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Check whether a given polygon in a 2D plane is convex; if not, return the index of a vertex that doesn't belong to the convex hull.

Spoken language:

An array A of points in a 2D plane is given. These points represent a polygon: every two consecutive points describe an edge of the polygon, and there is an edge connecting the last point and the first point in the array.

A set of points in a 2D plane, whose boundary is a straight line, is called a *semiplane*. More precisely, any set of the form {(x, y) : ax + by ≥ c} is a semiplane. The semiplane contains its boundary.

A polygon is *convex* if and only if, no line segment between two points on the boundary ever goes outside the polygon.

For example, the polygon consisting of vertices whose Cartesian coordinates are consecutively:

is convex.

The *convex hull* of a finite set of points in a 2D plane is the smallest convex polygon that contains all points in this set. For example, the convex hull of a set consisting of seven points whose Cartesian coordinates are:

is a polygon that has five vertices. When traversed clockwise, its vertices are:

If a polygon is concave (that is, it is not convex), it has a vertex which does not lie on its convex hull border. Your assignment is to find such a vertex.

Assume that the following declarations are given:

struct Point2D { int x; int y; };

Write a function:

int solution(struct Point2D A[], int N);

that, given a non-empty array A consisting of N elements describing a polygon, returns −1 if the polygon is convex. Otherwise, the function should return the index of any point that doesn't belong to the convex hull border. Note that consecutive edges of the polygon may be collinear (that is, the polygon might have 180−degrees angles).

To access the coordinates of the K-th point (where 0 ≤ K < N), use the following syntax:

A[K].xto access the x-coordinate,A[K].yto access the y-coordinate.

For example, given array A such that:

the function should return −1, as explained in the example above.

However, given array A such that:

the function should return either 2 or 6. These are the indices of the polygon lying strictly in its convex hull (that is, not on the convex hull border).

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [3..10,000];
- the coordinates of each point in array A are integers within the range [−1,000,000,000..1,000,000,000];
- no two edges of the polygon A intersect, other than meeting at their endpoints;
- array A does not contain duplicate points.

Copyright 2009–2021 by Codility Limited. All Rights Reserved. Unauthorized copying, publication or disclosure prohibited.

An array A of points in a 2D plane is given. These points represent a polygon: every two consecutive points describe an edge of the polygon, and there is an edge connecting the last point and the first point in the array.

A set of points in a 2D plane, whose boundary is a straight line, is called a *semiplane*. More precisely, any set of the form {(x, y) : ax + by ≥ c} is a semiplane. The semiplane contains its boundary.

A polygon is *convex* if and only if, no line segment between two points on the boundary ever goes outside the polygon.

For example, the polygon consisting of vertices whose Cartesian coordinates are consecutively:

is convex.

The *convex hull* of a finite set of points in a 2D plane is the smallest convex polygon that contains all points in this set. For example, the convex hull of a set consisting of seven points whose Cartesian coordinates are:

is a polygon that has five vertices. When traversed clockwise, its vertices are:

If a polygon is concave (that is, it is not convex), it has a vertex which does not lie on its convex hull border. Your assignment is to find such a vertex.

Assume that the following declarations are given:

struct Point2D { int x; int y; };

Write a function:

int solution(vector<Point2D> &A);

that, given a non-empty array A consisting of N elements describing a polygon, returns −1 if the polygon is convex. Otherwise, the function should return the index of any point that doesn't belong to the convex hull border. Note that consecutive edges of the polygon may be collinear (that is, the polygon might have 180−degrees angles).

To access the coordinates of the K-th point (where 0 ≤ K < N), use the following syntax:

A[K].xto access the x-coordinate,A[K].yto access the y-coordinate.

For example, given array A such that:

the function should return −1, as explained in the example above.

However, given array A such that:

the function should return either 2 or 6. These are the indices of the polygon lying strictly in its convex hull (that is, not on the convex hull border).

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [3..10,000];
- the coordinates of each point in array A are integers within the range [−1,000,000,000..1,000,000,000];
- no two edges of the polygon A intersect, other than meeting at their endpoints;
- array A does not contain duplicate points.

Copyright 2009–2021 by Codility Limited. All Rights Reserved. Unauthorized copying, publication or disclosure prohibited.

An array A of points in a 2D plane is given. These points represent a polygon: every two consecutive points describe an edge of the polygon, and there is an edge connecting the last point and the first point in the array.

