Local Concavity
Once the orientation of a polygon formed from an ordered set of vertices is known, the concavity of a local region of the polygon can be determined using a second orientation matrix. This matrix is composed of three consecutive vertices which are being examined for concavity. For example, in the polygon pictured above, if we wanted to know whether the sequence of points F-G-H is concave, convex, or collinear (flat), we construct the matrix
If the determinant of this matrix is 0, then the sequence is collinear - 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 | collinear sequence of points | collinear sequence of points |
Read more about this topic: Curve Orientation
Famous quotes containing the word local:
“These native villages are as unchanging as the woman in one of their stories. When she was called before a local justice he asked her age. “I have 45 years.” “But,” said the justice, “you were forty-five when you appeared before me two years ago.” “Señor Judge,” she replied proudly, drawing herself to her full height, “I am not of those who are one thing today and another tomorrow!””
—State of New Mexico, U.S. public relief program (1935-1943)