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Hyperbolic Triangle
In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called ''sides'' or ''edges'' and three point (geometry), points called ''angles'' or ''vertices''. Just as in the Euclidean space, Euclidean case, three points of a hyperbolic space of an arbitrary dimension (mathematics), dimension always lie on the same plane. Hence planar hyperbolic triangles also describe triangles possible in any higher dimension of hyperbolic spaces. Definition A hyperbolic triangle consists of three non-collinear points and the three segments between them. Properties Hyperbolic triangles have some properties that are analogous to those of triangles in Euclidean geometry: *Each hyperbolic triangle has an inscribed circle but not every hyperbolic triangle has a circumscribed circle (see below). Its vertices can lie on a horocycle or hypercycle (geometry), hypercycle. Hyperbolic triangles have some properties that are analogous to t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the ''base'', in which case the opposite vertex is called the ''apex''; the shortest segment between the base and apex is the ''height''. The area of a triangle equals one-half the product of height and base length. In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points that do not all lie on the same straight line determine a unique triangle situated w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horocycle
In hyperbolic geometry, a horocycle ( from Greek roots meaning "boundary circle"), sometimes called an oricycle or limit circle, is a curve of constant curvature where all the perpendicular geodesics ( normals) through a point on a horocycle are limiting parallel, and all converge asymptotically to a single ideal point called the '' centre'' of the horocycle. In some models of hyperbolic geometry, it looks like the two "ends" of a horocycle get closer and closer to each other and closer to its centre, but this is not true; the two "ends" of a horocycle get further and further away from each other and stay at an infinite distance off its centre. A horosphere is the 3-dimensional version of a horocycle. In Euclidean space, all curves of constant curvature are either straight lines (geodesics) or circles, but in a hyperbolic space of sectional curvature -1, the curves of constant curvature come in four types: geodesics with curvature \kappa = 0, hypercycles with curvature 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angle Of Parallelism
In hyperbolic geometry, angle of parallelism \Pi(a) is the angle at the non-right angle vertex of a right hyperbolic triangle having two limiting parallel, asymptotic parallel sides. The angle depends on the segment length ''a'' between the right angle and the vertex of the angle of parallelism. Given a point not on a line, drop a perpendicular to the line from the point. Let ''a'' be the length of this perpendicular segment, and \Pi(a) be the least angle such that the line drawn through the point does not intersect the given line. Since two sides are asymptotically parallel, : \lim_ \Pi(a) = \tfrac\pi\quad\text\quad\lim_ \Pi(a) = 0. There are five equivalent expressions that relate '' \Pi(a)'' and ''a'': : \sin\Pi(a) = \operatorname a = \frac =\frac \ , : \cos\Pi(a) = \tanh a = \frac \ , : \tan\Pi(a) = \operatorname a = \frac = \frac \ , : \tan \left( \tfrac\Pi(a) \right) = e^, : \Pi(a) = \tfrac\pi - \operatorname(a), where sinh, cosh, tanh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Circles
In geometry, tangent circles (also known as kissing circles) are circles in a common plane that intersect in a single point. There are two types of tangency: internal and external. Many problems and constructions in geometry are related to tangent circles; such problems often have real-life applications such as trilateration and maximizing the use of materials. Two given circles Two circles are mutually and externally tangent if distance between their centers is equal to the sum of their radii Steiner chains Pappus chains Three given circles: Apollonius' problem Apollonius' problem is to construct circles that are tangent to three given circles. Apollonian gasket If a circle is iteratively inscribed into the interstitial curved triangles between three mutually tangent circles, an Apollonian gasket results, one of the earliest fractals described in print. Malfatti's problem Malfatti's problem is to carve three cylinders from a triangular block of marble, using ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line (geometry)