A set of points in a 2D plane, whose boundary is a straight line, is called a *semiplane*. More precisely, any set of the form {(x, y) : ax + by ≥ c} is a semiplane. The semiplane contains its boundary.

A polygon is *convex* if and only if, no line segment between two points on the boundary ever goes outside the polygon.

For example, the polygon consisting of vertices whose Cartesian coordinates are consecutively:

is convex.

The *convex hull* of a finite set of points in a 2D plane is the smallest convex polygon that contains all points in this set. For example, the convex hull of a set consisting of seven points whose Cartesian coordinates are:

is a polygon that has five vertices. When traversed clockwise, its vertices are:

If a polygon is concave (that is, it is not convex), it has a vertex which does not lie on its convex hull border. Your assignment is to find such a vertex.

Assume that the following declarations are given:

class Point2D { public int x; public int y; };

Write a function:

class Solution { public int solution(Point2D[] A); }

that, given a non-empty array A consisting of N elements describing a polygon, returns −1 if the polygon is convex. Otherwise, the function should return the index of any point that doesn't belong to the convex hull border. Note that consecutive edges of the polygon may be collinear (that is, the polygon might have 180−degrees angles).

To access the coordinates of the K-th point (where 0 ≤ K < N), use the following syntax:

A[K].xto access the x-coordinate,A[K].yto access the y-coordinate.

For example, given array A such that:

the function should return −1, as explained in the example above.

However, given array A such that:

the function should return either 2 or 6. These are the indices of the polygon lying strictly in its convex hull (that is, not on the convex hull border).

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [3..10,000];
- the coordinates of each point in array A are integers within the range [−1,000,000,000..1,000,000,000];
- no two edges of the polygon A intersect, other than meeting at their endpoints;
- array A does not contain duplicate points.

Copyright 2009–2021 by Codility Limited. All Rights Reserved. Unauthorized copying, publication or disclosure prohibited.

*semiplane*. More precisely, any set of the form {(x, y) : ax + by ≥ c} is a semiplane. The semiplane contains its boundary.

*convex* if and only if, no line segment between two points on the boundary ever goes outside the polygon.

For example, the polygon consisting of vertices whose Cartesian coordinates are consecutively:

is convex.

*convex hull* of a finite set of points in a 2D plane is the smallest convex polygon that contains all points in this set. For example, the convex hull of a set consisting of seven points whose Cartesian coordinates are:

is a polygon that has five vertices. When traversed clockwise, its vertices are:

Assume that the following declarations are given:

type Point2D struct { X int Y int }

Write a function:

func Solution(A []Point2D) int

To access the coordinates of the K-th point (where 0 ≤ K < N), use the following syntax:

A[K].Xto access the x-coordinate,A[K].Yto access the y-coordinate.

For example, given array A such that:

the function should return −1, as explained in the example above.

However, given array A such that:

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [3..10,000];
- no two edges of the polygon A intersect, other than meeting at their endpoints;
- array A does not contain duplicate points.

*semiplane*. More precisely, any set of the form {(x, y) : ax + by ≥ c} is a semiplane. The semiplane contains its boundary.

*convex* if and only if, no line segment between two points on the boundary ever goes outside the polygon.

For example, the polygon consisting of vertices whose Cartesian coordinates are consecutively:

is convex.

*convex hull* of a finite set of points in a 2D plane is the smallest convex polygon that contains all points in this set. For example, the convex hull of a set consisting of seven points whose Cartesian coordinates are:

is a polygon that has five vertices. When traversed clockwise, its vertices are:

Assume that the following declarations are given:

class Point2D { public int x; public int y; }

Write a function:

class Solution { public int solution(Point2D[] A); }

To access the coordinates of the K-th point (where 0 ≤ K < N), use the following syntax:

A[K].xto access the x-coordinate,A[K].yto access the y-coordinate.

For example, given array A such that:

the function should return −1, as explained in the example above.

However, given array A such that:

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [3..10,000];
- no two edges of the polygon A intersect, other than meeting at their endpoints;
- array A does not contain duplicate points.

*semiplane*. More precisely, any set of the form {(x, y) : ax + by ≥ c} is a semiplane. The semiplane contains its boundary.

*convex* if and only if, no line segment between two points on the boundary ever goes outside the polygon.

For example, the polygon consisting of vertices whose Cartesian coordinates are consecutively:

is convex.

