# 469 Convex Polygon

### 469. Convex Polygon

• The polygon is entirely contained in a closed half-plane defined by each of its edges.

http://stackoverflow.com/questions/471962/how-do-determine-if-a-polygon-is-complex-convex-nonconvex

You can make things a lot easier than the Gift-Wrapping Algorithm... that's a good answer when you have a set of points w/o any particular boundary and need to find the convex hull.

A polygon is a set of points in a list where the consecutive points form the boundary. It is much easier to figure out whether a polygon is convex or not (and you don't have to calculate any angles, either):

For each consecutive pair of edges of the polygon (each triplet of points), compute the z-component of the cross product of the vectors defined by the edges pointing towards the points in increasing order. Take the cross product of these vectors:

The polygon is convex if the z-components of the cross products are either all positive or all negative. Otherwise the polygon is nonconvex.

given p[k], p[k+1], p[k+2] each with coordinates x, y:

dx1 = x[k+1]-x[k]

dy1 = y[k+1]-y[k]

dx2 = x[k+2]-x[k+1]

dy2 = y[k+2]-y[k+1]

zcrossproduct = dx1 * dy2 - dy1 * dx2

If there are N points, make sure you calculate N cross products, e.g. be sure to use the triplets (p[N-2],p[N-1],p[0]) and (p[N-1],p[0],p[1]).

class Solution(object):
def isConvex(self, points):
"""
:type points: List[List[int]]
:rtype: bool
"""
n = len(points)
zcrossproduct = None

for i in range(-2, n-2):
x = [ points[i][0], points[i+1][0], points[i+2][0] ]
y = [ points[i][1], points[i+1][1], points[i+2][1] ]

dx1 = x[1] - x[0]
dy1 = y[1] - y[0]

dx2 = x[2] - x[1]
dy2 = y[2] - y[1]

if not zcrossproduct:
zcrossproduct = dx1 * dy2 - dy1 * dx2
elif ( dx1 * dy2 - dy1 * dx2 ) * zcrossproduct < 0:
return False
return True