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 ...

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.

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.

The article Positively oriented curve at en.wikipedia.org was accessed 10 times in the last 30 days. (as of: 01/09/2014)

Search results from Google and Bing

1

>30

1

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 ...

en.wikipedia.org/wiki/Curve_orientation

2

>30

2

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 ...

mathworld.wolfram.com/CurveOrientation.html

3

>30

3

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 ...

math.stackexchange.com/questions/218849/rigorous-definition-of-positive-orientation-of-curve-in-mathbbrn

4

>30

4

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 ...

tutorial.math.lamar.edu/Classes/CalcIII/GreensTheorem.aspx

5

>30

5

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 ...

www.ccs.tulane.edu/~chiayu/CalculusIII2009/Lec33.pdf

6

>30

6

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 ...

stange317.wordpress.com/2010/11/04/greens-theorem-negatively-oriented/

7

>30

7

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.

answers.yahoo.com/question/index?qid=20110420145416AAuHWdb

8

>30

8

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 ...

math.berkeley.edu/~sean/H53/curves.pdf

9

>30

9

here

(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 ...

www.math.ou.edu/~forester/2443F09/s3.pdf

10

>30

10

Lec28

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 ...

math.harvard.edu/~ytzeng/worksheet/1114.pdf

Search results for "Positively oriented curve"

Google: approx. 36.400.000

Positively oriented curve in science

Green's theorem - Wikipedia, the free encyclopedia

[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 ...

[PDF]LECTURE NOTES: THEOREM OF TURNING ANGLE OF A CURVE ...

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

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

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

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, ...

Differential Geometry

This classic work is now available in an unabridged paperback edition.

Spectral Theory of Operators on Hilbert Spaces

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 ...

Development of Google searches

Online sources for the term

Positively oriented curve

Positively oriented curve

http://www.math.hmc.edu/faculty/gu/curves_and_surfaces/curves/_topology.html

www.math.hmc.edu/faculty/gu/curves_and_surfaces/curves/_topology.html
Curve orientation at MathWorld

mathworld.wolfram.com/CurveOrientation.html
Blog posts on the term

Positively oriented curve

Positively oriented curve

Green’s Theorem Negatively Oriented | Math 317 Virtual Office Hours

Fall 2010 UBC

stange317.wordpress.com/2010/11/04/greens-theorem-negatively-oriented/
Consistent monodromies | The Electric Handle Slide

by PLC | June 2, 2013 · 12:31 pm.

electrichandleslide.wordpress.com/2013/06/02/consistent-monodromies/
A Quick and Dirty Introduction to the Geometry of Curves » CS 177: Discrete Differential Geometry

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 an interval \(I = [0,T] \subset \mathbb{R}\) of the real line to \(\mathbb{R}^3\).

brickisland.net/cs177fa12/?p=187
differential geometry - Inversion of Hopf's Umlaufsatz - Mathematics Stack Exchange
math.stackexchange.com/questions/275930/inversion-of-hopfs-umlaufsatz

Math2111: Chapter 5: Integral theorems. Section 2: Stokes’ theorem | Quasi-Random Ideas.

This is a blog on my research, courses and related topics. By Josef Dick @ the University of New South Wales.

quasirandomideas.wordpress.com/2010/05/16/math2111-chapter-5-integral-theorems-section-1-stokes-theorem/
The Math and ActionScript of Curves: Gradients and Normals - Tuts+
hub.tutsplus.com/tutorials/the-math-and-actionscript-of-curves-gradients-and-normals--active-11739?request_uri=%2Ftutorials%2Factionscript%2Fthe-math-and-actionscript-of-curves-gradients-and-normals%2F

Can America’s Youth be a Focal Point for Bending Healthcare’s Cost Curve? – TripleTree Blog
www.triple-tree.com/blog/2012/11/05/can-americas-youth-be-a-focal-point-for-bending-healthcares-cost-curve/

A Quick and Dirty Introduction to the Geometry of Curves » CS 177 (Fall 2011)

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 \(\mathbb{R}^3\).

brickisland.net/cs177/?p=121
BLOG.CSHARPHELPER.COM: Draw a conic section from its polynomial equation in C#
blog.csharphelper.com/2013/06/24/draw-a-conic-section-from-its-polynomial-equation-in-c.aspx

GRE Practice: GR9768.66: Green's Theorem
sfmathgre.blogspot.com/2010/05/gr976866-complex-integration.html

123

More cyclopaedia entries

10. Potworow Commune

11. Power Athlete

19. Powiat of Olesno

21. Praful Bidwai

22. Praga Khan

24. Pseudalus

33. Psycho Cop

34. Psycho Fox

35. Psycho Path