The Isoperimetric Problem of the Convex Hull of a Closed Space Curve

1960 ◽  
Vol 11 (2) ◽  
pp. 265 ◽  
Author(s):  
Z. A. Melzak
Keyword(s):  
Convex Hull ◽  
Isoperimetric Problem ◽  
Space Curve ◽  
Closed Space

Closed polylines inscribed in a closed space curve

2011 ◽  
Vol 175 (5) ◽  
pp. 556-558
Author(s):  
V. V. Makeev
Keyword(s):  
Space Curve ◽  
Closed Space

COMPUTING THE WRITHE OF A KNOT

2001 ◽  
Vol 10 (03) ◽  
pp. 387-395 ◽  
Author(s):  
DAVID CIMASONI
Keyword(s):  
Space Curve ◽  
Closed Formula ◽  
Closed Space ◽  
Polygonal Curves

We study the variation of the Tait number of a closed space curve according to its different projections. The results are used to compute the writhe of a knot, leading to a closed formula in case of polygonal curves.

Structure of the convex hull of a space curve

1986 ◽  
Vol 33 (4) ◽  
pp. 1140-1153 ◽  
Author(s):  
V. D. Sedykh
Keyword(s):  
Convex Hull ◽  
Space Curve

scholarly journals Geometry of Călugăreanu's theorem

2005 ◽  
Vol 461 (2062) ◽  
pp. 3245-3254 ◽  
Author(s):  
M.R Dennis ◽  
J.H Hannay
Keyword(s):  
Crossing Number ◽  
Space Curve ◽  
Closed Space ◽  
Linking Number ◽  
Space Geometry ◽  
The Common ◽  
Central Result

A central result in the space geometry of closed twisted ribbons is Călugăreanu's theorem (also known as White's formula, or the Călugăreanu–White–Fuller theorem). This enables the integer linking number of the two edges of the ribbon to be written as the sum of the ribbon twist (the rate of rotation of the ribbon about its axis) and its writhe. We show that twice the twist is the average, over all projection directions, of the number of places where the ribbon appears edge-on (signed appropriately)—the ‘local’ crossing number of the ribbon edges. This complements the common interpretation of writhe as the average number of signed self-crossings of the ribbon axis curve. Using the formalism we develop, we also construct a geometrically natural ribbon on any closed space curve—the ‘writhe framing’ ribbon. By definition, the twist of this ribbon compensates its writhe, so its linking number is always zero.

On the convex hull of a space curve

2012 ◽  
Vol 12 (1) ◽  
Author(s):  
Kristian Ranestad ◽  
Bernd Sturmfels
Keyword(s):  
Convex Hull ◽  
Space Curve

The range of the (n + 1)th moment for distributions on [0, 1]

1967 ◽  
Vol 4 (03) ◽  
pp. 543-552 ◽  
Author(s):  
Morris Skibinsky
Keyword(s):  
Positive Integer ◽  
Convex Hull ◽  
Higher Order ◽  
Unit Interval ◽  
Space Curve ◽  
Order Moment ◽  
Probability Measures ◽  
Image Position ◽  
Moment Spaces

Let p denote the class of all probability measures defined on the Borel subsets of the unit interval I = [0, 1]. For each positive integer n, take Mn is convex, closed, bounded, and n-dimensional; the convex hull of the space curve {(t,t2, …, tn ): 0 ≦ t ≦ 1}; e.g., see Theorems 7.2, 7.3 of [1]. At each point (c1, C2, …, cn ) of Mn , define Note that v −, v + depend only on C1, C2, …, Cn− 1; Vm only on cn ; We shall as notational convenience dictates and as will be apparent from the context regard v ± n as functions on Mn− 1 or on higher order moment spaces and also regard Vn as a function on moment spaces of order higher than n.

Framed curves and knotted DNA

2013 ◽  
Vol 41 (2) ◽  
pp. 635-638 ◽  
Author(s):  
Gregory S. Chirikjian
Keyword(s):  
Space Curve ◽  
Closed Space ◽  
Global Geometry

The present mini-review covers the local and global geometry of framed curves and the computation of twist and writhe in knotted DNA circles. Classical inequalities relating the total amount of bending of a closed space curve and associated knot parameters are also explained.

The range of the (n + 1)th moment for distributions on [0, 1]

1967 ◽  
Vol 4 (3) ◽  
pp. 543-552 ◽  
Author(s):  
Morris Skibinsky
Keyword(s):  
Positive Integer ◽  
Convex Hull ◽  
Higher Order ◽  
Unit Interval ◽  
Space Curve ◽  
Order Moment ◽  
Probability Measures ◽  
Image Position ◽  
Moment Spaces

Let p denote the class of all probability measures defined on the Borel subsets of the unit interval I = [0, 1]. For each positive integer n, take Mn is convex, closed, bounded, and n-dimensional; the convex hull of the space curve {(t,t2, …, tn): 0 ≦ t ≦ 1}; e.g., see Theorems 7.2, 7.3 of [1]. At each point (c1, C2, …, cn) of Mn, define Note that v−, v+ depend only on C1, C2, …, Cn− 1; Vm only on cn; We shall as notational convenience dictates and as will be apparent from the context regard v±n as functions on Mn− 1 or on higher order moment spaces and also regard Vn as a function on moment spaces of order higher than n.

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