*convex hull* of a finite set of points in a 2D plane is the smallest convex polygon that contains all points in this set. For example, the convex hull of a set consisting of seven points whose Cartesian coordinates are:

is a polygon that has five vertices. When traversed clockwise, its vertices are:

Assume that the following declarations are given:

class Point2D { public int x; public int y; }

Write a function:

class Solution { public int solution(Point2D[] A); }

To access the coordinates of the K-th point (where 0 ≤ K < N), use the following syntax:

A[K].xto access the x-coordinate,A[K].yto access the y-coordinate.

For example, given array A such that:

the function should return −1, as explained in the example above.

However, given array A such that:

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [3..10,000];
- no two edges of the polygon A intersect, other than meeting at their endpoints;
- array A does not contain duplicate points.

*semiplane*. More precisely, any set of the form {(x, y) : ax + by ≥ c} is a semiplane. The semiplane contains its boundary.

*convex* if and only if, no line segment between two points on the boundary ever goes outside the polygon.

For example, the polygon consisting of vertices whose Cartesian coordinates are consecutively:

is convex.

*convex hull* of a finite set of points in a 2D plane is the smallest convex polygon that contains all points in this set. For example, the convex hull of a set consisting of seven points whose Cartesian coordinates are:

is a polygon that has five vertices. When traversed clockwise, its vertices are:

Assume that the following declarations are given:

// Point2D obj is an Object with attributes // obj.x - type: int // obj.y - type: int

Write a function:

function solution(A);

To access the coordinates of the K-th point (where 0 ≤ K < N), use the following syntax:

A[K].xto access the x-coordinate,A[K].yto access the y-coordinate.

For example, given array A such that:

the function should return −1, as explained in the example above.

However, given array A such that:

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [3..10,000];
- no two edges of the polygon A intersect, other than meeting at their endpoints;
- array A does not contain duplicate points.

*semiplane*. More precisely, any set of the form {(x, y) : ax + by ≥ c} is a semiplane. The semiplane contains its boundary.

*convex* if and only if, no line segment between two points on the boundary ever goes outside the polygon.

For example, the polygon consisting of vertices whose Cartesian coordinates are consecutively:

is convex.

*convex hull* of a finite set of points in a 2D plane is the smallest convex polygon that contains all points in this set. For example, the convex hull of a set consisting of seven points whose Cartesian coordinates are:

is a polygon that has five vertices. When traversed clockwise, its vertices are:

Assume that the following declarations are given:

-- Point2D obj is an object with attributes -- obj.x - type: int -- obj.y - type: int

Write a function:

function solution(A)

To access the coordinates of the K-th point (where 0 ≤ K < N), use the following syntax:

A[K].xto access the x-coordinate,A[K].yto access the y-coordinate.

For example, given array A such that:

the function should return −1, as explained in the example above.

However, given array A such that:

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [3..10,000];
- no two edges of the polygon A intersect, other than meeting at their endpoints;
- array A does not contain duplicate points.

Note: All arrays in this task are zero-indexed, unlike the common Lua convention. You can use `#A` to get the length of the array A.

*semiplane*. More precisely, any set of the form {(x, y) : ax + by ≥ c} is a semiplane. The semiplane contains its boundary.

*convex* if and only if, no line segment between two points on the boundary ever goes outside the polygon.

For example, the polygon consisting of vertices whose Cartesian coordinates are consecutively:

is convex.

*convex hull* of a finite set of points in a 2D plane is the smallest convex polygon that contains all points in this set. For example, the convex hull of a set consisting of seven points whose Cartesian coordinates are:

is a polygon that has five vertices. When traversed clockwise, its vertices are:

Assume that the following declarations are given:

@interface Point2D : NSObject { @public int x; int y; } @end @implementation Point2D @end

Write a function:

int solution(NSMutableArray *A);

To access the coordinates of the K-th point (where 0 ≤ K < N), use the following syntax:

((Point2D *)[A objectAtIndex: K])->xto access the x-coordinate,((Point2D *)[A objectAtIndex: K])->yto access the y-coordinate.

For example, given array A such that:

the function should return −1, as explained in the example above.

However, given array A such that:

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [3..10,000];
- no two edges of the polygon A intersect, other than meeting at their endpoints;
- array A does not contain duplicate points.

*semiplane*. More precisely, any set of the form {(x, y) : ax + by ≥ c} is a semiplane. The semiplane contains its boundary.

