Transition Maps
A transition map provides a way of comparing two charts of an atlas. To make this comparison, we consider the composition of one chart with the inverse of the other. This composition is not well-defined unless we restrict both charts to the intersection of their domains of definition. (For example, if we have a chart of Europe and a chart of Russia, then we can compare these two charts on their overlap, namely the European part of Russia.)
To be more precise, suppose that and are two charts for a manifold M such that is non-empty. The transition map is the map defined by
Note that since and are both homeomorphisms, the transition map is also a homeomorphism.
Read more about this topic: Atlas (topology)
Famous quotes containing the words transition and/or maps:
“The god or hero of the sculptor is always represented in a transition from that which is representable to the senses, to that which is not.”
—Ralph Waldo Emerson (1803–1882)
“Living in cities is an art, and we need the vocabulary of art, of style, to describe the peculiar relationship between man and material that exists in the continual creative play of urban living. The city as we imagine it, then, soft city of illusion, myth, aspiration, and nightmare, is as real, maybe more real, than the hard city one can locate on maps in statistics, in monographs on urban sociology and demography and architecture.”
—Jonathan Raban (b. 1942)