In differential geometry, the Henneberg surface is a non-orientable minimal surface [1] named after Lebrecht Henneberg.
It has parametric equation
and can be expressed as an order-15 algebraic surface. [2] It can be viewed as an immersion of a punctured projective plane. [3] Up until 1981 it was the only known non-orientable minimal surface. [4]
The surface contains a semicubical parabola ("Neile's parabola") and can be derived from solving the corresponding Björling problem. [5] [6]
In physics and geometry, a catenary is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field.
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t) respectively.
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.
In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis. It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally described in 1744 by the mathematician Leonhard Euler.
In mathematics and physics, n-dimensional anti-de Sitter space (AdSn) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter (1872–1934), professor of astronomy at Leiden University and director of the Leiden Observatory. Willem de Sitter and Albert Einstein worked together closely in Leiden in the 1920s on the spacetime structure of the universe. Paul Dirac was the first person to rigorously explore anti-de Sitter space, doing so in 1963.
The sine-Gordon equation is a nonlinear hyperbolic partial differential equation for a function dependent on two variables typically denoted and , involving the wave operator and the sine of .
In mathematics, a Dupin cyclide or cyclide of Dupin is any geometric inversion of a standard torus, cylinder or double cone. In particular, these latter are themselves examples of Dupin cyclides. They were discovered c. 1802 by Charles Dupin, while he was still a student at the École polytechnique following Gaspard Monge's lectures. The key property of a Dupin cyclide is that it is a channel surface in two different ways. This property means that Dupin cyclides are natural objects in Lie sphere geometry.
In mathematics, an involute is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from or wrapped around the curve.
In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic functions as coordinates. In mathematics, hyperbolic angle is an invariant measure as it is preserved under hyperbolic rotation.
In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers.
In mathematics, the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of differential geometry.
In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci and are generally taken to be fixed at and , respectively, on the -axis of the Cartesian coordinate system.
Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci and in bipolar coordinates become a ring of radius in the plane of the toroidal coordinate system; the -axis is the axis of rotation. The focal ring is also known as the reference circle.
In mathematics, a Scherk surface is an example of a minimal surface. Scherk described two complete embedded minimal surfaces in 1834; his first surface is a doubly periodic surface, his second surface is singly periodic. They were the third non-trivial examples of minimal surfaces. The two surfaces are conjugates of each other.
In differential geometry, a breather surface is a one-parameter family of mathematical surfaces which correspond to breather solutions of the sine-Gordon equation, a differential equation appearing in theoretical physics. The surfaces have the remarkable property that they have constant curvature , where the curvature is well-defined. This makes them examples of generalized pseudospheres.
In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigonometry. It can also be related to the relativistic velocity addition formula.
Ordinary trigonometry studies triangles in the Euclidean plane . There are a number of ways of defining the ordinary Euclidean geometric trigonometric functions on real numbers, for example right-angled triangle definitions, unit circle definitions, series definitions, definitions via differential equations, and definitions using functional equations. Generalizations of trigonometric functions are often developed by starting with one of the above methods and adapting it to a situation other than the real numbers of Euclidean geometry. Generally, trigonometry can be the study of triples of points in any kind of geometry or space. A triangle is the polygon with the smallest number of vertices, so one direction to generalize is to study higher-dimensional analogs of angles and polygons: solid angles and polytopes such as tetrahedrons and n-simplices.
In differential geometry, Catalan's minimal surface is a minimal surface originally studied by Eugène Charles Catalan in 1855.
In differential geometry, the Björling problem is the problem of finding a minimal surface passing through a given curve with prescribed normal. The problem was posed and solved by Swedish mathematician Emanuel Gabriel Björling, with further refinement by Hermann Schwarz.
In geometry, focal conics are a pair of curves consisting of either