*convex* if and only if, no line segment between two points on the boundary ever goes outside the polygon.

For example, the polygon consisting of vertices whose Cartesian coordinates are consecutively:

is convex.

*convex hull* of a finite set of points in a 2D plane is the smallest convex polygon that contains all points in this set. For example, the convex hull of a set consisting of seven points whose Cartesian coordinates are:

is a polygon that has five vertices. When traversed clockwise, its vertices are:

Assume that the following declarations are given:

Point2D = record x : longint; y : longint; end;

Write a function:

function solution(A: array of Point2D; N: longint): longint;

To access the coordinates of the K-th point (where 0 ≤ K < N), use the following syntax:

A[K].xto access the x-coordinate,A[K].yto access the y-coordinate.

For example, given array A such that:

the function should return −1, as explained in the example above.

However, given array A such that:

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [3..10,000];
- no two edges of the polygon A intersect, other than meeting at their endpoints;
- array A does not contain duplicate points.

*semiplane*. More precisely, any set of the form {(x, y) : ax + by ≥ c} is a semiplane. The semiplane contains its boundary.

*convex* if and only if, no line segment between two points on the boundary ever goes outside the polygon.

For example, the polygon consisting of vertices whose Cartesian coordinates are consecutively:

is convex.

*convex hull* of a finite set of points in a 2D plane is the smallest convex polygon that contains all points in this set. For example, the convex hull of a set consisting of seven points whose Cartesian coordinates are:

is a polygon that has five vertices. When traversed clockwise, its vertices are:

Assume that the following declarations are given:

class Point2D { var $x = 0; var $y = 0; }

Write a function:

function solution($A);

To access the coordinates of the K-th point (where 0 ≤ K < N), use the following syntax:

$A[$K]->xto access the x-coordinate,$A[$K]->yto access the y-coordinate.

For example, given array A such that:

the function should return −1, as explained in the example above.

However, given array A such that:

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [3..10,000];
- no two edges of the polygon A intersect, other than meeting at their endpoints;
- array A does not contain duplicate points.

*semiplane*. More precisely, any set of the form {(x, y) : ax + by ≥ c} is a semiplane. The semiplane contains its boundary.

*convex* if and only if, no line segment between two points on the boundary ever goes outside the polygon.

For example, the polygon consisting of vertices whose Cartesian coordinates are consecutively:

is convex.

*convex hull* of a finite set of points in a 2D plane is the smallest convex polygon that contains all points in this set. For example, the convex hull of a set consisting of seven points whose Cartesian coordinates are:

is a polygon that has five vertices. When traversed clockwise, its vertices are:

Assume that the following declarations are given:

# Point2D is a dictionary with keys # 'x' - type: int # 'y' - type: int

Write a function:

sub solution { my (@A)=@_; ... }

To access the coordinates of the K-th point (where 0 ≤ K < N), use the following syntax:

$A[$K]->{x}to access the x-coordinate,$A[$K]->{y}to access the y-coordinate.

For example, given array A such that:

the function should return −1, as explained in the example above.

However, given array A such that:

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [3..10,000];
- no two edges of the polygon A intersect, other than meeting at their endpoints;
- array A does not contain duplicate points.

*semiplane*. More precisely, any set of the form {(x, y) : ax + by ≥ c} is a semiplane. The semiplane contains its boundary.

*convex* if and only if, no line segment between two points on the boundary ever goes outside the polygon.

For example, the polygon consisting of vertices whose Cartesian coordinates are consecutively:

is convex.

*convex hull* of a finite set of points in a 2D plane is the smallest convex polygon that contains all points in this set. For example, the convex hull of a set consisting of seven points whose Cartesian coordinates are:

is a polygon that has five vertices. When traversed clockwise, its vertices are:

Assume that the following declarations are given:

class Point2D(object): x = 0 y = 0

Write a function:

def solution(A)

To access the coordinates of the K-th point (where 0 ≤ K < N), use the following syntax:

A[K].xto access the x-coordinate,A[K].yto access the y-coordinate.

For example, given array A such that:

the function should return −1, as explained in the example above.

However, given array A such that:

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [3..10,000];
- no two edges of the polygon A intersect, other than meeting at their endpoints;
- array A does not contain duplicate points.

*semiplane*. More precisely, any set of the form {(x, y) : ax + by ≥ c} is a semiplane. The semiplane contains its boundary.

