CounterClockWise

Goal : Determine what direction to take when multiple points are given.

CounterClockWise(CCW) is based on Shoelace formula by Carl Friedrich Gauss.

1. Take the first x-coordinate and multiply it by the second y-value, then take the second x-coordinate and multiply it by the third y-value, and repeat as many times until it is done for all wanted points.
2. If the points are labeled sequentially in the counterclockwise direction, then the sum of the above determinants is positive and the absolute value signs can be omitted if they are labeled in the clockwise direction, the sum of the determinants will be negative. This is because the formula can be viewed as a special case of Green's theorem.

Here's an implementation in Swift that should be easy to understand:

func ccw(points: [Point]) -> Int{
  let polygon = points.count
  var orientation = 0

  for i in 0..<polygon{
    orientation += (points[i%polygon].x*points[(i+1)%polygon].y
      - points[(i+1)%polygon].x*points[i%polygon].y)
  }
    
  switch orientation {
  case Int.min..<0:
    return -1 // if operation < 0 : ClockWise
  case 0:
    return 0 // if operation == 0 : Parallel
  default:
    return 1 // if operation > 0 : CounterClockWise
  }
}

Put this code in a playground and test it like so:

var p1 = Point(x: 5, y: 8)
var p2 = Point(x: 9, y: 1)
var p3 = Point(x: 3, y: 6)

print(ccw(points: [p1,p2,p3])) // -1 means ClockWise

Here's how it works. When given an [Photo], ccw(points:) calculates the direction of the given points according to the Shoelaces formula.

orientation is less than 0, the direction is clockwise.

orientation is equal to 0, the direction is parallel.

orientation is greater than 0, the direction is counterclockwise.

An example

In Triangle

var p1 = Point(x: 5, y: 8)
var p2 = Point(x: 9, y: 1)
var p3 = Point(x: 3, y: 6)

print(ccw(points: [p1,p2,p3])) // -1 means ClockWise

In Quadrilateral

var p4 = Point(x: 5, y: 8)
var p5 = Point(x: 2, y: 3)
var p6 = Point(x: 6, y: 1)
var p7 = Point(x: 9, y: 3)

print(ccw(points: [p4,p5,p6,p7])) // 1 means CounterClockWise

In Pentagon

var p8 = Point(x: 5, y: 11)
var p9 = Point(x: 3, y: 4)
var p10 = Point(x: 5, y: 6)
var p11 = Point(x: 9, y: 5)
var p12 = Point(x: 12, y: 8)

print(ccw(points: [p8,p9,p10,p11,p12])) // 1 means CounterClockWise

You probably won't need to use the CCW in any real-world problems, but it's cool to play around with geometry algorithm. The formula was described by Meister (1724-1788) in 1769 and by Gauss in 1795. It can be verified by dividing the polygon into triangles, and can be considered to be a special case of Green's theorem.

Written for Swift Algorithm Club by TaeJoong Yoon