Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T06:54:45.129Z Has data issue: false hasContentIssue false

Parametric surfaces

I. Compactness

Published online by Cambridge University Press:  24 October 2008

A. S. Besicovitch
Affiliation:
Trinity CollegeCambridge

Extract

We shall first give some definitions concerning parametric surfaces. Denote by H a closed circle (disk) and by M a variable point on it. Let P = Ф(M) be a continuous function on H whose value P is a point in three-dimensional space. The symbols Ф(E), Ф−1(P), where E is a set of points on H and P a point in the three-dimensional space, will have their usual meaning. Ф−1(P) is a closed set. Any saturated continuum in Ф−1(P) or any point of Ф−1(P) that does not belong to such continua is called a Ф-element of H. Thus to any continuous function Ф(M) corresponds a representation of H in the form of the sum σQ of Ф-elements. The set of the pairs (P, Q), where Q runs through all Ф-elements of H and, for any Q, P = Ф(Q), is called a parametric surface, and any pair (P, Q) is called a point of the parametric surface. We shall often speak of a point Ф(M) of the parametric surface, by which we shall mean either the point (P, Q), where P = Ф(M) and Q is the Ф-element containing M, or the point P = Ф(M) of the three-dimensional space. The exact meaning will always be clear from the context. If there are exactly k points of the parametric surface whose first member is P0 we say that P0 is a point of multiplicity k. If k = 1, P0 is a simple point.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1949

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

E denotes the number of elements of E.