*convex* if and only if, no line segment between two points on the boundary ever goes outside the polygon.

For example, the polygon consisting of vertices whose Cartesian coordinates are consecutively:

is convex.

*convex hull* of a finite set of points in a 2D plane is the smallest convex polygon that contains all points in this set. For example, the convex hull of a set consisting of seven points whose Cartesian coordinates are:

is a polygon that has five vertices. When traversed clockwise, its vertices are:

Assume that the following declarations are given:

class Point2D attr_accessor :x, :y end

Write a function:

def solution(a)

To access the coordinates of the K-th point (where 0 ≤ K < N), use the following syntax:

A[K].xto access the x-coordinate,A[K].yto access the y-coordinate.

For example, given array A such that:

the function should return −1, as explained in the example above.

However, given array A such that:

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [3..10,000];
- no two edges of the polygon A intersect, other than meeting at their endpoints;
- array A does not contain duplicate points.

*semiplane*. More precisely, any set of the form {(x, y) : ax + by ≥ c} is a semiplane. The semiplane contains its boundary.

*convex* if and only if, no line segment between two points on the boundary ever goes outside the polygon.

For example, the polygon consisting of vertices whose Cartesian coordinates are consecutively:

is convex.

*convex hull* of a finite set of points in a 2D plane is the smallest convex polygon that contains all points in this set. For example, the convex hull of a set consisting of seven points whose Cartesian coordinates are:

is a polygon that has five vertices. When traversed clockwise, its vertices are:

Assume that the following declarations are given:

class Point2D(var x: Int, var y: Int)

Write a function:

object Solution { def solution(a: Array[Point2D]): Int }

To access the coordinates of the K-th point (where 0 ≤ K < N), use the following syntax:

A(K).xto access the x-coordinate,A(K).yto access the y-coordinate.

For example, given array A such that:

the function should return −1, as explained in the example above.

However, given array A such that:

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [3..10,000];
- no two edges of the polygon A intersect, other than meeting at their endpoints;
- array A does not contain duplicate points.

*semiplane*. More precisely, any set of the form {(x, y) : ax + by ≥ c} is a semiplane. The semiplane contains its boundary.

*convex* if and only if, no line segment between two points on the boundary ever goes outside the polygon.

For example, the polygon consisting of vertices whose Cartesian coordinates are consecutively:

is convex.

*convex hull* of a finite set of points in a 2D plane is the smallest convex polygon that contains all points in this set. For example, the convex hull of a set consisting of seven points whose Cartesian coordinates are:

is a polygon that has five vertices. When traversed clockwise, its vertices are:

Assume that the following declarations are given:

public struct Point2D { public var x: Int = 0; public var y: Int = 0; public init() {}; }

Write a function:

public func solution(_ A : inout [Point2D]) -> Int

To access the coordinates of the K-th point (where 0 ≤ K < N), use the following syntax:

A[K].xto access the x-coordinate,A[K].yto access the y-coordinate.

For example, given array A such that:

the function should return −1, as explained in the example above.

However, given array A such that:

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [3..10,000];
- no two edges of the polygon A intersect, other than meeting at their endpoints;
- array A does not contain duplicate points.

*semiplane*. More precisely, any set of the form {(x, y) : ax + by ≥ c} is a semiplane. The semiplane contains its boundary.

*convex* if and only if, no line segment between two points on the boundary ever goes outside the polygon.

For example, the polygon consisting of vertices whose Cartesian coordinates are consecutively:

is convex.

*convex hull* of a finite set of points in a 2D plane is the smallest convex polygon that contains all points in this set. For example, the convex hull of a set consisting of seven points whose Cartesian coordinates are:

is a polygon that has five vertices. When traversed clockwise, its vertices are:

Assume that the following declarations are given:

Class Point2D Public x As Integer Public y As Integer End Class

Write a function:

Private Function solution(A As Point2D()) As Integer

To access the coordinates of the K-th point (where 0 ≤ K < N), use the following syntax:

A(K).xto access the x-coordinate,A(K).yto access the y-coordinate.

For example, given array A such that:

the function should return −1, as explained in the example above.

However, given array A such that:

Write an ** efficient** algorithm for the following assumptions:

- N is an integer within the range [3..10,000];
- no two edges of the polygon A intersect, other than meeting at their endpoints;
- array A does not contain duplicate points.