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Secant Plane
A secant plane is a plane containing a nontrivial section of a sphere or an ellipsoid, or such a plane that a sphere is projected onto. Secant planes are similar to tangent planes, which contact the sphere's surface at a point, while secant planes contact the surface along curves. The two-dimensional representations of secant planes are secant lines, the lines that join two distinct points on a curve. Applications Secant planes are used in map projections. The secant plane intersects a globe along a small circle with no distortion, forming a standard parallel which has true scale. See also * Circular segment * Spherical cap * Tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ... References {{Reflist External links Tangents and Secants in Map Projections ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ellipsoid Circular Sections
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties. Every planar cross section is either an ellipse, or is empty, or is reduced to a single point (this explains the name, meaning "ellipse-like"). It is bounded, which means that it may be enclosed in a sufficiently large sphere. An ellipsoid has three pairwise perpendicular axes of symmetry which intersect at a center of symmetry, called the center of the ellipsoid. The line segments that are delimited on the axes of symmetry by the ellipsoid are called the ''principal axes'', or simply axes of the ellipsoid. If the three axes have different lengths, the figure is a triaxial ellipsoid (r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane (geometry)
In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with one of a room's walls, infinitely extended, or they may enjoy an independent existence in their own right, as in the setting of two-dimensional Euclidean geometry. Sometimes the word ''plane'' is used more generally to describe a two-dimensional surface, for example the hyperbolic plane and elliptic plane. When working exclusively in two-dimensional Euclidean space, the definite article is used, so ''the'' plane refers to the whole space. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory, and graphing are performed in a two-dimensional space, often in the plane. Euclidean geometry Euclid set forth the first great landmark of mathematical thought, an axio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross Section (geometry)
In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross-sections. The boundary of a cross-section in three-dimensional space that is parallel to two of the axes, that is, parallel to the plane determined by these axes, is sometimes referred to as a contour line; for example, if a plane cuts through mountains of a raised-relief map parallel to the ground, the result is a contour line in two-dimensional space showing points on the surface of the mountains of equal elevation. In technical drawing a cross-section, being a projection of an object onto a plane that intersects it, is a common tool used to depict the internal arrangement of a 3-dimensional object in two dimensions. It is traditionally crosshatched with the style of crosshatching often indicating the types of materials being used. With com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties. Every planar cross section is either an ellipse, or is empty, or is reduced to a single point (this explains the name, meaning "ellipse-like"). It is bounded, which means that it may be enclosed in a sufficiently large sphere. An ellipsoid has three pairwise perpendicular axes of symmetry which intersect at a center of symmetry, called the center of the ellipsoid. The line segments that are delimited on the axes of symmetry by the ellipsoid are called the ''principal axes'', or simply axes of the ellipsoid. If the three axes have different lengths, the figure is a triaxial elli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secant Line
Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to: * a secant line, in geometry * the secant variety, in algebraic geometry * secant (trigonometry) (Latin: secans), the multiplicative inverse (or reciprocal) trigonometric function of the cosine * the secant method In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function ''f''. The secant method can be thought of as a finite-difference approximation o ..., a root-finding algorithm in numerical analysis, based on secant lines to graphs of functions * a secant ogive in nose cone design {{mathdab sr:Секанс ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Map Projection
In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitude, of locations from the surface of the globe are transformed to coordinates on a plane. Projection is a necessary step in creating a two-dimensional map and is one of the essential elements of cartography. All projections of a sphere on a plane necessarily distort the surface in some way and to some extent. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. The study of map projections is primarily about the characterization of their distortions. There is no limit to the number of possible map projections. More generally, projections are considered in sev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Parallel (map Projections)
In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitude, of locations from the surface of the globe are transformed to coordinates on a plane. Projection is a necessary step in creating a two-dimensional map and is one of the essential elements of cartography. All projections of a sphere on a plane necessarily distort the surface in some way and to some extent. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. The study of map projections is primarily about the characterization of their distortions. There is no limit to the number of possible map projections. More generally, projections are considered in several f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circular Segment
In geometry, a circular segment (symbol: ), also known as a disk segment, is a region of a disk which is "cut off" from the rest of the disk by a secant or a chord. More formally, a circular segment is a region of two-dimensional space that is bounded by a circular arc (of less than π radians by convention) and by the circular chord connecting the endpoints of the arc. Formulae Let ''R'' be the radius of the arc which forms part of the perimeter of the segment, ''θ'' the central angle subtending the arc in radians, ''c'' the chord length, ''s'' the arc length, ''h'' the sagitta (height) of the segment, ''d'' the apothem of the segment, and ''a'' the area of the segment. Usually, chord length and height are given or measured, and sometimes the arc length as part of the perimeter, and the unknowns are area and sometimes arc length. These can't be calculated simply from chord length and height, so two intermediate quantities, the radius and central angle are usually calc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Cap
In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane. It is also a spherical segment of one base, i.e., bounded by a single plane. If the plane passes through the center of the sphere (forming a great circle), so that the height of the cap is equal to the radius of the sphere, the spherical cap is called a '' hemisphere''. Volume and surface area The volume of the spherical cap and the area of the curved surface may be calculated using combinations of * The radius r of the sphere * The radius a of the base of the cap * The height h of the cap * The polar angle \theta between the rays from the center of the sphere to the apex of the cap (the pole) and the edge of the disk forming the base of the cap If \phi denotes the latitude in geographic coordinates, then \theta+\phi = \pi/2 = 90^\circ\,, and \cos \theta = \sin \phi. The relationship between h and r is relevant as long as 0\le h\le2r. For example, the red section of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold. Informal description In differential geometry, one can attach to every point x of a differentiable manifold a ''tangent space''—a real vector space that intuitively contains the possible directions in which one can tangentially pass through x . The elements of the tangent space at x are called the ''tangent vectors'' at x . This is a generalization of the notion of a vector, based at a given initial point, in a Euclidean space. The dimension of the tangent space at every point of a connected manifold is the same as that of the manifold itself. For example, if the given manifold is a 2 -sphere, then one can picture t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
