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. Practical considerations In practical applications, the following considerations are commonly taken into account. One does not need to construct the convex hull of a polygon to find a suitable vertex. A common choice is the vertex of the polygon with the smallest X-coordinate. If there are several of them, the one with the smallest Y-coordinate is picked. It is guaranteed to be a vertex of the convex hull of the polygon. Alternatively, the vertex with the smallest Y-coordinate among the ones with the largest X-coordinates or the vertex with the smallest X-coordinate among the ones with the largest Y-coordinates (or any other of 8 "smallest, largest" X/Y combinations) will do as well. Once a vertex of the convex hull is chosen, one can then apply the formula using the previous and next vertices, even if those are not on the convex hull, as there can be no local concavity on this vertex. If the orientation of a convex polygon is sought, then, of course, any vertex may be picked. For numerical reasons, the following equivalent formula for the determinant is commonly used: The latter formula has four multiplications less. What is more important in computer computations involved in most practical applications, such as computer graphics or CAD, the absolute values of the multipliers are usually smaller (e.g., when A, B, C are within the same quadrant), thus giving a smaller numerical error or, in the extreme cases, avoiding the arithmetic overflow. When it is not known in advance that the sequence of points defines a simple polygon, the following things must be kept in mind. For a self-intersecting polygon (complex polygon) (or for any self-intersecting curve) there is no natural notion of the "interior", hence the orientation is not defined. At the same time, in geometry and computer graphics there are a number of concepts to replace the notion of the "interior" for closed non-simple curves; see, e.g., "flood fill" and "winding number". In "mild" cases of self-intersection, with degenerate vertices when three consecutive points are allowed be on the same straight line and form a zero-degree angle, the concept of "interior" still makes sense, but an extra care must be taken in selection of the tested angle.