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Understanding Convex Hull in Plane Geometry

Convex hull in plane geometry refers to the smallest convex polygon that encloses a set of points. This concept finds applications in collision avoidance, image processing, statistics, and more. The convex hull of a set of points can be obtained by identifying extreme points and using analytical geometry principles. The convex hull of a set of n points not all on the same line forms a convex polygon with vertices at some of the points. Implementing algorithms for convex hull involves partitioning the plane, checking point positions, and determining line equations. Time complexity for this process is O(n^3) due to drawing and complexity tests.

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Understanding Convex Hull in Plane Geometry

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  1. CONVEX HULL

  2. CONVEX: A set of points (finite or infinite) in the plane is called convex if for any two points p and q in the set, the entire line segment with the endpoints at p and q belongs to the set. CONVEX HULL : Convex Hull of set of points is defined as smallest convex polygon that encloses all the points in the set APPLICATION : 1. Collision Avoidance 5. Image Processing 2. Smallest Box 6. Statistics 3. Shape Analysis 7. Geographic Information System 4. Pattern Recognition 8. Game Theory

  3. REPRESENTATION: 1. Obtain the vertices of convex hull [Extreme points] 2. Obtain vertices of convex hull in some order [eg: Clockwise] Convex Hull of a set of points S is the boundary of the smallest convex region that contains all points in S inside it or on its boundary

  4. If S is convex, its convex hull is obviously S itself. If S is a set of two points, its convex hull is the line segment connecting these points. If S is a set of three points not on the same line, its convex hull is the triangle with the vertices at the three points given if the three points do lie on the same line, the convex hull is the line segment with its endpoints at the two points that are farthest apart THEOREM : The convex hull of any set S of n > 2 points not all on the same line is a convex polygon with the vertices at some of the points of S. Extreme Points : p1, p5, p6, p7, and p3.

  5. A few elementary facts from analytical geometry are needed to implement this algorithm. 1. The straight line through two points (x1, y1), (x2, y2) in the coordinate plane can be defined by the equation ax + by = c, where a = y2− y1, b = x1− x2, c = x1y2− y1x2. 2. Such a line divides the plane into two half-planes ◦ for all the points in one of them, ax + by > c ◦ while for all the points in the other, ax + by < c. (For the points on the line itself, of course, ax + by = c.) 3. To check whether certain points lie on the same side of the line, we can simply check whether the expression ax + by − c has the same sign for each of these points

  6. Time Complexity = O(n3) where O(n2) :- drawing by taking each point O(n) :- complexity test to check whether line lies inside the polygon

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