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Whitney Umbrella
frame, Section of the surface In geometry, the Whitney umbrella (or Whitney's umbrella, named after American mathematician Hassler Whitney, and sometimes called a Cayley umbrella) is a specific self-intersecting ruled surface placed in three dimensions. It is the union of all straight lines that pass through points of a fixed parabola and are perpendicular to a fixed straight line which is parallel to the axis of the parabola and lies on its perpendicular bisecting plane. Formulas Whitney's umbrella can be given by the parametric equations in Cartesian coordinates : \left\{\begin{align} x(u, v) &= uv, \\ y(u, v) &= u, \\ z(u, v) &= v^2, \end{align}\right. where the parameters ''u'' and ''v'' range over the real numbers. It is also given by the implicit equation : x^2 - y^2 z = 0. This formula also includes the negative ''z'' axis (which is called the ''handle'' of the umbrella). Properties Whitney's umbrella is a ruled surface and a right conoid. It is important in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitney Umbrella
frame, Section of the surface In geometry, the Whitney umbrella (or Whitney's umbrella, named after American mathematician Hassler Whitney, and sometimes called a Cayley umbrella) is a specific self-intersecting ruled surface placed in three dimensions. It is the union of all straight lines that pass through points of a fixed parabola and are perpendicular to a fixed straight line which is parallel to the axis of the parabola and lies on its perpendicular bisecting plane. Formulas Whitney's umbrella can be given by the parametric equations in Cartesian coordinates : \left\{\begin{align} x(u, v) &= uv, \\ y(u, v) &= u, \\ z(u, v) &= v^2, \end{align}\right. where the parameters ''u'' and ''v'' range over the real numbers. It is also given by the implicit equation : x^2 - y^2 z = 0. This formula also includes the negative ''z'' axis (which is called the ''handle'' of the umbrella). Properties Whitney's umbrella is a ruled surface and a right conoid. It is important in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitney Umbrella
frame, Section of the surface In geometry, the Whitney umbrella (or Whitney's umbrella, named after American mathematician Hassler Whitney, and sometimes called a Cayley umbrella) is a specific self-intersecting ruled surface placed in three dimensions. It is the union of all straight lines that pass through points of a fixed parabola and are perpendicular to a fixed straight line which is parallel to the axis of the parabola and lies on its perpendicular bisecting plane. Formulas Whitney's umbrella can be given by the parametric equations in Cartesian coordinates : \left\{\begin{align} x(u, v) &= uv, \\ y(u, v) &= u, \\ z(u, v) &= v^2, \end{align}\right. where the parameters ''u'' and ''v'' range over the real numbers. It is also given by the implicit equation : x^2 - y^2 z = 0. This formula also includes the negative ''z'' axis (which is called the ''handle'' of the umbrella). Properties Whitney's umbrella is a ruled surface and a right conoid. It is important in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitney Unbrella
Whitney may refer to: Film and television * ''Whitney'' (2015 film), a Whitney Houston biopic starring Yaya DaCosta * ''Whitney'' (2018 film), a documentary about Whitney Houston * ''Whitney'' (TV series), an American sitcom that premiered in 2011 Firearms *Whitney Wolverine, a semi-automatic, .22 LR caliber pistol *Whitney revolver, a gun carried by Powell when he attempted to assassinate Secretary of State William Seward Music * Whitney Houston, sometimes eponymously known as 'Whitney' ** ''Whitney'' (album), an album by Whitney Houston * Whitney (band), an American rock band Places Canada * Whitney, Ontario United Kingdom * Witney, Oxfordshire ** Witney (UK Parliament constituency), a constituency for the House of Commons * Whitney-on-Wye, Herefordshire United States * Whitney, Alabama * Whitney, California, a community in Placer County * Whitney, California, former name of Lone Pine Station, California * Whitney, Idaho * Whitney, Maine * Whitney, Michigan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Property
In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points). Properties of a point on a function Perhaps the best-known example of the idea of locality lies in the concept of local minimum (or local maximum), which is a point in a function whose functional value is the smallest (resp., largest) within an immediate neighborhood of points. This is to be contrasted with the idea of global minimum (or global maximum), which corresponds to the minimum (resp., maximum) of the function across its entire domain. Properties of a single space A topological space is sometimes said to exhibit a property locally, if the property is exhibited "near" each point in one of the following ways: # Each point has a neighborhood exhibiting the property; # Each point has a neighborhood base of sets exhibiting the prop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ruled Surface
In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the right conoid, the helicoid, and the tangent developable of a smooth curve in space. A ruled surface can be described as the set of points swept by a moving straight line. For example, a cone is formed by keeping one point of a line fixed whilst moving another point along a circle. A surface is ''doubly ruled'' if through every one of its points there are two distinct lines that lie on the surface. The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces. The plane is the only surface which contains at least three distinct lines through each of its points . The properties of being ruled or doubly ruled are preserved by projective maps, and therefore are concepts of projective geometry. In algebrai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right Conoid
In geometry, a right conoid is a ruled surface generated by a family of straight lines that all intersect perpendicularly to a fixed straight line, called the ''axis'' of the right conoid. Using a Cartesian coordinate system in three-dimensional space, if we take the to be the axis of a right conoid, then the right conoid can be represented by the parametric equations: :x=v\cos u :y=v\sin u :z=h(u) where is some function for representing the ''height'' of the moving line. Examples A typical example of right conoids is given by the parametric equations : x=v\cos u, y=v\sin u, z=2\sin u The image on the right shows how the coplanar lines generate the right conoid. Other right conoids include: *Helicoid: x=v\cos u, y=v\sin u, z=cu. *Whitney umbrella: x=vu, y=v, z=u^2. * Wallis's conical edge: x=v\cos u, y=v \sin u, z=c\sqrt. * Plücker's conoid: x=v\cos u, y=v\sin u, z=c\sin nu. *hyperbolic paraboloid: x=v, y=u, z=uv (with x-axis and y-axis as its axes). See also * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
F-theory
In theoretical physics, F-theory is a branch of string theory developed by Iranian physicist Cumrun Vafa. The new vacua described by F-theory were discovered by Vafa and allowed string theorists to construct new realistic vacua — in the form of F-theory compactified on elliptically fibered Calabi–Yau four-folds. The letter "F" supposedly stands for "Father". Compactifications F-theory is formally a 12-dimensional theory, but the only way to obtain an acceptable background is to compactify this theory on a two-torus. By doing so, one obtains type IIB superstring theory in 10 dimensions. The SL(2,Z) S-duality symmetry of the resulting type IIB string theory is manifest because it arises as the group of large diffeomorphisms of the two-dimensional torus. More generally, one can compactify F-theory on an elliptically fibered manifold (elliptic fibration), i.e. a fiber bundle whose fiber is a two-dimensional torus (also called an elliptic curve). For example, a subcl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
