PBL), also known as the atmospheric boundary layer (ABL) or peplosphere, is the lowest part of the atmosphere and its behaviour is directly influenced by its contact with a planetary surface. On Earth it usually responds to changes in surface radiative forcing in an hour or less. In this layer physical quantities such as flow velocity, temperature, and moisture display rapid fluctuations (turbulence) and vertical mixing is strong.
Differential geometryDifferential 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.
PerpendicularIn elementary geometry, two geometric objects are perpendicular if their intersection forms right angles (angles that are 90 degrees or π/2 radians wide) at the point of intersection called a foot. The condition of perpendicularity may be represented graphically using the perpendicular symbol, ⟂. Perpendicular intersections can happen between two lines (or two line segments), between a line and a plane, and between two planes.
Projective geometryIn mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice-versa.
Linear equationIn mathematics, a linear equation is an equation that may be put in the form where are the variables (or unknowns), and are the coefficients, which are often real numbers. The coefficients may be considered as parameters of the equation, and may be arbitrary expressions, provided they do not contain any of the variables. To yield a meaningful equation, the coefficients are required to not all be zero. Alternatively, a linear equation can be obtained by equating to zero a linear polynomial over some field, from which the coefficients are taken.
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.
Intersection theoryIn mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form. There is yet an ongoing development of intersection theory. Currently the main focus is on: virtual fundamental cycles, quantum intersection rings, Gromov-Witten theory and the extension of intersection theory from schemes to stacks.
Intersection (geometry)In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines, which either is one point or does not exist (if the lines are parallel). Other types of geometric intersection include: Line–plane intersection Line–sphere intersection Intersection of a polyhedron with a line Line segment intersection Intersection curve Determination of the intersection of flats – linear geometric objects embedded in a higher-dimensional space – is a simple task of linear algebra, namely the solution of a system of linear equations.
Line–line intersectionIn Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or another line. Distinguishing these cases and finding the intersection have uses, for example, in computer graphics, motion planning, and collision detection. In three-dimensional Euclidean geometry, if two lines are not in the same plane, they have no point of intersection and are called skew lines.
Line (geometry)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 embedded in two, three, or higher dimensional spaces. The word line may also refer to a line segment in everyday life that has two points to denote its ends (endpoints). A line can be referred to by two points that lie on it (e.g. ) or by a single letter (e.g. ).
