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Ellipsoidal Normal Vector
The ''n''-vector representation (also called geodetic normal or ellipsoid normal vector) is a three-parameter non-singular representation well-suited for replacing geodetic coordinates (latitude and longitude) for horizontal position representation in mathematical calculations and computer algorithms. Geometrically, the ''n''-vector for a given position on an ellipsoid is the outward-pointing unit vector that is normal in that position to the ellipsoid. For representing horizontal positions on Earth, the ellipsoid is a reference ellipsoid and the vector is decomposed in an Earth-centered Earth-fixed coordinate system. It behaves smoothly at all Earth positions, and it holds the mathematical one-to-one property. More in general, the concept can be applied to representing positions on the boundary of a strictly convex bounded subset of ''k''-dimensional Euclidean space, provided that that boundary is a differentiable manifold. In this general case, the ''n''-vector consists of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Singularity
In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. For example, the real function : f(x) = \frac has a singularity at x = 0, where the numerical value of the function approaches \pm\infty so the function is not defined. The absolute value function g(x) = , x, also has a singularity at x = 0, since it is not differentiable there. The algebraic curve defined by \left\ in the (x, y) coordinate system has a singularity (called a cusp) at (0, 0). For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory. Real analysis In real analysis, singularities are either discontinuities, or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives). There are four kinds of discon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Vector
In geometry, a normal is an object such as 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 (a unit vector) or its length may represent the curvature of the object (a ''curvature vector''); its algebraic sign may indicate sides (interior or exterior). In three dimensions, a surface normal, or simply normal, to a surface at point P is a vector perpendicular to the tangent plane of the surface at P. The word "normal" is also used as an adjective: a line ''normal'' to a plane, the ''normal'' component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality (right angles). The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at point P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Singularity
In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. For example, the real function : f(x) = \frac has a singularity at x = 0, where the numerical value of the function approaches \pm\infty so the function is not defined. The absolute value function g(x) = , x, also has a singularity at x = 0, since it is not differentiable there. The algebraic curve defined by \left\ in the (x, y) coordinate system has a singularity (called a cusp) at (0, 0). For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory. Real analysis In real analysis, singularities are either discontinuities, or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives). There are four kinds of discon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geographical Pole
A geographical pole or geographic pole is either of the two points on Earth where its axis of rotation intersects its surface. The North Pole lies in the Arctic Ocean while the South Pole is in Antarctica. North and South poles are also defined for other planets or satellites in the Solar System, with a North pole being on the same side of the invariable plane as Earth's North pole. Relative to Earth's surface, the geographic poles move by a few metres over periods of a few years. This is a combination of Chandler wobble, a free oscillation with a period of about 433 days; an annual motion responding to seasonal movements of air and water masses; and an irregular drift towards the 80th west meridian. As cartography requires exact and unchanging coordinates, the averaged locations of geographical poles are taken as fixed ''cartographic poles'' and become the points where the body's great circles of longitude intersect. See also * Earth's rotation * Polar motion * Poles of as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalar Resolute
The vector projection of a vector on (or onto) a nonzero vector , sometimes denoted \operatorname_\mathbf \mathbf (also known as the vector component or vector resolution of in the direction of ), is the orthogonal projection of onto a straight line parallel to . It is a vector parallel to , defined as: \mathbf_1 = a_1\mathbf where a_1 is a scalar, called the scalar projection of onto , and is the unit vector in the direction of . In turn, the scalar projection is defined as: a_1 = \left\, \mathbf\right\, \cos\theta = \mathbf\cdot\mathbf where the operator ⋅ denotes a dot product, ‖a‖ is the length of , and ''θ'' is the angle between and . Which finally gives: \mathbf_1 = \left(\mathbf \cdot \mathbf\right) \mathbf = \frac \frac = \frac = \frac ~ . The scalar projection is equal to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of . The vector component or vector resolute of perpendi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow. Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, ''Vector Analysis''. In the conventional form using cross products, vector calculus does not generalize to higher dimensions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Vector
In 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 \overrightarrow . A vector is what is needed to "carry" the point ''A'' to the point ''B''; the Latin word ''vector'' means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from ''A'' to ''B''. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two '' directed lines'' in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative. Quaternions are generally represented in the form :a + b\ \mathbf i + c\ \mathbf j +d\ \mathbf k where , and are real numbers; and , and are the ''basic quaternions''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horizontal Position Representation
A position representation is the parameters used to express a position relative to a reference. When representing positions relative to the Earth, it is often most convenient to represent ''vertical position'' (height or depth) separately, and to use some other parameters to represent horizontal position. There are also several applications where only the horizontal position is of interest, this might e.g. be the case for ships and ground vehicles/cars. It is a type of geographic coordinate system. There are several options for horizontal position representations, each with different properties which makes them appropriate for different applications. Latitude/longitude and UTM are common horizontal position representations. The horizontal position has two degrees of freedom, and thus two parameters are sufficient to uniquely describe such a position. However, similarly to the use of Euler angles as a formalism for representing rotations, using only the minimum number of paramet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation Formalisms In Three Dimensions
In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational (or angular) kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space. According to Euler's rotation theorem the rotation of a rigid body (or three-dimensional coordinate system with the fixed origin) is described by a single rotation about some axis. Such a rotation may be uniquely described by a minimum of three real parameters. However, for various reasons, there are several ways to represent it. Many of these representations use more than the necessary minimum of three parameters, although each of them still has only three degrees ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degrees Of Freedom (mechanics)
In physics, the degrees of freedom (DOF) of a mechanical system is the number of independent parameters that define its configuration or state. It is important in the analysis of systems of bodies in mechanical engineering, structural engineering, aerospace engineering, robotics, and other fields. The position of a single railcar (engine) moving along a track has one degree of freedom because the position of the car is defined by the distance along the track. A train of rigid cars connected by hinges to an engine still has only one degree of freedom because the positions of the cars behind the engine are constrained by the shape of the track. An automobile with highly stiff suspension can be considered to be a rigid body traveling on a plane (a flat, two-dimensional space). This body has three independent degrees of freedom consisting of two components of translation and one angle of rotation. Skidding or drifting is a good example of an automobile's three independent degrees ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
