|
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 generally, 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 o ... [...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 reciprocal function f(x) = 1/x has a singularity at x = 0, where the value of the function is not defined, as involving a division by zero. The absolute value function g(x) = , x, also has a singularity at x = 0, since it is not differentiable there. The algebraic curve In 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 cu ... defined by \left\ in the (x, y) coordinate system has a singularity (called a cusp (singularity), cusp) at (0, 0). For singularities in algebraic geometry, see singular point of an algebraic variety. For singul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
Normal Vector
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 straight line perpendicular to the tangent line to the curve at the point. A normal vector is a vector perpendicular to a given object at a particular point. A normal vector of length one is called a unit normal vector or normal direction. A curvature vector is a normal vector whose length is the curvature of the object. Multiplying a normal vector by results in the opposite vector, which may be used for indicating sides (e.g., interior or exterior). In three-dimensional space, a surface normal, or simply normal, to a surface at point is a vector perpendicular to the tangent plane of the surface at . The vector field of normal directions to a surface is known as '' Gauss map''. The word "normal" is also used as an adjective: a line ''normal'' to a plane, the ''normal'' component of a for ... [...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 reciprocal function f(x) = 1/x has a singularity at x = 0, where the value of the function is not defined, as involving a division by zero. The absolute value function g(x) = , x, also has a singularity at x = 0, since it is not differentiable there. The algebraic curve In 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 cu ... defined by \left\ in the (x, y) coordinate system has a singularity (called a cusp (singularity), cusp) at (0, 0). For singularities in algebraic geometry, see singular point of an algebraic variety. For singul ... [...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 (geography), 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 mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
Atan2
In computing and mathematics, the function (mathematics), function atan2 is the 2-Argument of a function, argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi 0, \\[5mu] \arctan\left(\frac y x\right) + \pi &\text x < 0 \text y \ge 0, \\[5mu] \arctan\left(\frac y x\right) - \pi &\text x < 0 \text y < 0, \\[5mu] +\frac &\text x = 0 \text y > 0, \\[5mu] -\frac &\text x = 0 \text y < 0, \\[5mu] \text &\text x = 0 \text y = 0. \end Instead of the tangent, it can be convenient to use the half-tangent as a representation of an angle, partly because the angle has a unique half-tangent, (See tangent half-angle formula.) The expression with in the denominator should be used when and to avoid possible loss of significance in computing . When an function is unavailable, it can be computed as twice the arctangent of the half-tangent . That is, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
Scalar Resolute
The vector projection (also known as the vector component or vector resolution) of a vector on (or onto) a nonzero vector is the orthogonal projection of onto a straight line parallel to . The projection of onto is often written as \operatorname_\mathbf \mathbf or . The vector component or vector resolute of perpendicular to , sometimes also called the vector rejection of ''from'' (denoted \operatorname_ \mathbf or ), is the orthogonal projection of onto the plane (or, in general, hyperplane) that is orthogonal to . Since both \operatorname_ \mathbf and \operatorname_ \mathbf are vectors, and their sum is equal to , the rejection of from is given by: \operatorname_ \mathbf = \mathbf - \operatorname_ \mathbf. To simplify notation, this article defines \mathbf_1 := \operatorname_ \mathbf and \mathbf_2 := \operatorname_ \mathbf. Thus, the vector \mathbf_1 is parallel to \mathbf, the vector \mathbf_2 is orthogonal to \mathbf, and \mathbf = \mathbf_1 + \mathbf_2. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
Vector Calculus
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-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 the theory of quaternions 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'', though earlier mathematicians such as Isaac Newton pioneered ... [...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. Euclidean vectors can be added and scaled to form a vector space. A '' vector quantity'' is a vector-valued physical quantity, including units of measurement and possibly a support, formulated as a '' directed line segment''. A vector is frequently depicted graphically as an arrow connecting an ''initial point'' ''A'' with a ''terminal point'' ''B'', and denoted by \stackrel \longrightarrow. A vector is what is needed to "carry" the point ''A'' to the point ''B''; the Latin word 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 numbe ... [...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. The algebra of quaternions is often denoted by (for ''Hamilton''), or in blackboard bold by \mathbb H. Quaternions are not a field, because multiplication of quaternions is not, in general, commutative. Quaternions provide a definition of the quotient of two vectors in a three-dimensional space. Quaternions are generally represented in the form a + b\,\mathbf i + c\,\mathbf j +d\,\mathbf k, where the coefficients , , , are real numbers, and , are the ''basis vectors'' or ''basis elements''. 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, robotics, magnetic resonance i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
Horizontal Position Representation
A '' position representation'' is a set of parameters used to express a position relative to a reference frame. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Rotation Formalisms In Three Dimensions
In geometry, there exist various rotation formalisms 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 a 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
Degrees Of Freedom (mechanics)
In classical mechanics, physics, the number of degrees of freedom (DOF) of a mechanical system is the number of independent parameters required to completely specify its configuration or state. That number is an important property in the analysis of systems of bodies in mechanical engineering, structural engineering, aerospace engineering, robotics, and other fields. As an example, the position of a single railcar (engine) moving along a track has one degree of freedom because the position of the car can be completely specified by a single number expressing its distance along the track from some chosen origin. 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. For a second example, an automobile with a very stiff suspension can be considered to be a rigid body traveling on a plane (a flat, two-dimensional space). This body has three independe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |