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Property of a planar simple closed curve

In mathematics, an orientation of a curve is the choice of one of the two possible directions for travelling on the curve. For example, for Cartesian coordinates, the x axis is traditionally oriented toward the right, and the y axis is upward oriented. In the case of 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), 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 traveling on it. Otherwise, that is if left and right are exchanged, the curve is negatively oriented or clockwise oriented. 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. In trigonometry, the unit circle is traditionally oriented counterclockwise. 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.). 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.

In two dimensions, given an ordered set of three or more connected vertices (points) (such as in connect the dots) which forms a simple polygon, the orientation of the resulting polygon is directly related to the sign of the angle at any vertex of the convex hull of the polygon, for example, of the angle ABC in the picture. In computations, the sign of the smaller angle formed by a pair of vectors is typically determined by the sign of the cross product of the vectors. The latter one may be calculated as the sign of the determinant of their orientation matrix. In the particular case when the two vectors are defined by two line segments with common endpoint, such as the sides BA and BC of the angle ABC in our example, the orientation matrix may be defined as follows:

A formula for its determinant may be obtained, e. g., using the method of cofactor expansion:

If the determinant is negative, then the polygon is oriented clockwise. If the determinant is positive, the polygon is oriented counterclockwise. The determinant is non zero if points A, B, and C are non collinear. In the above example, with points ordered A, B, C, etc., the determinant is negative, and therefore the polygon is clockwise.

In two dimensions, given an ordered set of three or more connected vertices (points) (such as in connect the dots) which forms a simple polygon, the orientation of the resulting polygon is directly related to the sign of the angle at any vertex of the convex hull of the polygon, for example, of the angle ABC in the picture. In computations, the sign of the smaller angle formed by a pair of vectors is typically determined by the sign of the cross product of the vectors. The latter one may be calculated as the sign of the determinant of their orientation matrix. In the particular case when the two vectors are defined by two line segments with common endpoint, such as the sides BA and BC of the angle ABC in our example, the orientation matrix may be defined as follows:

A formula for its determinant may be obtained, e. g., using the method of cofactor expansion:

If the determinant is negative, then the polygon is oriented clockwise. If the determinant is positive, the polygon is oriented counterclockwise. The determinant is non zero if points A, B, and C are non collinear. In the above example, with points ordered A, B, C, etc., the determinant is negative, and therefore the polygon is clockwise.