Definition
Intuitively, a continuous curve in 2 or 3 (or higher) dimensions can be thought of as the path of a continuously moving point. To eliminate the inherent vagueness of this notion Jordan in 1887 introduced the following rigorous definition, which has since been adopted as the precise description of the notion of a continuous curve:
- A curve (with endpoints) is a continuous function whose domain is the unit interval .
In the most general form, the range of such a function may lie in an arbitrary topological space, but in the most commonly studied cases, the range will lie in a Euclidean space such as the 2-dimensional plane (a planar curve) or the 3-dimensional space (space curve).
Sometimes, the curve is identified with the range or image of the function (the set of all possible values of the function), instead of the function itself. It is also possible to define curves without endpoints to be a continuous function on the real line (or on the open unit interval (0, 1)).
Read more about this topic: Space-filling Curve
Famous quotes containing the word definition:
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)