|
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] |
Superellipse Rounded Diamond
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 Star
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 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] |
Circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is called the radius. The length of a line segment connecting two points on the circle and passing through the centre is called the diameter. A circle bounds a region of the plane called a Disk (mathematics), disc. The circle has been known since before the beginning of recorded history. Natural circles are common, such as the full moon or a slice of round fruit. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Terminology * Annulus (mathematics), Annulus: a ring-shaped object, the region bounded by two concentric circles. * Circular arc, Arc: any Connected ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombus
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a "diamond", after the Diamonds (suit), diamonds suit in playing cards which resembles the projection of an Octahedron#Orthogonal projections, octahedral diamond, or a lozenge (shape), lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (which some authors call a calisson after calisson, the French sweet—also see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle. Every rhombus is simple polygon, simple (non-self-intersecting), and is a special case of a parallelogram and a Kite (geometry), kite. A rhombus with right angles is a square. Etymology The word "rhombus" comes from , meaning something that spins, which derives from the verb , roman ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Astroid
In mathematics, an astroid is a particular type of roulette curve: a hypocycloid with four cusp (singularity), cusps. Specifically, it is the Locus (mathematics), locus of a point on a circle as it Rolling, rolls inside a fixed circle with four times the radius. By double generation, it is also the locus of a point on a circle as it rolls inside a fixed circle with 4/3 times the radius. It can also be defined as the Envelope (mathematics), envelope of a line segment of fixed length that moves while keeping an end point on each of the axes. It is therefore the Envelope (mathematics), envelope of the moving bar in the Trammel of Archimedes. Its modern name comes from the Greek word for "star". It was proposed, originally in the form of "Astrois", by Joseph Johann von Littrow in 1838. The curve had a variety of names, including tetracuspid (still used), cubocycloid, and paracycle. It is nearly identical in form to the evolute of an ellipse. Equations If the radius of the fixed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norm (mathematics)
In mathematics, a norm is a function (mathematics), function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the Origin (mathematics), origin: it Equivariant map, commutes with scaling, obeys a form of the triangle inequality, and zero is only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the #Euclidean norm, Euclidean norm, the #p-norm, 2-norm, or, sometimes, the magnitude or length of the vector. This norm can be defined as the square root of the inner product of a vector with itself. A seminorm satisfies the first two properties of a norm but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a ''seminormed vector space''. The term pseudonorm has been used for several related meaning ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gabriel Lamé
Gabriel Lamé (22 July 1795 – 1 May 1870) was a French mathematician who contributed to the theory of partial differential equations by the use of curvilinear coordinates, and the mathematical theory of elasticity (for which linear elasticity and finite strain theory elaborate the mathematical abstractions). Biography Lamé was born in Tours, in today's ''département'' of Indre-et-Loire. He became well known for his general theory of curvilinear coordinates and his notation and study of classes of ellipse-like curves, now known as Lamé curves or superellipses, and defined by the equation: : \left, \,\,\^n + \left, \,\,\^n = 1 where ''n'' is any positive real number. He is also known for his running time analysis of the Euclidean algorithm, marking the beginning of computational complexity theory. In 1844, using Fibonacci numbers, he proved that when finding the greatest common divisor of integers ''a'' and ''b'', the algorithm runs in no more than 5''k'' steps, where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenization of a polynomial, homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse function, inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. If the defining polynomial of a plane algebraic curve is irreducible polynomial, irreducible, then one has an ''irreducible plane algebraic curve''. Otherwise, the algebraic curve is the union of one or several irreducible curves, called its ''Irreduc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ball (mathematics)
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ''ball'' in dimensions is called a hyperball or -ball and is bounded by a ''hypersphere'' or ()-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the planar region bounded by a circle. In Euclidean 3-space, a ball is taken to be the region of space bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment. In other contexts, such as in Euclidean geometry and informal use, ''sphere'' is sometimes used to mean ''ball''. In the field of topology the closed n-dimensional ball is often denoted as B^n or D^n while the open n-dimensional ball is \o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat Curve
In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (''X'':''Y'':''Z'') by the Fermat equation: :X^n + Y^n = Z^n.\ Therefore, in terms of the affine plane its equation is: :x^n + y^n = 1.\ An integer solution to the Fermat equation would correspond to a nonzero rational number solution to the affine equation, and vice versa. But by Fermat's Last Theorem it is now known that (for ''n'' > 2) there are no nontrivial integer solutions to the Fermat equation; therefore, the Fermat curve has no nontrivial rational points. The Fermat curve is non-singular and has genus: :(n - 1)(n - 2)/2.\ This means genus 0 for the case ''n'' = 2 (a conic) and genus 1 only for ''n'' = 3 (an elliptic curve). The Jacobian variety of the Fermat curve has been studied in depth. It is isogenous to a product of simple abelian varieties with complex multiplication. The Fermat curve also has gonality: :n-1.\ Fermat va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
