Curve orientation

In mathematics, an orientation of a curve (including polygonal curves) is the choice of one of the two possible senses for travelling on the curve, as in forward and backward. For example, for Cartesian coordinates on the Euclidean plane, the x-axis is traditionally oriented toward the right, and the y-axis is upward oriented. At each point along the curve, there are two tangent directions possible, colinear and opposite to each other.

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 orientation is said to be positive or counterclockwise, if one always has the curve interior to the left (and consequently, the curve exterior to the right), when traveling on it. Otherwise, that is if left and right are exchanged, the curve orientation is negative or clockwise. This definition relies on the fact that every simple closed curve admits a well-defined interior, which follows from the Jordan curve theorem. For example, the inner loop of a beltway road in a country where people drive on the right side of the road is a negatively oriented (clockwise) curve. In trigonometry, the unit circle is traditionally oriented counterclockwise.

Orientation of a curve is associated with parametrization of its points by a real variable. A curve may have equivalent parametrizations when there is a continuous increasing monotonic function relating the parameter of one curve to the parameter of the other. When there is a decreasing continuous function relating the parameters, then the parametric representations are opposite and the orientation of the curve is reversed.

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.