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In the case of a plane 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), the curve is said to be positively oriented or counterclockwise oriented, if one always has the curve interior to the left (and consequently, the curve exterior to the right), when ...
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 ...
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 ...
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 fact that this may be positive or negative has the intuitive meaning that v and w may be oriented in a counterclockwise or clockwise sense as the vertices of the parallelogram they define. Such an area is called the signed area of the parallelogram: the absolute value of the signed area is the ordinary area, and the sign determines its ...
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]
In mathematics and physics, the right-hand rule is a convention and a mnemonic, utilized to define the orientation of axes in three-dimensional space and to determine the direction of the cross product of two vectors, as well as to establish the direction of the force on a current-carrying conductor in a magnetic field.
In geometry, a geodesic (/ ˌ dʒ iː. ə ˈ d ɛ s ɪ k,-oʊ-,-ˈ d iː s ɪ k,-z ɪ k /) [1] [2] is a curve representing in some sense the locally [a] shortest [b] path between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection.