Leetcode:convex Polygon

if  This  3 and at the mosta -10,000 to a. Assume the polygon formed by given points are always a simple polygon (simple polygon definition). In the other words, we ensure that exactly the edges intersect at all vertex, and that edges otherwise don' t intersect E Ach other. Example 1: [[0,0],[0,1],[1,1],[1,0]]answer:true

Explanation:

2: [[0,0],[0,10],[10,10],[10,0],[5,5

Explanation:

Https://discuss.leetcode.com/topic/70706/beyond-my-knowledge-java-solution-with-in-line-explanation

https://discuss.leetcode.com/topic/70664/ C-7-line-o-n-solution-to-check-convexity-with-cross-product-of-adajcent-vectors-detailed-explanation

The key observation for convexity are that vector pi+1-pi all turns to the same direction to Pi+2-pi formed by any 3 Sequentially adjacent vertices, i.e., cross product (PI+1-PI) x (PI+2-PI) does no change sign when traversing sequential Ly along polygon vertices.

Note that for any 2D vectors v1, v2,

• V1 x v2 = det ([v1, v2])

Which is the determinant of the 2x2 matrix [v1, v2]. And the sign of Det ([v1, v2]) represents the positive z-direction of right-hand system from v1 to v2 . So det ([v1, v2]) ≥0 if and only if v1 turns at most of the degrees counterclockwise to v2< /c11>.

1  Public classSolution {2      Public BooleanIsconvex (list<list<integer>>points) {3         //for each set of three adjacent points A, B, C, find the cross product AB Bc. If the sign of4         //All the same, the angles be all positive or negative (depending on the5         //order in which we visit them) so the polygon is convex.6         BooleanGotnegative =false;7         BooleanGotpositive =false;8         intNumpoints =points.size ();9         intB, C;Ten          for(intA = 0; A < numpoints; a++) { One             //Trick to calc the last 3 points:n-1, 0 and 1. AB = (A + 1)%numpoints; -C = (B + 1)%numpoints; -      the             intCrossproduct = - Crossproductlength ( -Points.get (a). Get (0), Points.get (a). Get (1), -Points.get (b). Get (0), Points.get (b). Get (1), +Points.get (c). Get (0), Points.get (c). Get (1)); -             if(Crossproduct < 0) { +Gotnegative =true; A             } at             Else if(Crossproduct > 0) { -Gotpositive =true; -             } -             if(Gotnegative && gotpositive)return false; -         } -      in         //If We got this far, the polygon is convex. -         return true; to     } +      -     //Return The cross product AB x BC. the     //The Cross product are a vector perpendicular to AB and BC have length | ab| * | bc| * Sin (theta) and *     //With direction given by the right-hand rule. For both vectors in the X-y plane, the result is a $     //vector with X and Y are 0 so the Z component gives the vector ' s length and direction.Panax Notoginseng     Private intCrossproductlength (intAxintAy,intBxintBy,intCx,intCy) -     { the         //Get the vectors ' coordinates. +         intABx = Bx-Ax; A         intABy = by-Ay; the         intBCx = Cx-Bx; +         intBcy = Cy-by ; -      $         //Calculate The Z coordinate of the cross product. $         return(ABX * Bcy-aby *BCx); -     } -}

Leetcode:convex Polygon