Translation (geometry) of the object. The translation vector formula_8 common to all points of the object describes a particular type of displacement of the object, usually called a "linear" displacement to distinguish it from displacements involving rotation, called "angular" displacements. When considering spacetime, a change of time coordinate is considered to be a translation. For example, the Galilean group and the Poincaré group include translations with respect to time. Translation (geometry) In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction. In Euclidean geometry a
Melnikov distance formula_4. Suppose for formula_5 the system has a hyperbolic fixed point x and a homoclinic orbit formula_6 corresponding to this fixed point. Then for sufficiently small formula_7 there exists a "T"-periodic hyperbolic solution. The stable and unstable manifolds of this periodic solution intersect transversally. The distance between these manifolds measured along a direction that is perpendicular to the unperturbed homoclinc orbit formula_6 is called the Melnikov distance. If formula_9 denotes this distance, then formula_10. The function formula_11 is called the Melnikov function. Melnikov’s distance can be used to predict chaotic vibrations. In this method, critical amplitude is found by setting