Polygons and Convex Hull

Objectives

The the ccw function to check if a polygon is convex or not.
Compute the perimeter and area or an arbitrary polygon
Find a polygon (convex hull) that fits a set of points.
Most code samples from Competitive Programming 3.

Getting the Area

$A = {1\over 2} \left[\begin{array}{ll} x_0 & y_0 \\ x_1 & y_1 \\ x_2 & y_2 \\ \cdots & \cdots \\ x_n & y_n \\ \end{array} \right]$

$= \begin{array}{l} x_0 y_1 + x_1 y_2 + \cdots + x_n y_0 \\ - x_1 y_0 - x_2 y_1 - \cdots - x_0 y_n \end{array}$

double area(const vector<point> &P) {
   double result = 0.0, x1, y1, x2, y2;
   for (int i = 0; i < (int)P.size()-1; i++) {
      x1 = P[i].x; x2 = P[i+1].x;
      y1 = P[i].y; y2 = P[i+1].y;
      result += (x1 * y2 - x2 * y1);
   }
   return fabs(result) / 2.0;
}

Convex Hull

Given: a bunch of points
We want: a minimal convex polygon to contain them.
Basic algorithm:
- Pick lowest point (and rightmost, if tie) as a pivot
- Sort the points by the angle to the pivot. Call these $p_1, p_2$ , etc.
- Push pivot, $p_1$ , to a stack.
- Repeat from p2:
  - Push point to stack
  - If (angle of top three points) is clockwise, delete midpoint from stack.

Polygons and Convex Hull

Table of Contents

Objectives

Polygons

Representation

Getting the Area

Is it Convex?

Convex Hull

Example

Sorting by Angle

Convex Hull

Convex Hull Code, 2