You are keen to determine if a given 3D point lies inside a convex hull defined by a set of points. Checking for point membership in a convex hull can be a computational challenge, especially in 3D space. Understanding the mechanics of this problem can be very beneficial in fields such as computer graphics, computational geometry, and geographic information systems. In this article, we delve into the most efficient approaches to solve this.

Convex Hull Basics

A convex hull of a set of points is the smallest convex shape that encloses all the points. In 3D, this would be akin to wrapping the points with a tight, non-flexible surface. Mathematically, given a set of points $P = { p_1, p_2, \ldots, p_n }$, the convex hull is the smallest convex set $H$ such that every point in $P$ is either on the boundary or inside $H$.

Problem Overview

Given a set $P$ of points in 3D and a test point $T$, we want to quickly determine if $T$ lies inside the convex hull of $P$.

Algorithms for Checking Point in Convex Hull

There are different algorithms to handle this efficiently. Below are some of the most effective:

1. Direct Use of Existing Convex Hull Algorithm

Leverage existing algorithms to first compute the convex hull of $P$. Once the convex hull is computed, checking if $T$ is inside can be done in constant time $O(1)$:

• Implementation: Use a 3D Convex Hull algorithm like Quickhull or Chan's algorithm (which operates on $O(n \log n)$ time complexity) to construct the convex hull. • Interior Check: After constructing the hull, verify if $T$ is within the boundary defined by the hull facets.

2. Half-Space Intersection

The half-space intersection method can determine point concurrency:

• For every face in the convex hull, create a half-space that encompasses the convex hull exterior. • A point $T$ is inside the convex hull if and only if it lies on the "inner" side of all the half-space boundaries.

3. Binary Space Partitioning (BSP)

A Binary Space Partitioning tree can rapidly aid in checking the position of a point relative to a convex polyhedron:

• Construct a BSP tree for splitting space into convex sets. • Test the point $T$ by traversing the tree, quickly eliminating half-spaces.

4. Ray Casting Algorithm

Adapt the 2D ray casting strategy into 3D:

• Cast a ray from $T$ in random directions. • Count the intersections with the convex hull's surfaces. If the number of intersections is odd, $T$ is inside; otherwise, it is outside.

Implementing Quickhull Algorithm:

Here's a more detailed walkthrough using the popular Quickhull algorithm: