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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 travelling 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. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...
In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ...
Crucial to this definition is the fact that every simple closed curve admits a well-defined interior; that follows from the Jordan curve theorem. In topology, the Jordan curve theorem states that every non-self-intersecting loop in the plane divides the plane into an inside and an outside. It was proved by Oswald Veblen in 1905. ...
A circle oriented counterclockwise is an example of a positively oriented curve. The same circle oriented clockwise would be a negatively oriented curve. A circle, in Euclidean geometry, is the set of all points at a fixed distance, called the radius, from a fixed point, the centre. ...
A clockwise motion is one that proceeds like the clocks hands: from the top to the right, then down and then to the left, and back to the top. ...
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 orentation on the curve is then the orientation it inherits as the boundary of its interior; the negative orientation is inherited from the exterior. In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ...
A manifold is a mathematical space which is constructed, like a patchwork, by gluing and bending together copies of simple spaces. ...
An open surface with X-, Y-, and Z-contours shown. ...
In mathematics, a hypersurface is some kind of submanifold. ...
See also
In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. ...
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