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A fundamental parallelogram is drawn. SnapPea is free software designed to help mathematicians, in particular low-dimensional topologists, study hyperbolic 3-manifolds. The primary developer is Jeffrey Weeks, who created the first version [1] as part of his doctoral thesis, [2] supervised by William Thurston.
The metric of the model on the half-plane, { , >}, is: = + ()where s measures the length along a (possibly curved) line. The straight lines in the hyperbolic plane (geodesics for this metric tensor, i.e., curves which minimize the distance) are represented in this model by circular arcs perpendicular to the x-axis (half-circles whose centers are on the x-axis) and straight vertical rays ...
In biomechanics, Hill's muscle model refers to the 3-element model consisting of a contractile element (CE) in series with a lightly-damped elastic spring element (SE) and in parallel with lightly-damped elastic parallel element (PE). Within this model, the estimated force-velocity relation for the CE element is usually modeled by what is ...
A radial hyperbolic trajectory is a non-periodic trajectory on a straight line where the relative speed of the two objects always exceeds the escape velocity. There are two cases: the bodies move away from each other or towards each other. This is a hyperbolic orbit with semi-minor axis = 0 and eccentricity = 1.
Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. 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. Laplace's equations are important in many areas of physics, including ...
Truncated trioctagonal tiling. In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto ...
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces [a] with various additional structures, most often, a Riemannian metric. [b] Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting ...
Hyperboloid model. Red circular arc is geodesic in Poincaré disk model; it projects to the brown geodesic on the green hyperboloid. Animation of partial {7,3} hyperbolic tiling of the hyperboloid rotated into the Poincare perspective. In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of n ...