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Squircle
A squircle is a shape intermediate between a square and a circle. There are at least two definitions of "squircle" in use, one based on the superellipse, the other arising from work in optics. The word "squircle" is a portmanteau of the words "square" and "circle". Squircles have been applied in design and optics. Superellipse-based squircle In a Cartesian coordinate system, the superellipse is defined by the equation \left, \frac\^n + \left, \frac\^n = 1, where and are the semi-major and semi-minor axes, and are the and coordinates of the centre of the ellipse, and is a positive number. The squircle is then defined as the superellipse where and . Its equation is:\left, x - a\^4 + \left, y - b\^4 = r^4where is the radius of the squircle. Compare this to the equation of a circle. When the squircle is centred at the origin, then , and it is called Lamé's special quartic. The area inside the squircle can be expressed in terms of the beta function or the gamma funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superellipse Chamfered Square
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 defined by an equation that allows for various shapes between a rectangle and an ellipse. In two dimensional Cartesian coordinate system, a superellipse is defined as the set of all points (x,y) on the curve that satisfy the equation\left, \frac\^n\!\! + \left, \frac\^n\! = 1,where a and b are positive numbers referred to as semi-diameters or semi-axes of the superellipse, and n is a positive parameter that defines the shape. When n=2, the superellipse is an ordinary ellipse. For n>2, the shape is more rectangular with rounded corners, and for 0 [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superellipse
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 defined by an equation that allows for various shapes between a rectangle and an ellipse. In two dimensional Cartesian coordinate system, a superellipse is defined as the set of all points (x,y) on the curve that satisfy the equation\left, \frac\^n\!\! + \left, \frac\^n\! = 1,where a and b are positive numbers referred to as Semidiameter, semi-diameters or Semi-major and semi-minor axes, semi-axes of the superellipse, and n is a positive parameter that defines the shape. When n=2, the superellipse is an ordinary ellipse. For n>2, the shape is more rectangular with rounded corners, and for 00, where m either equals to or differs from ''n''. If m=n, it is the Lamé's superellipses. If m\neq n, the curve possesses more flexibility of behavior, and is better possible fit to des ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square (geometry)
In geometry, a square is a regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal sides. As with all rectangles, a square's angles are right angles (90 degrees, or /2 radians), making adjacent sides perpendicular. The area of a square is the side length multiplied by itself, and so in algebra, multiplying a number by itself is called squaring. Equal squares can tile the plane edge-to-edge in the square tiling. Square tilings are ubiquitous in tiled floors and walls, graph paper, image pixels, and game boards. Square shapes are also often seen in building floor plans, origami paper, food servings, in graphic design and heraldry, and in instant photos and fine art. The formula for the area of a square forms the basis of the calculation of area and motivates the search for methods for squaring the circle by compass and straightedge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lp Space
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbaki group they were first introduced by Frigyes Riesz . spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. Preliminaries The -norm in finite dimensions The Euclidean length of a vector x = (x_1, x_2, \dots, x_n) in the n-dimensional real vector space \Reals^n is given by the Euclidean norm: \, x\, _2 = \left(^2 + ^2 + \dotsb + ^2\right)^. The Euclidean distance between two points x and y is the length \, x - y\, _2 of the straight line b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Desmos
Desmos is an advanced graphing calculator implemented as a web application and a mobile application written in TypeScript and JavaScript. History Desmos was founded by Eli Luberoff, a math and physics double major from Yale University, and was launched as a startup at TechCrunch's Disrupt New York conference in 2011. , it had received around 1 million US dollars of funding from Kapor Capital, Learn Capital, Kindler Capital, Elm Street Ventures and Google Ventures. The name ''Desmos'' came from the Greek word which means a bond or a tie. In May 2022, Amplify acquired the Desmos curriculum and teacher.desmos.com. Some 50 employees joined Amplify. Desmos Studio was spun off as a separate public benefit corporation focused on building calculator products and other math tools. In May 2023, Desmos released a beta for a remade Geometry Tool. In it, geometrical shapes can be made, as well as expressions from the normal graphing calculator, with extra features. In September 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit (mathematics)
In mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value. Limits of functions are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. The limit inferior and limit superior provide generalizations of the concept of a limit which are particularly relevant when the limit at a point may not exist. Notation In formulas, a limit of a function is usually written as : \lim_ f(x) = L, and is read as "the limit of of as approaches equals ". This means that the value of the function can be made arbitrarily close to , by choosing sufficiently close to . Alternatively, the fact that a function approaches the limit as approaches is sometimes denoted by a right arrow (→ or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometry
Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation. Trigonometry is known for its many identities. These trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation. History Sumerian astronomers studied angle me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indeterminate Form
Indeterminate form is a mathematical expression that can obtain any value depending on circumstances. In calculus, it is usually possible to compute the limit of the sum, difference, product, quotient or power of two functions by taking the corresponding combination of the separate limits of each respective function. For example, \begin \lim_ \bigl(f(x) + g(x)\bigr) &= \lim_ f(x) + \lim_ g(x), \\ mu\lim_ \bigl(f(x)g(x)\bigr) &= \lim_ f(x) \cdot \lim_ g(x), \end and likewise for other arithmetic operations; this is sometimes called the algebraic limit theorem. However, certain combinations of particular limiting values cannot be computed in this way, and knowing the limit of each function separately does not suffice to determine the limit of the combination. In these particular situations, the limit is said to take an indeterminate form, described by one of the informal expressions \frac 00,~ \frac,~ 0\times\infty,~ \infty - \infty,~ 0^0,~ 1^\infty, \text \infty^0, among a wide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinity
Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophical nature of infinity has been the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including Guillaume de l'Hôpital, l'Hôpital and Johann Bernoulli, Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or Magnitude (mathematics), magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parametrization (geometry)
In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation. The inverse process is called implicitization. "To parameterize" by itself means "to express in terms of parameters". Parametrization is a mathematical process consisting of expressing the state of a system, process or model as a function of some independent quantities called parameters. The state of the system is generally determined by a finite set of coordinates, and the parametrization thus consists of one function of several real variables for each coordinate. The number of parameters is the number of degrees of freedom of the system. For example, the position of a point that moves on a curve in three-dimensional space is determined by the time needed to reach the point when starting from a fixed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
