Orientation (topology) Totally Explained

Everything about Orientation Topology totally explained

In topology, orientation is the direction in which a simple closed curve made up of three or more vertices is traversed. All closed curves can be classified as negatively oriented (clockwise), positively oriented (counterclockwise), or non-orientable. The inner loop of a beltway road in the United States (or other countries where people drive on the right side of the road) would be an example of a negatively oriented (clockwise) curve.

Determining orientation for a simple closed polygon

In two dimensions, given an ordered set of connected points (such as in connect-the-dots), it's possible to calculate the orientation of the resulting polygon using the determinant of an orientation matrix. To do this, an endpoint on the convex hull of the polygon is selected as a reference point. An endpoint that lies on the bounding box of the polygon is also acceptable, since any endpoints on the bounding box will also lie on the convex hull. In this example, point B is chosen as the reference point. Then, two points which are non-collinear with the reference point, one before the reference point in the sequence and one after (A and C, respectively) are chosen and used to construct the orientation matrix ť

mathbf

If the determinant of this matrix is 0, then the sequence is flat - neither concave nor convex. If the determinant has the same sign as that of the orientation matrix for the entire polygon, then the sequence is convex. If the signs differ, then the sequence is concave. In this example, the polygon is negatively oriented, but the determinant for the points F-G-H is positive, and so the sequence F-G-H is concave.
The following table illustrates rules for determining whether a sequence of points is convex, concave, or flat:

	Negatively oriented polygon (clockwise)	Positively oriented polygon (counterclockwise)
determinant of orientation matrix for local points is negative	convex sequence of points	concave sequence of points
determinant of orientation matrix for local points is positive	concave sequence of points	convex sequence of points
determinant of orientation matrix for local points is 0	flat sequence of points	flat sequence of points

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