Translation (geometry) 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 transformation is a one-to-one correspondence between two sets of points or a mapping from one plane to another. A translation can be described as a rigid motion: the other rigid motions are rotations, reflections and glide reflections. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. A translation operator is an
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