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Supertoroid
In geometry and computer graphics, a supertoroid or supertorus is usually understood to be a family of doughnut-like surface (geometry), surfaces (technically, a topology, topological torus (mathematics), torus) whose shape is defined by mathematical formulas similar to those that define the superquadrics. The plural of "supertorus" is either supertori or supertoruses. The family was described and named by Alan H.Barr, Alan Barr in 1994.Alan H. Barr (1981) ''Superquadrics and Angle-Preserving Transformations''. IEEE Computer Graphics and Applications, volume 1 issue 1. pp. 11-23. Barr's supertoroids have been fairly popular in computer graphics as a convenient model for many objects, such as smooth frames for rectangular things. One quarter of a supertoroid can provide a smooth and seamless 90-degree joint between two superquadric cylinder (geometry), cylinders. However, they are not algebraic surfaces (except in special cases). Formulas Alan Barr's supertoroids are defined by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superquadric
In mathematics, the superquadrics or super-quadrics (also superquadratics) are a family of geometry, geometric shapes defined by formulas that resemble those of ellipsoids and other quadrics, except that the square (algebra), squaring operations are replaced by arbitrary powers. They can be seen as the three-dimensional relatives of the superellipses. The term may refer to the solid object or to its Surface (mathematics), surface, depending on the context. The equations below specify the surface; the solid is specified by replacing the equality signs by less-than-or-equal signs. The superquadrics include many shapes that resemble cubes, octahedron, octahedra, Cylinder (geometry), cylinders, Lozenge (shape), lozenges and spindle torus, spindles, with rounded or sharp corners. Because of their flexibility and relative simplicity, they are popular geometric modeling tools, especially in computer graphics. Some authors, such as Alan H. Barr, Alan Barr, define "superquadrics" as includ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superquadric
In mathematics, the superquadrics or super-quadrics (also superquadratics) are a family of geometry, geometric shapes defined by formulas that resemble those of ellipsoids and other quadrics, except that the square (algebra), squaring operations are replaced by arbitrary powers. They can be seen as the three-dimensional relatives of the superellipses. The term may refer to the solid object or to its Surface (mathematics), surface, depending on the context. The equations below specify the surface; the solid is specified by replacing the equality signs by less-than-or-equal signs. The superquadrics include many shapes that resemble cubes, octahedron, octahedra, Cylinder (geometry), cylinders, Lozenge (shape), lozenges and spindle torus, spindles, with rounded or sharp corners. Because of their flexibility and relative simplicity, they are popular geometric modeling tools, especially in computer graphics. Some authors, such as Alan H. Barr, Alan Barr, define "superquadrics" as includ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superellipsoid
In mathematics, a superellipsoid (or super-ellipsoid) is a solid whose horizontal sections are superellipses (Lamé curves) with the same exponent ''r'', and whose vertical sections through the center are superellipses with the same exponent ''t''. Superellipsoids as computer graphics primitives were popularized by Alan H. Barr (who used the name "superquadrics" to refer to both superellipsoids and supertoroids).Barr, A.H. (January 1981), ''Superquadrics and Angle-Preserving Transformations''. IEEE_CGA vol. 1 no. 1, pp. 11–23Barr, A.H. (1992), ''Rigid Physically Based Superquadrics''. Chapter III.8 of ''Graphics Gems III'', edited by D. Kirk, pp. 137–159 However, while some superellipsoids are superquadrics, neither family is contained in the other. Special cases A handful of notable mathematical figures can arise as special cases of superellipsoids given the correct set of values, which are depicted in the above graphic: * Cylinder * Sphere * Steinmetz solid * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superegg
In geometry, a superegg is a solid of revolution obtained by rotating an elongated superellipse with exponent greater than 2 around its longest axis. It is a special case of superellipsoid. Unlike an elongated ellipsoid, an elongated superegg can stand upright on a flat surface, or on top of another superegg. This is due to its curvature being zero at the tips. The shape was popularized by Danish poet and scientist Piet Hein (1905–1996). Supereggs of various materials, including brass, were sold as novelties or "executive toys" in the 1960s. Mathematical description The superegg is a superellipsoid whose horizontal cross-sections are circles. It is defined by the inequality :\left, \frac\^p + \left, \frac\^p \leq 1 where ''R'' is the horizontal radius at the "equator" (the widest part), and ''h'' is one half of the height. The exponent ''p'' determines the degree of flattening at the tips and equator. Hein's choice was ''p'' = 2.5 (the same one he used for the Ser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GNU Octave
GNU Octave is a high-level programming language primarily intended for scientific computing and numerical computation. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with MATLAB. It may also be used as a batch-oriented language. As part of the GNU Project, it is free software under the terms of the GNU General Public License. History The project was conceived around 1988. At first it was intended to be a companion to a chemical reactor design course. Full development was started by John W. Eaton in 1992. The first alpha release dates back to 4 January 1993 and on 17 February 1994 version 1.0 was released. Version 7.1.0 was released on Apr 6, 2022. The program is named after Octave Levenspiel, a former professor of the principal author. Levenspiel was known for his ability to perform quick back-of-the-envelope calculations. Development history Developments In addition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lamé Curve
A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but a different overall shape. In the Cartesian coordinate system, the set of all points (x,y) on the curve satisfy the equation :\left, \frac\^n\!\! + \left, \frac\^n\! = 1, where n,a and b are positive numbers, and the vertical bars around a number indicate the absolute value of the number. Specific cases This formula defines a closed curve contained in the rectangle −''a'' ≤ ''x'' ≤ +''a'' and −''b'' ≤ ''y'' ≤ +''b''. The parameters ''a'' and ''b'' are called the ''semi-diameters'' of the curve. The overall shape of the curve is determined by the value of the exponent ''n'', as shown in the following table: If ''n'' 2, a hyperellipse. When ''n'' ≥ 1 and ''a'' = ''b'', the superell ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval (mathematics)
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other examples of intervals are the set of numbers such that , the set of all real numbers \R, the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element). Real intervals play an important role in the theory of integration, because they are the simplest sets whose "length" (or "measure" or "size") is easy to define. The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the Lebesgue measure. Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, and is usually used to refer to an object that is invariant under some transformations; including translation, reflection, rotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are intricately related, and hence are discussed together in this article. Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, including theoretic models, language, and music. This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that category was abolished in 1995). The radian is defined in the SI as being a dimensionless unit, with 1 rad = 1. Its symbol is accordingly often omitted, especially in mathematical writing. Definition One radian is defined as the angle subtended from the center of a circle which intercepts an arc equal in length to the radius of the circle. More generally, the magnitude in radians of a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, \theta = \frac, where is the subtended angle in radians, is arc length, and is radius. A right angle is exactly \frac radians. The rotation angle (360°) corresponding to one complete revolution is the length of the circumference divided by the radius, which i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product of multiplying bases: b^n = \underbrace_. The exponent is usually shown as a superscript to the right of the base. In that case, is called "''b'' raised to the ''n''th power", "''b'' (raised) to the power of ''n''", "the ''n''th power of ''b''", "''b'' to the ''n''th power", or most briefly as "''b'' to the ''n''th". Starting from the basic fact stated above that, for any positive integer n, b^n is n occurrences of b all multiplied by each other, several other properties of exponentiation directly follow. In particular: \begin b^ & = \underbrace_ \\[1ex] & = \underbrace_ \times \underbrace_ \\[1ex] & = b^n \times b^m \end In other words, when multiplying a base raised to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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