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Slab (geometry)
In geometry, a slab is a region between two parallel lines in the Euclidean plane, or between two parallel planes or hyperplanes in higher dimensions. Set definition A slab can also be defined as a set of points: \, where n is the normal vector of the planes n \cdot x = \alpha and n \cdot x = \beta. Or, if the slab is centered around the origin: \, where \theta = , \alpha - \beta, is the thickness of the slab. See also * Bounding slab * Convex polytope * Half-plane * Hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hype ... * Prismatoid * Slab decomposition * Spherical shell References Elementary geometry Geometric shapes Spherical geometry {{geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. 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. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of parallel lines, and also metrical notions of distance, circles, and angle measurement. The set \mathbb^2 of pairs of real numbers (the real coordinate plane) augmented by appropriate structure often serves as the canonical example. History Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics. Later, the plane was described in a so-called ''Cartesian coordinate system'', a coordinate system that specifies each point uniquely in a plane ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines. This notion can be used in any general space in which the concept of the dimension of a subspace is defined. In different settings, hyperplanes may have different properties. For instance, a hyperplane of an -dimensional affine space is a flat subset with dimension and it separates the space into two half spaces. While a hyperplane of an -dimensional projective space does not have this property. The difference in dimension between a subspace and its ambient space is known as the codimension of with respect to . Therefore, a necessary and sufficient condition for to be a hyperplane in is for to have codimension one in . Technical description In geometry, a hyperplane of an ''n''-di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounding Volume
In computer graphics and computational geometry, a bounding volume for a set of objects is a closed volume that completely contains the union of the objects in the set. Bounding volumes are used to improve the efficiency of geometrical operations by using simple volumes to contain more complex objects. Normally, simpler volumes have simpler ways to test for overlap. A bounding volume for a set of objects is also a bounding volume for the single object consisting of their union, and the other way around. Therefore, it is possible to confine the description to the case of a single object, which is assumed to be non-empty and bounded (finite). Uses Bounding volumes are most often used to accelerate certain kinds of tests. In ray tracing, bounding volumes are used in ray-intersection tests, and in many rendering algorithms, they are used for viewing frustum tests. If the ray or viewing frustum does not intersect the bounding volume, it cannot intersect the object contained ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum and Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines. This notion can be used in any general space in which the concept of the dimension of a subspace is defined. In different settings, hyperplanes may have different properties. For instance, a hyperplane of an -dimensional affine space is a flat subset with dimension and it separates the space into two half spaces. While a hyperplane of an -dimensional projective space does not have this property. The difference in dimension between a subspace and its ambient space is known as the codimension of with respect to . Therefore, a necessary and sufficient condition for to be a hyperplane in is for to have codimension one in . Technical description In geometry, a hyperplane of an ''n''-di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prismatoid
In geometry, a prismatoid is a polyhedron whose vertices all lie in two parallel planes. Its lateral faces can be trapezoids or triangles. If both planes have the same number of vertices, and the lateral faces are either parallelograms or trapezoids, it is called a prismoid. Volume If the areas of the two parallel faces are and , the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is , and the height (the distance between the two parallel faces) is , then the volume of the prismatoid is given by V = \fracB. E. Meserve, R. E. Pingry: ''Some Notes on the Prismoidal Formula''. The Mathematics Teacher, Vol. 45, No. 4 (April 1952), pp. 257-263 (This formula follows immediately by integrating the area parallel to the two planes of vertices by Simpson's rule, since that rule is exact for integration of polynomials of degree up to 3, and in this case the area is at most a quadratic function in the height.) Prismatoid famili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point Location
The point location problem is a fundamental topic of computational geometry. It finds applications in areas that deal with processing geometrical data: computer graphics, geographic information systems (GIS), motion planning, and computer aided design (CAD). In its most general form, the problem is, given a partition of the space into disjoint regions, to determine the region where a query point lies. As an example application, each time one clicks a mouse to follow a link in a web browser, this problem must be solved in order to determine which area of the computer screen is under the mouse pointer. A simple special case is the point in polygon problem. In this case, one needs to determine whether the point is inside, outside, or on the boundary of a single polygon. In many applications, one needs to determine the location of several different points with respect to the same partition of the space. To solve this problem efficiently, it is useful to build a data structure tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Shell
In geometry, a spherical shell is a generalization of an annulus to three dimensions. It is the region of a ball between two concentric spheres of differing radii. Volume The volume of a spherical shell is the difference between the enclosed volume of the outer sphere and the enclosed volume of the inner sphere: : V=\frac\pi R^3- \frac\pi r^3 : V=\frac\pi (R^3-r^3) : V=\frac\pi (R-r)(R^2+Rr+r^2) where is the radius of the inner sphere and is the radius of the outer sphere. Approximation An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness of the shell: : V \approx 4 \pi r^2 t, when is very small compared to (t \ll r). The total surface area of the spherical shell is 4 \pi r^2. See also * Spherical pressure vessel * Ball * Solid torus * Bubble * Sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. 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. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Shapes
Geometric Shapes is a Unicode block of 96 symbols at code point range U+25A0–25FF. U+25A0–U+25CF The BLACK CIRCLE is displayed when typing in a password field, in order to hide characters from a screen recorder or shoulder surfing. U+25D0–U+25FF The CIRCLE WITH LEFT HALF BLACK is used to represent the contrast ratio of a screen. Font coverage Font sets like Code2000 and the DejaVu family include coverage for each of the glyphs in the Geometric Shapes range. Unifont also contains all the glyphs. Among the fonts in widespread use, full implementation is provided by Segoe UI Symbol and significant partial implementation of this range is provided by Arial Unicode MS and Lucida Sans Unicode, which include coverage for 83% (80 out of 96) and 82% (79 out of 96) of the symbols, respectively. Block Emoji The Geometric Shapes block contains eight emoji: U+25AA–U+25AB, U+25B6, U+25C0 and U+25FB–U+25FE. The block has sixteen standardized variants defined to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
