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This definition relies on the fact that every simple closed curve admits a well-defined interior, which follows from the Jordan curve theorem. The inner loop of a beltway road in a country where people drive on the right side of the road is an example of a negatively oriented ( clockwise ) curve.
The orientation of a real vector space or simply orientation of a vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional Euclidean space , right-handed bases are typically declared to be positively oriented, but the choice is arbitrary, as they may also ...
A torus is an orientable surface The Möbius strip is a non-orientable surface. Note how the disk flips with every loop. The Roman surface is non-orientable.. In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". [1]
Lemma 1 (Decomposition Lemma) — Assume is a rectifiable, positively oriented Jordan curve in the plane and let be its inner region. For every positive real δ {\displaystyle \delta } , let F ( δ ) {\displaystyle {\mathcal {F}}(\delta )} denote the collection of squares in the plane bounded by the lines x = m δ , y = m δ {\displaystyle x=m ...
An example of a 1-dimensional manifold is an interval [a, b], and intervals can be given an orientation: they are positively oriented if a < b, and negatively oriented otherwise. If a < b then the integral of the differential 1 -form f ( x ) dx over the interval [ a , b ] (with its natural positive orientation) is
Serpentine lines from Hogarth's The Analysis of Beauty. Line of beauty is a term and a theory in art or aesthetics used to describe an S-shaped curved line (a serpentine line) appearing within an object, as the boundary line of an object, or as a virtual boundary line formed by the composition of several objects.
A trihedron is said to be adapted to a surface if P always lies on the surface and e 3 is the oriented unit normal to the surface at P. In the case of the Darboux frame along an embedded curve, the quadruple (P(s) = γ(s), e 1 (s) = T(s), e 2 (s) = t(s), e 3 (s) = u(s)) defines a tetrahedron adapted to the surface into which the curve is embedded.
The convention for positive linking number is based on a right-hand rule. The winding number of an oriented curve in the x-y plane is equal to its linking number with the z-axis (thinking of the z-axis as a closed curve in the 3-sphere). More generally, if either of the curves is simple, then the first homology group of its complement is ...