Introduction

When working with computational geometry, one frequent problem is determining whether a given point lies inside or on the boundary of a convex polygon formed by a set of points, known as the convex hull. While one common approach is to first compute the convex hull and then check if the point is inside it, this may not be necessary or efficient for some applications. Here, we will explore an approach that allows us to determine the point's location relative to a convex hull without computing the hull explicitly.

Convex Hull and Basic Definitions

A convex hull is the smallest convex shape that encloses a set of points. For a set of points in two dimensions, the convex hull can be envisioned as the shape formed by a rubber band stretched around them. Sometimes, it's not necessary to compute the hull to know if a point is inside it.

Convex Hull Properties

1. Convexity: For any two points within the hull, the line segment connecting them remains entirely within the hull.
2. Unique Minimum: Among possible encloses, convex hull is minimal in area.

Problem Formulation

Given:
• A set of points $P = {p_1, p_2, \ldots, p_n}$ in a 2D plane. • A test point $q$.

To Determine:
• Whether the point $q$ lies inside the convex polygon that can be formed as the convex hull of $P$.

Approach Without Computing the Hull

To find if a point $q$ is inside the potential convex hull of $P$, we use a technique based on the properties of orientations to check if point $q$ is inside all triangles formed by splitting the convex hull. Here, determining the orientation can be crucial.

Orientation Test

For three points, $a$, $b$, and $c$, the orientation can be calculated using the determinant:

$$\text{orientation}(a, b, c) = (b\_x - a\_x) \cdot (c\_y - a\_y) - (b\_y - a\_y) \cdot (c\_x - a\_x)$$

• If this value is positive, then $c$ is counterclockwise to the line from $a$ to $b$. • If zero, then $c$ is collinear with $a$ and $b$. • If negative, then $c$ is clockwise to the line from $a$ to $b$.

Ray Casting Method

In the context of triangulation:

1. For each point $p_i$ in $P$, check the orientation of triplets $(p_i, q, p_{i+1})$ for all $i$ (taking indices modulo $n$ to wrap around the convex hull).
2. If all orientations have the same sign or are zero (collinear), $q$ is inside or on the edge of the polygon.

Example Discussion

Consider the set of points $P = {(0, 0), (4, 0), (4, 4), (0, 4)}$ forming a convex square. We want to check the position of $q = (2, 2)$.

1. Calculate orientation for $(q, p_1, p_2)$, $(q, p_2, p_3)$, $(q, p_3, p_4)$, and $(q, p_4, p_1)$.
2. Each of these computations should yield a consistent non-negative value (if $q$ is inside) or zero (if on the boundary).

If all orientation checks pass, $q$ is inside the convex shape defined by $P$.

Complexity and Performance

• Time Complexity: $O(n)$ for $n$ points, since we check each segment defined by $p_i$ and $p_{i+1}$. • Space Complexity: $O(1)$ additional space, aside from input.

Comparisons

Conclusion

Checking if a point lies within the convex hull of a set of points is possible without calculating the hull itself by leveraging geometric properties and orientation tests. This approach is more direct, with lower computational overhead, and is particularly beneficial when working with large datasets where explicit hull computation is impractical. Understanding these techniques extends the toolkit available for effective solutions in computational geometry, offering both performance and simplicity.