Convex Polygon

Convex Polygon - Problem

You are given an array of points on the X-Y plane points where points[i] = [xi, yi]. The points form a polygon when joined sequentially.

Return true if this polygon is convex and false otherwise.

You may assume the polygon formed by given points is always a simple polygon. In other words, we ensure that exactly two edges intersect at each vertex and that edges otherwise don't intersect each other.

Note: A polygon is convex if all interior angles are less than 180 degrees, which means all cross products of consecutive edge vectors have the same sign.

Input & Output

Example 1 — Square (Convex)

$ Input: points = [[0,0],[0,1],[1,1],[1,0]]

› Output: true

💡 Note: A square is convex. All interior angles are 90 degrees (less than 180), and all cross products have the same sign when traversed counterclockwise.

Example 2 — Concave Shape

$ Input: points = [[0,0],[0,10],[10,10],[10,0],[5,5]]

› Output: false

💡 Note: This forms a concave polygon with an inward dent. The interior angle at [5,5] is greater than 180 degrees, making it non-convex.

Example 3 — Triangle (Convex)

$ Input: points = [[0,0],[3,0],[1,2]]

› Output: true

💡 Note: Any triangle is always convex. All three interior angles are less than 180 degrees.

Constraints

3 ≤ points.length ≤ 10⁴
points[i].length == 2
-10⁴ ≤ x_i, y_i ≤ 10⁴
All the given points are unique

Visualization

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Asked in

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The key insight is that a polygon is convex if and only if all cross products of consecutive edge vectors have the same sign. The best approach uses cross product sign checking in a single pass. Time: O(n), Space: O(1).

Common Approaches

✓ Cross Product Sign Check

⏱️ Time: O(n) Space: O(1)

For each three consecutive points, compute the cross product of the two edge vectors. A polygon is convex if all cross products have the same sign (all positive or all negative).

Optimized Cross Product Check

⏱️ Time: O(n) Space: O(1)

Enhanced version of cross product checking with early termination when inconsistent signs are found and proper handling of integer overflow for large coordinates.

Cross Product Sign Check — Algorithm Steps

Step 1: Iterate through all consecutive triplets of points
Step 2: Calculate cross product for each triplet
Step 3: Check if all cross products have consistent sign

Visualization

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Step-by-Step Walkthrough

Take 3 Points

Consider consecutive triplets of vertices

Calculate Cross Product

Find cross product of edge vectors

Check Sign Consistency

All cross products must have same sign

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <stdbool.h>

bool solution(int points[][2], int n) {
    if (n < 3) return true;
    
    int sign = 0;
    for (int i = 0; i < n; i++) {
        // Get three consecutive points (with wraparound)
        int *p1 = points[i];
        int *p2 = points[(i + 1) % n];
        int *p3 = points[(i + 2) % n];
        
        // Calculate cross product of vectors (p1->p2) and (p2->p3)
        int v1x = p2[0] - p1[0], v1y = p2[1] - p1[1];
        int v2x = p3[0] - p2[0], v2y = p3[1] - p2[1];
        long long cross = (long long)v1x * v2y - (long long)v1y * v2x;
        
        if (cross != 0) {
            if (sign == 0) {
                sign = cross > 0 ? 1 : -1;
            } else if ((cross > 0 && sign < 0) || (cross < 0 && sign > 0)) {
                return false;
            }
        }
    }
    
    return true;
}

int main() {
    char line[10000];
    fgets(line, sizeof(line), stdin);
    
    // Parse 2D array manually
    int points[1000][2];
    int n = 0;
    
    char *ptr = strchr(line, '[');
    while (ptr && *ptr) {
        if (*ptr == '[' && *(ptr+1) != '[') {
            sscanf(ptr, "[%d,%d]", &points[n][0], &points[n][1]);
            n++;
        }
        ptr++;
    }
    
    bool result = solution(points, n);
    printf(result ? "true\n" : "false\n");
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(n)

Single pass through all points to check each angle

✓ Linear Growth

Space Complexity

O(1)

Only using variables for calculations, no extra data structures

✓ Linear Space

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Smart Actions

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Input & Output

Constraints

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Related Problems

Common Approaches

Cross Product Sign Check — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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