unique description of a convex quadrilateral I’m looking for a unique way to describe a convex quadrilateral.
The common way to express a quadrilateral is to give the four vertices: $[(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4)]$. The problem I have is that if we know that we are looking for a convex quadrilateral, the order of the points doesn’t matter and thus $[(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4)]$ describes the same quadrilateral than $[(x_2, y_2), (x_1, y_1), (x_3, y_3), (x_4, y_4)]$. Is there a better notation for a convex quadrilateral?
Make sure to order the four vertices in the same traversal order (it is a simple matter to determine the convex hull of four points). Doing so, you reduce the number of permutations from $24$ to $4$.



To describe the quadrilateral, you need 8 parameters, as there are 8 degrees of freedom. The following ones are order independent:


coordinates of the centroid (the mass can be uniform, uniform along edges or concentrated at vertices);
higher order moments;
coordinates of the intersection of the diagonals;
area;
perimeter;
shortest and longest side;
shortest and longest diagonal;
ratios of the above;
directions of the above segments (in range $[0,\pi)$).
For a convex quadrilateral, an unordered set of four points $\{(x_1,y_1),(x_2,y_2), (x_3,y_3), (x_4,y_4) \}$ defines a quadrilateral uniquely, so this notation is fine. 

For a concave quadrilateral however, four points do not determine the quadrilateral uniquely, so you can do a couple of things.

1) You can order the points $((x_1,y_1),(x_2,y_2), (x_3,y_3), (x_4,y_4) )$ and define the quadrilateral they represent by starting at the first point, then drawing a line to the second, then to the third, to the fourth, and then back to the first. 
or,

2) You can represent the edges of the quadrilateral (but this is a bit longer) in the form $\{ \{(x_1,y_1), (x_2,y_2)\},\{(x_2,y_2), (x_3,y_3)\},\{(x_3,y_3), (x_4,y_4)\},\{(x_4,y_4), (x_1,y_1)\} \}$. (You only need to store 3 edges in fact, as the last will be uniquely determined)