Examples
The map between the Euclidean plane and the plane with the Manhattan distance that sends every point to itself is a quasi-isometry: in it, distances are multiplied by a factor of at most .
The map (both with the Euclidean metric) that sends every -tuple of integers to itself is a quasi-isometry: distances are preserved exactly, and every real tuple is within distance of an integer tuple. In the other direction, the discontinuous function that rounds every tuple of real numbers to the nearest integer tuple is also a quasi-isometry: each point is taken by this map to a point within distance of it, so rounding changes the distance between pairs of points by adding or subtracting at most .
Every pair of finite or bounded metric spaces is quasi-isometric. In this case, every function from one space to the other is a quasi-isometry.
Read more about this topic: Quasi-isometry
Famous quotes containing the word examples:
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)