In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray (optics), ray of light. Lines are space (mathematics), spaces of dimension one, which may be Embedding, embedded in spaces of dimension two, three, or higher. The word ''line'' may also refer, in everyday life, to a line segment, which is a part of a line delimited by two Point (geometry), points (its ''endpoints''). Euclid's Elements, Euclid's ''Elements'' defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several postulates as basic unprovable properties on which the rest of geometry was established. ''Euclidean line'' and ''Euclidean geometry'' are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as Non-Euclidean geometry, non-Euclidean, Project ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero
0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and complex numbers, as well as other algebraic structures. Multiplying any number by 0 results in 0, and consequently division by zero has no meaning in arithmetic. As a numerical digit, 0 plays a crucial role in decimal notation: it indicates that the power of ten corresponding to the place containing a 0 does not contribute to the total. For example, "205" in decimal means two hundreds, no tens, and five ones. The same principle applies in place-value notations that uses a base other than ten, such as binary and hexadecimal. The modern use of 0 in this manner derives from Indian mathematics that was transmitted to Europe via medieval Islamic mathematicians and popularized by Fibonacci. It was independently used by the Maya. Common name ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limiting Parallel
In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line l through a point P not on line R; however, in the plane, two parallels may be closer to l than all others (one in each direction of R). Thus it is useful to make a new definition concerning parallels in neutral geometry. If there are closest parallels to a given line they are known as the limiting parallel, asymptotic parallel or horoparallel (horo from — border). For rays, the relation of limiting parallel is an equivalence relation, which includes the equivalence relation of being coterminal. If, in a hyperbolic triangle, the pairs of sides are limiting parallel, then the triangle is an ideal triangle. Definition A ray Aa is a limiting parallel to a ray Bb if they are coterminal or if they lie on distinct lines not equal to the line AB, they do not meet, and every ray in the interior of the angle BAa meets the ray Bb. Properties Distinct lines carrying lim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal Circles
Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered in abstract algebra * Ideal, special subsets of a Semigroup#Subsemigroups and ideals, semigroup * Ideal (order theory), special kind of lower sets of an order * Ideal (set theory), a collection of sets regarded as "small" or "negligible" * Ideal (Lie algebra), a particular subset in a Lie algebra * Ideal point, a boundary point in hyperbolic geometry * Ideal triangle, a triangle in hyperbolic geometry whose vertices are ideal points Science * Ideal chain, in science, the simplest model describing a polymer * Ideal gas law, in physics, governing the pressure of an ideal gas * Ideal transformer, an electrical transformer having zero resistance and perfect magnetic threading * Ideal final result, in TRIZ methodology, the best possible soluti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal Point
In hyperbolic geometry, an ideal point, omega point or point at infinity is a well-defined point outside the hyperbolic plane or space. Given a line ''l'' and a point ''P'' not on ''l'', right- and left-limiting parallels to ''l'' through ''P'' converge to ''l'' at ''ideal points''. Unlike the projective case, ideal points form a boundary, not a submanifold. So, these lines do not intersect at an ideal point and such points, although well-defined, do not belong to the hyperbolic space itself. The ideal points together form the Cayley absolute or boundary of a hyperbolic geometry. For instance, the unit circle forms the Cayley absolute of the Poincaré disk model and the Klein disk model. The real line forms the Cayley absolute of the Poincaré half-plane model. Pasch's axiom and the exterior angle theorem still hold for an omega triangle, defined by two points in hyperbolic space and an omega point. Properties * The hyperbolic distance between an ideal point and any other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Because of this, the elliptic geometry described in this article is sometimes referred to as ''single elliptic geometry'' whereas spherical geometry is sometimes referred to as ''double elliptic geometry''. The appearance of this geometry in the nineteenth century stimulated the development of non-Euclidean geometry generally, including hyperbolic geometry. Elliptic geometry has a variety of properties that differ from those of classical Euclidean plane geometry. For example, the sum of the interior angles of any triangle is always greater than 180°. Definitions Elliptic geometry may be derived from spherical geometry by identifying antipodal points of the sphere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
