Positively oriented curve
In mathematics, a positively oriented curve is a planar simple closed curve (that is, a curve in the plane whose starting point is also the end point and which has no other self-intersections) such that when traveling on it one always has the curve interior to the left (and consequently, the curve exterior to the right). If in the above definition one interchanges left and right, one obtains a negatively oriented curve.
Crucial to this definition is the fact that every simple closed curve admits a well-defined interior; that follows from the Jordan curve theorem.
All simple closed curves can be classified as negatively oriented (clockwise), positively oriented (counterclockwise), or non-orientable. The inner loop of a beltway road in the United States (or other countries where people drive on the right side of the road) would be an example of a negatively oriented (clockwise) curve. A circle oriented counterclockwise is an example of a positively oriented curve. The same circle oriented clockwise would be a negatively oriented curve.
The concept of orientation of a curve is just a particular case of the notion of orientation of a manifold (that is, besides orientation of a curve one may also speak of orientation of a surface, hypersurface, etc.). Here, the interior and the exterior of a curve both inherit the usual orientation of the plane. The positive orientation on the curve is then the orientation it inherits as the boundary of its interior; the negative orientation is inherited from the exterior.

This is an excerpt from the article Positively oriented curve from the Wikipedia free encyclopedia. A list of authors is available at Wikipedia.
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Curve orientation - Wikipedia, the free encyclopedia
In mathematics, a Positively oriented curve is a planar simple closed curve (that is , a curve in the plane whose starting point is also the end point and which has ...
Curve Orientation -- from Wolfram MathWorld
Curve Orientation. A curve has positive orientation if a region R is on the left when traveling around the outside of R , or on the right when traveling around the ...
Rigorous definition of positive orientation of curve in $\mathbb{R}^n
Oct 22, 2012 ... When one formulates the Green Theorem the phrase "curve positively oriented" comes up. After a thorough google search the only description ...
Pauls Online Notes : Calculus III - Green's Theorem
We will use the convention here that the curve C has a positive orientation if it is ... Let C be a positively oriented, piecewise smooth, simple, closed curve and let ...
Def: A simple closed curve C is defined to be positive orientation if ...
orientation if the travel direction on C is counterclockwise along C. Theorem: Let C be a positively oriented, piecewise-smooth, simple closed curve in the plane ...
Green's Theorem Negatively Oriented | Math 317 Virtual Office Hours
Green's Theorem Negatively Oriented. November 4, 2010. Green's Theorem says that when your curve is positively oriented (and all the other hypotheses are ...
What is the line integral along the given positively oriented ...
integral of y^3dx - x^3dy, C is the circle x^2 + y^2 = 4. We're using ... ∫{y^3*dx - x ^3*dy} = ∫∫{(3y^2 +9x^2)*dx*dy from Green's Theorem.
On parameterizations of oriented curves, and - Mathematics
Apr 11, 2012 ... positive orientation of C relative to S. (Unlike in R2, the surface bounded by C is not unique.) Remark 1.4. A curve in R2 that is closed and ...
(0,0), (2,2), and (2,4), oriented positively. 1(b) Give a vector field F(a:,y) with the property that for any positively oriented simple closed curve C, the line integral f0 ...
Green's Theorem: Suppose C is a positively oriented, piecewise-smooth, simple closed curve in the plane that bounds a region D. If P and Q have continuous ...
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Positively oriented curve in science
Green's theorem - Wikipedia, the free encyclopedia
Let C be a positively oriented, piecewise smooth, simple closed curve in a plane, .... S.J. Bence, Cambridge University Press, 2010, ISBN 978-0-521-86153-3 ...
[PDF]Math 12, Fall 2006 - Lecture 31 - Dartmouth College
Nov 28, 2006 ... university-logo ... Orientation of a curve induced by a parametrization r(t) ... Same idea as positive orientation for a plane curve. If we.
[PDF]A Unified Latent Curve, Latent State-Trait Analysis of the ...
Developmental Trajectories and. Correlates of Positive Orientation. Guido Alessandri and Gian Vittorio Caprara. Sapienza, University of Rome. John Tisak.
Positively oriented ideal triangulations on hyperbolic three-manifolds
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK ... We call such a solution a positively oriented ideal triangulation of M3 and denote the ... In particular, it coincides with the algebraic intersection pairing of curves on ∂M3:.
On the evolution of curves via a function of curvature. I. The classical ...
Department of Electrical Engineering, McGill University, Montréal, Québec, Canada ... o Let the positive orientation of a curve be defined so that the interior is to the ... 0 Next we define the orientation of a curve as the angle the tangent makes ...
[PDF]Stokes' theorem - University of Alberta
University of Alberta. Edmonton, Fall ... orientation of S induces the positive orientation of C: if you ... simple, closed, piecewise smooth curve C with positive.
[PDF]17.4 - Penn Math - University of Pennsylvania
Dec 10, 2009 ... Green's Theorem. It is about a line integral. ∫. C. Pdx + Qdy along a. 1 positively oriented,. 2 piecewise-smooth,. 3 simple, closed curve C.
Orientation - WikEd
Feb 4, 2010 ... Orientation (mental); Sexual orientation; Orientation (rigid body); Orientation week, the first week of a university year in several countries ... orientability and Curve orientation (such as a Positively oriented curve).
[PDF]The Surveyor's Area Formula - Mathematical Association of America
University. ... curve in parametric form as an integral around its boundary. ... counterclockwise, all the triangles inherit this positive orientation; so the oriented ...
Let x : U → R3 be a smooth regular surface patch with a fixed orientation (given ... be the unit vector ¯n × ¯v, which makes {¯v, ˜v, ¯n} forms a positive base of R3. .... Department of Mathematics, University of Oregon, Eugene, Oregon, 97403.
Books on the term Positively oriented curve
General Theory of Irregular Curves
General Theory of Irregular Curves
A. Aleksandr Danilovich Aleksandrov, I︠U︡riĭ Grigorʹevich Reshetni︠a︡k, 1989
The definition, in particular, affords that any plane polygonal line having no points of return is an f.c.t. curve. Let K be an arbitrary locally convex curve on a plane, not lying in one straight line. The curve K will be termed positively oriented if its ...
Projective Geometry
Projective Geometry
Oswald Veblen, John Wesley Young, 1918
By the theorems above, this gives a unique determination of the positive and negative sense-classes on any curve of the given convex region. A curve associated with its positive sense-class is called a Positively oriented curve, and a curve ...
Shaping Preserving Representations in Computer Aided Geometric Design
Shaping Preserving Representations in Computer Aided Geometric Design
Juan M. Peña, 1999
Let us remark that the fact that a polygon or curve is positively oriented has different meanings depending on the dimension. ... In the 2-dimensional case (s = 2), the fact that a curve or polygon is positively oriented with respect to C implies, ...
Spectral Theory of Operators on Hilbert Spaces
Spectral Theory of Operators on Hilbert Spaces
Carlos S. Kubrusly, 2012
A finite union F I of disjoint closed rectifiable oriented simple curves is called a path, and its winding number wF about Q G (C\F is defined by w,_ I A path P is positively oriented if for every Q G (C\P the winding number wF is either 0 or 1, and ...
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Positively oriented curve
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The picture we looked at for surfaces is actually a nice way of thinking about manifolds of any dimension. For instance, we can think of a one-dimensional curve as a map \(\gamma: I \rightarrow \mathbb{R}^3\) from a subset \(I = [0,T] \subset \mathbb{R}\) of the real line to \(\mathbb{R}^3\).
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