Hyperbolic Trajectory

1. A spacecraft making a fly-by of a planet follows a hyperbolic trajectory.
2. In astrodynamics or celestial mechanics a hyperbolic trajectory is an orbit with the eccentricity greater than 1.
3. In practice, the hyperbolic excess velocity (V hyp) is attained relatively soon after injection into a hyperbolic trajectory.

Hyperbolic Surface

1. Example: the manifold of unit vectors of the tangent bundle of a hyperbolic surface.
2. In a sense, there's an excess of area around a point on a hyperbolic surface.

Hyperboloid

1. In some sources that use the Hyperboloid model of the hyperbolic plane, the hyperboloid is referred to as a pseudosphere[ 3].

Hyperbolic Geometry

1. In spite of its positive curvature, the hyperboloid of two sheets with another suitably chosen metric can also be used as a model for hyperbolic geometry.
2. Hyperbolic geometry is more interesting than flat (Euclidean) geometry, leading to more possibilities as well.
3. Euclidean, elliptical and hyperbolic geometry.

Figure

1. Figure 5.4a may help you recall the proof of this theorem - and see why it is false in Hyperbolic Geometry.
2. Figure 2. Crochet stitches for the hyperbolic plane.
3. Figure 9. Hyperbolic surface of revolution.

Shows

1. The image shows a Poincaré disk model projection of the hyperbolic plane.
2. By itself, the distance, d, shows a hyperbolic geometry.

Numerical Relativity

1. Many equations arising from Numerical Relativity are not hyperbolic, but elliptic equations.

Application

1. The application comprises algebraic, trigonometric, hyperbolic and transcendental functions.
2. I will also present another application of this work that is related to the module category for vertex algebras associated with hyperbolic even lattices.
3. As an application we prove that the 3-dimensional Lie group SOL is not quasiconformaly equivalent to the hyperbolic space H 3.

Something

1. Something like Convex uniform honeycombs in hyperbolic space might be a bit more evocative, I think.

Larger

1. The path of the object is a hyperbolic trajectory if the speed of the smaller object relative to the larger object is more than escape velocity.
2. The larger the eccentricity, the flatter the hyperbolic curve.

Call

1. Today we call these three geometries Euclidean, hyperbolic, and absolute.
2. The result is an impulse Gauss curvature that is negative, and we will call the vertex a hyperbolic cone point.

Types

1. In, the possible geometries include Euclidean, hyperbolic, and elliptic, but also include five other types.

Act

1. There are four closely related Lie groups that act on the upper half-plane by fractional linear transformations and preserve the hyperbolic distance.
2. The act of constructing the surface will give a feel for hyperbolic plane that is difficult to get any other way.

Invariant

1. Indeed, the area of any hyperbolic sector is invariant under squeezing.

Euclidean Geometry

1. The continuity problem is solved here, Switch between elliptic, hyperbolic and Euclidean geometry by one click.
2. Thus, as distances get smaller, the hyperbolic plane behaves more and more like Euclidean geometry.

Parallel

1. Because there is no precise hyperbolic analogue to Euclidean parallel lines, the hyperbolic use of parallel and related terms varies among writers.
2. One of the first things proved in hyperbolic geometry is that through a point P not on a line l, there are infinitely many lines parallel to l.
3. In Euclidean and hyperbolic geometry, the two lines are then parallel.

Klein

1. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way.
2. Despite the naming, the two disc models and the half-plane model were introduced as models of hyperbolic space by Beltrami, not by Poincaré or Klein.

Convex

1. There are infinitely many uniform tilings of convex regular polygons on the hyperbolic plane, each based on a different reflective symmetry group (p q r).
2. These constraints allow for 21 forms: 6 are convex, 10 are nonconvex, one is a Euclidean 3-space honeycomb, and 4 are hyperbolic honeycombs.

Similarly

1. Similarly to parabolic trajectory all hyperbolic trajectories are also escape trajectories.
2. Angle deficit is defined similarly on a pseudosphere, and is likewise proportional to area in hyperbolic geometry.
3. Similarly, the corresponding transformation for hyperbolic orbits is found by equating Eq.

Trigonometric Functions

1. Therefore, it is possible to define hyperbolic and trigonometric functions on time scales in a standard way.

Hyperbolic Functions

1. Note that, by convention, sinh 2 x means (sinh x) 2, not sinh(sinh x); similarly for the other hyperbolic functions when used with positive exponents.
2. Thus, hyperbolic functions are periodic with respect to the imaginary component,with period 2π i (π i for hyperbolic tangent and cotangent).
3. The exponential function also gives rise to the trigonometric functions (as can be seen from Euler's formula) and to the hyperbolic functions.

Exponential

1. Limits and continuity of functions; trigonometric, exponential, logarithmic and hyperbolic functions, uniform continuity, properties of continuous functions.

System

1. Hyperbolic geometric space is the target embedding space for the BBS (Hyperbolic) system.
2. The UV cutoff for hyperbolic angle measured in the rest system of a particle emitting virtual particle is the basic ad hoc element of the model.

Classification

1. Classification Some linear, second-order partial differential equations can be classified as parabolic, hyperbolic or elliptic.
2. There are a great variety of hyperbolic 3-manifolds, and their classification is not completely understood.

Solution

1. Because the Fokker-Planck equation is a first order partial differential equation of hyperbolic type, usually the solution can not be derived in closed form.

Points

1. For an elliptic [hyperbolic] cusp, the nearby bitangent planes touch the surface at elliptic [hyperbolic] points.
2. The hyperbolic distance between two points on the hyperboloid can then be identified with the relative rapidity between the two corresponding observers.
3. It also denotes the entire set of points of an infinite hyperbolic space which is one of the three models of Riemannian geometry.

Nodes

1. Figure 16 shows the signature plots of the nodes with maximum rrl for BBS (Hyperbolic) TP and LRN embeddings.
2. The bending becomes larger when space curvature increases, and thus, Hyperbolic distance between two nodes increases.

Terms

1. This flow can be understood in terms of the flow on the tangent bundle of the Poincare half-plane model of hyperbolic geometry.
2. The full quantum version of the special vertex operator $e^varphi$ in terms of free field exponentials is constructed in the hyperbolic sector.
3. Nevertheless, they don't seem to have noticed the formula in terms of integrals of harmonic functions over hyperbolic triangles.

Analogous

1. Analogous conditions define hyperbolic and elliptic spaces of higher dimension in the class of higher-dimensional spaces of constant Riemannian curvature.

Circle

1. Examples include the product of a hyperbolic surface with a circle, or more generally the mapping torus of an isometry of a hyperbolic surface.

Perpendicular

1. In this model, hyperbolic lines are represented by circular arcs that are perpendicular to the bounding circle, including diameters.
2. The ultraparallel theorem states that there is a unique line in the hyperbolic plane that is perpendicular to each of a given pair of ultraparallel lines.

Construction

1. The construction of the extended space gives rise to a complex valued geometry consistent with both the hyperbolic and de Sitter space.
2. Note that the construction of a hyperbolic plane is dependent on r (the radius of the annuli), which is often called the radius of the hyperbolic plane.
3. We will call the results of this construction the annular hyperbolic plane.

Constructions

1. New constructions of fundamental polyhedra in complex hyperbolic space, with E. Falbel and J.Paupert, Acta Math.

Definition

1. Recall that if a semigroup is analytic, then is hyperbolic if and only if (see, e.g., [ 11, Definition 1.14 and Proposition 1.15, page 305]).
2. Kleinian groups are, by definition, discrete subgroups of the isometry group of hyperbolic 3-space.
3. Fuchsian groups are, by definition, discrete subgroups of the isometry group of the hyperbolic plane.

Definitions

1. Since the exponential function can be defined for any complex argument, we can extend the definitions of the hyperbolic functions also to complex arguments.
2. Here we will establish definitions and concepts that we can apply, via analogy, to our discussion of hyperbolic geometry.
3. In this lesson: The notations; Definitions; Derivatives and Indefinite integrals of inverse hyperbolic functions.

Unit Disk

1. The only hyperbolic, simply connected Riemann surface is the open unit disk.
2. The authors begin with rigid motions in the plane, which are used as motivation for a full development of hyperbolic geometry in the unit disk.
3. The open unit disk is commonly used as a model for the hyperbolic plane, by introducing a new metric on it, the Poincaré metric.

Boundary

1. The boundary of hyperbolic space is best thought of as the boundary of the disk in the Poincare model.
2. Let n 2. Mostow's rigidity Theorem asserts that two compact hyperbolic n-manifolds without boundary having the same fundamental group are isometric.
3. Complex hyperbolic space is the upper hemisphere of a sphere of radius one with equator the boundary of real hyperbolic space.

Universe

1. A simply connected Euclidean or hyperbolic universe would indeed be infinite.
2. It reaches a similar conclusion that the Universe is Open and space is hyperbolic in its pattern of expansion.
3. Because the universe expands (see the hubble constant), the space where no matter exists could be described by using a hyperbolic model.

Motion

1. Hyperbolic motion In physics, hyperbolic motion is the motion of an object with constant acceleration in special relativity.
2. Hyperbolic motion is the motion of an object with constant proper acceleration in special relativity.

Theory

1. These discussions are followed by an introduction to the theory of hyperbolic systems, with emphasis on the quintessential role of the geodesic flow.
2. He applied his results in the theory of hyperbolic and parabolic partial differential equations.
3. The first was Thurston's hyperbolization theorem which introduced the theory of hyperbolic 3-manifolds into knot theory and made it of prime importance.

Relativity

1. Hyperbolic Geometry is a "curved" space, and plays an important role in Einstein's General theory of Relativity.
2. There are alternative ways to set up a physical model of hyperbolic geometry in Einstein's Special Theory of Relativity.
3. Hyperbolic motion (relativity) Hyperbolic motion is the motion of an object with constant proper acceleration in special relativity.

Demonstration

1. Cabri constructions for the demonstration of the basic concepts of hyperbolic geometry in the Poincare disc model.

Symmetry Group

1. DODECAHEDRON symmetry group This is the symmetry group of translations of hyperbolic 3 space tiled with right-angled dodecahedra.
2. Each triangle group is the symmetry group of a tiling of the Euclidean plane, the sphere or the hyperbolic plane by congruent triangles.

Reflections

1. There are also examples of discrete groups generated by reflections on the other simply connected space of constant curvature, hyperbolic n -space, [H.sup.
2. Hyperbolic Geometry is a Hilbert Geometry in which there exist reflections at all straight lines.

Mathematics

1. In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.
2. The classification into elliptic, parabolic, and hyperbolic is pervasive in mathematics, and often divides a field into sharply distinct subfields.
3. Euclidean and hyperbolic groups of three-dimensional space (1927) both in the Annals of Mathematics.

Differential Geometry

1. Types of metric geometries: see differential geometry, euclidean [including hyperbolic], lorentzian, riemannian geometry.
2. The author then introduces the methods of differential geometry and develops them toward the goal of constructing models of the hyperbolic plane.
3. Book Info Discusses differential geometry and hyperbolic geometry.

Categories

1. Elliptic
2. Information > Science > Mathematics > Geometry
3. Euclidean
4. Topology > Topological Spaces > Manifolds > Surfaces
5. Science > Mathematics > Calculus > Partial Differential Equations

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