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Spherical Segment
In geometry, a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes. It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum. The surface of the ''spherical segment'' (excluding the bases) is called spherical zone. If the radius of the sphere is called , the radii of the spherical segment bases are and , and the height of the segment (the distance from one parallel plane to the other) called , then the volume of the spherical segment is : V = \frac h \left(3 a^2 + 3 b^2 + h^2\right). For the special case of the top plane being tangent to the sphere, we have b = 0 and the solid reduces to a spherical cap. The equation above for volume of the spherical segment can be arranged to : V = \biggl \pi a^2 \left (\frac \biggr ) \right + \biggl \pi b^2 \left ( \frac \biggr ) \right + \biggl \frac \pi \left( \frac \right)^3 \biggr /math> Thus, the segment volume equals the su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface (mathematics)
In mathematics, a surface is a mathematical model of the common concept of a surface. It is a generalization of a plane, but, unlike a plane, it may be curved; this is analogous to a curve generalizing a straight line. There are several more precise definitions, depending on the context and the mathematical tools that are used for the study. The simplest mathematical surfaces are planes and spheres in the Euclidean 3-space. The exact definition of a surface may depend on the context. Typically, in algebraic geometry, a surface may cross itself (and may have other singularities), while, in topology and differential geometry, it may not. A surface is a topological space of dimension two; this means that a moving point on a surface may move in two directions (it has two degrees of freedom). In other words, around almost every point, there is a '' coordinate patch'' on which a two-dimensional coordinate system is defined. For example, the surface of the Earth resembles ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Wedge
A sphere (from Greek , ) is a surface analogous to the circle, a curve. In solid geometry, 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 ''center'' of the sphere, and the distance 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 surface 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 ea ... [...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 (mathematics), ball cut off by a plane (mathematics), plane. It is also a spherical segment of one base, i.e., bounded by a single plane. If the plane passes through the center (geometry), 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 ''Sphere#Hemisphere, 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 Spherical coordinate system, 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 (mathematics), disk forming the base of the cap. These variables are inter-related through the formulas a = r \sin \theta, h = r ( 1 - \cos \theta ), 2hr = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Area
The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration. A general definition of surface area was sought by Henri Lebesgue and Hermann Minkowski at the turn of the twentieth century. Their work led to the development of geometric measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volume
Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The definition of length and height (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. By metonymy, the term "volume" sometimes is used to refer to the corresponding region (e.g., bounding volume). In ancient times, volume was measured using similar-shaped natural containers. Later on, standardized containers were used. Some simple three-dimensional shapes can have their volume easily calculated using arithmetic formulas. Volumes of more complicated shapes can be calculated with integral calculus if a formula exists for the shape ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radius
In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is the line segment or distance from its center to any of its Vertex (geometry), vertices. The name comes from the Latin ''radius'', meaning ray but also the spoke of a chariot wheel.Definition of Radius at dictionary.reference.com. Accessed on 2009-08-08. The typical abbreviation and mathematical symbol for radius is ''R'' or ''r''. By extension, the diameter ''D'' is defined as twice the radius:Definition of radius at mathwords.com. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Parameters For Spherical Segment
Geometry (; ) 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. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained unsolved for several centuries. During t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frustum
In geometry, a ; (: frusta or frustums) is the portion of a polyhedron, solid (normally a pyramid (geometry), pyramid or a cone (geometry), cone) that lies between two parallel planes cutting the solid. In the case of a pyramid, the base faces are polygonal and the side faces are trapezoidal. A ''right frustum'' is a right pyramid or a right cone truncation (geometry), truncated perpendicularly to its axis; otherwise, it is an ''oblique frustum''. In a ''truncated cone'' or ''truncated pyramid'', the truncation plane is necessarily parallel to the cone's base, as in a frustum. If all its edges are forced to become of the same length, then a frustum becomes a ''Prism (geometry), prism'' (possibly oblique or/and with irregular bases). Elements, special cases, and related concepts A frustum's axis is that of the original cone or pyramid. A frustum is circular if it has circular bases; it is right if the axis is perpendicular to both bases, and oblique otherwise. The height of a f ... [...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 (mathematics), ball cut off by a plane (mathematics), plane. It is also a spherical segment of one base, i.e., bounded by a single plane. If the plane passes through the center (geometry), 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 ''Sphere#Hemisphere, 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 Spherical coordinate system, 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 (mathematics), disk forming the base of the cap. These variables are inter-related through the formulas a = r \sin \theta, h = r ( 1 - \cos \theta ), 2hr = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel (geometry)
In geometry, parallel lines are coplanar infinite straight line (geometry), lines that do not intersecting lines, intersect at any point. Parallel planes are plane (geometry), planes in the same three-dimensional space that never meet. ''Parallel curves'' are curves that do not tangent, touch each other or intersect and keep a fixed minimum distance. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called ''skew lines''. Line segments and Euclidean vectors are parallel if they have the same direction (geometry), direction or opposite direction (geometry), opposite direction (not necessarily the same length). Parallel lines are the subject of Euclid's parallel postulate. Parallelism is primarily a property of affine geometry, affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines can have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
