The hyperbolic functions take a real argument called a hyperbolic angle. For points on the hyperbola below the x-axis, the area is considered negative (see animated version with comparison with the trigonometric (circular) functions). area hyperbolic cosine " arcosh" (also denoted " cosh −1", " acosh" or sometimes " arccosh")Ī ray through the unit hyperbola x 2 − y 2 = 1 at the point (cosh a, sinh a), where a is twice the area between the ray, the hyperbola, and the x-axis.area hyperbolic sine " arsinh" (also denoted " sinh −1", " asinh" or sometimes " arcsinh").hyperbolic cotangent " coth" ( / ˈ k ɒ θ, ˈ k oʊ θ/), Ĭorresponding to the derived trigonometric functions.hyperbolic secant " sech" ( / ˈ s ɛ tʃ, ˈ ʃ ɛ k/),.hyperbolic cosecant " csch" or " cosech" ( / ˈ k oʊ s ɛ tʃ, ˈ k oʊ ʃ ɛ k/ ).hyperbolic tangent " tanh" ( / ˈ t æ ŋ, ˈ t æ n tʃ, ˈ θ æ n/),.hyperbolic cosine " cosh" ( / ˈ k ɒ ʃ, ˈ k oʊ ʃ/),.hyperbolic sine " sinh" ( / ˈ s ɪ ŋ, ˈ s ɪ n tʃ, ˈ ʃ aɪ n/),.Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. 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. 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. ![]() In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Every hyperbolic translation has a designated axis of translation which is fixed by that operation, but since there is no family of parallel lines in the Euclidean sense, you don't have parallel lines fixed as well."Hyperbolic curve" redirects here. Note that the Euclidean concept of a “translation by a vector” doesn't make sense in hyperbolic geometry. Otherwise there will be two ideal fixed points denoting the endpoints of the axis of glide reflection. This class of Möbius transformations is called parabolic.įor ani-Möbius transformations, there may be a whole geodesic of fixed points, in which case one has a reflection. In the limit between these two situations, there is a single ideal fixed point with multiplicity two, and the operation is a limit rotation which fixes horocycles through the fixed ideal point. The former is usually called a hyperbolic Möbius transformation, the latter an elliptic one. A translation has two ideal fixed points, while a rotation has a finite fixed point and its mirror image after reflection in the unit circle (resp. In that case the distinction would be for Möbius transformations which exchange the two mirror images, as these would be the orientation-reversing ones. ![]() As an alternative one could avoid dealing with anti-Möbius transformations and instead consider equivalence classes of points related by inversion in the unit circle, or use the half plane model and consider the lower half plane to be a mirror image of the upper half plane. ![]() My target categories are translations, rotations, limit rotations, reflections and glide reflections.įirstly I'd distinguish Möbius transformations from anti-Möbius transformations. ![]() thesis I'm classifying hyperbolic isometries in The Poincaré disk model based on their generalized Möbius transformation matrices in section 2.2.4. Will those results of linear algebra for the inner product still be valid in $\mathbb)$ there are two conjugacy classes, one rotating "positively" around the fixed point at infinity and the other rotating "negatively". I then proved that the isometries of the hyperbolic plane are Möbius Transforms, but I don't see how to apply this in the structure of the euclidean classification I did before. (iv) A glide reflection through a line and glided by a vector.Īnd then my teacher asked me to study and try to classify the isometries of the Hyperbolic plane in the same way, using the matrices of the Möbius transform. The theorem says that every isometry of the euclidean plane is exactly one of the following: ( pages 25 to 38, sorry for it to be in portuguese, but I believe it is the standard proof using linear algebra). I just did the classification of the isometries of the Euclidean Plane using some results of linear algebra and euclidean geometry, guiding myself through this article:
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