Introduction

Testing whether a point lies within a 3D shape defined by a point cloud is a common problem in computational geometry and computer graphics. This process involves using the sparse collection of points known as a point cloud to determine the volumetric properties of a shape. Point clouds are often obtained from 3D scanning devices such as LiDAR, which offer high precision but pose a challenge for modeling solid objects due to their discrete nature.

Approaches to Determine Point Containment

There are several methods to determine if a point lies within a 3D shape defined by a point cloud. Below we explore the technical aspects of the most common approaches.

1. Convex Hull Method

Overview

The convex hull is the smallest convex shape that can enclose all the points in the point cloud. If a point lies within this convex hull, it is said to be inside the 3D shape.

Method

1. Construct the Convex Hull: Use algorithms such as Quickhull or Delaunay triangulation to create the convex hull from the point cloud.
2. Test Point Inclusion: Determine if the test point lies inside the convex hull. For points located on the convex hull boundary, adjustments may be necessary depending on the application requirements.

Example

Consider a point cloud extracted from a 3D model and we would like to check if a point $P(x, y, z)$ is within this model. The convex hull encloses all the vertices, and using a library like scipy.spatial in Python, convex hull creation and point checking can be simplified.

• Computational Cost: Each method's efficiency and computational demands vary significantly. Convex hull calculations are inexpensive for convex shapes but ineffective for non-convex and complex geometries.
• Resolution and Accuracy: Voxelization introduces discretization errors, leading to a trade-off between accuracy and resolution.
• Algorithmm Complexity: Surface reconstruction processes might demand significant computational resources and pre-processing.