منابع مشابه
On the Unfolding of Simple Closed Curves
We show that every rectifiable simple closed curve in the plane can be continuously deformed into a convex curve in a motion which preserves arc length and does not decrease the Euclidean distance between any pair of points on the curve. This result is obtained by approximating the curve with polygons and invoking the result of Connelly, Demaine, and Rote that such a motion exists for polygons....
متن کاملContinuous Maps on Digital Simple Closed Curves
We give digital analogues of classical theorems of topology for continuous functions defined on spheres, for digital simple closed curves. In particular, we show the following: 1) A digital simple closed curve S of more than 4 points is not contractible, i.e., its identity map is not nullhomotopic in S ; 2) Let X and Y be digital simple closed curves, each symmetric with respect to the origin, ...
متن کاملVeech surfaces and simple closed curves
We study the SL(2,R)–infimal lengths of simple closed curves on half-translation surfaces. Our main result is a characterization of Veech surfaces in terms of these lengths. We also revisit the “no small virtual triangles” theorem of Smillie and Weiss and establish the following dichotomy: the virtual triangle area spectrum of a half-translation surface either has a gap above zero or is dense i...
متن کاملClosed Curves and Space Curves
So far we have discussed only ‘local’ properties of (plane) curves. These properties depend only on the behavior of a curve near a given point, and not on the ‘global’ shape of the curve. Now let us look at some global results about curves. The most famous, and perhaps the oldest, of these is the ‘isoperimetric inequality’, which relates the length of a certain ‘closed’ curve to the area it con...
متن کاملOn the Decompositions of Intervals and Simple Closed Curves
Let us note that every simple closed curve is non trivial. Let T be a non empty topological space. One can check that there exists a subset of T which is non empty, compact, and connected. Let us observe that every element of the carrier of I is real. Next we state two propositions: (1) Let X be a non empty set and A, B be non empty subsets of X. If A ⊂ B, then there exists an element p of X su...
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ژورنال
عنوان ژورنال: Proceedings of the National Academy of Sciences
سال: 1929
ISSN: 0027-8424,1091-6490
DOI: 10.1073/pnas.15.2.94