Efficient Querying of Triangles within Radius Point in 3D (2024)

Abstract: This article discusses methods for efficiently querying triangles within a given radius of a 3D point in a triangle mesh.

2024-04-24 by DevCodeF1 Editors

Efficient Querying of Triangles within Radius of Point in 3D Space

In many 3D graphics and geometric algorithms, it is often necessary to efficiently query a set of triangles within a certain radius of a given point in 3D space. This article will discuss the key concepts, algorithms, and techniques for performing such queries, specifically for triangular meshes that may be unoriented, non-manifold, or a soup of unconnected triangles.

Problem Definition

Given a triangular mesh (which may be unoriented, non-manifold, or a soup) and an arbitrary 3D point p with a radius r, the goal is to query and identify all the triangles whose distance to the point p is less than or equal to the radius r.

Key Concepts

The main challenge in efficiently querying triangles within a radius of a point in 3D space is the high computational cost associated with calculating the distance between the point and each triangle in the mesh. To address this challenge, we can employ the following concepts and techniques:

• Spatial partitioning data structures, such as octrees, k-d trees, and bounding volume hierarchies (BVHs), to reduce the number of distance calculations required.
• Approximate distance calculations, such as bounding sphere distances, to quickly filter out triangles that are clearly outside the query radius.
• Efficient distance calculation algorithms, such as the point-to-triangle distance formula or the closest point on triangle algorithm.

Spatial Partitioning Data Structures

Spatial partitioning data structures are used to divide the 3D space into smaller regions, allowing for more efficient queries of geometric objects within those regions. For the problem of querying triangles within a radius of a point in 3D space, we can use the following spatial partitioning data structures:

• Octrees: An octree is a tree data structure that recursively subdivides 3D space into eight octants. By building an octree from the triangular mesh, we can efficiently query triangles within a given radius of a point by traversing the octree and only considering the triangles within the relevant octants.
• k-d Trees: A k-d tree is a binary space partitioning tree that recursively divides 3D space along a plane perpendicular to one of the coordinate axes. By building a k-d tree from the triangular mesh, we can efficiently query triangles within a given radius of a point by traversing the k-d tree and only considering the triangles within the relevant subspaces.
• Bounding Volume Hierarchies (BVHs): A BVH is a tree data structure that recursively subdivides 3D space into bounding volumes, such as spheres or axis-aligned bounding boxes (AABBs). By building a BVH from the triangular mesh, we can efficiently query triangles within a given radius of a point by traversing the BVH and only considering the triangles within the relevant bounding volumes.

Approximate Distance Calculations

To further reduce the number of distance calculations required, we can use approximate distance calculations, such as bounding sphere distances, to quickly filter out triangles that are clearly outside the query radius. A bounding sphere is a sphere that encloses a set of triangles, and its center and radius can be calculated from the triangles' vertices. By calculating the distance between the query point and the bounding sphere, we can quickly determine if the triangles within the sphere are likely to be within the query radius.

Efficient Distance Calculation Algorithms

Once we have identified the triangles that are likely to be within the query radius using spatial partitioning data structures and approximate distance calculations, we can use efficient distance calculation algorithms to accurately determine the distance between the query point and each triangle. Two common algorithms for calculating the distance between a point and a triangle are:

• The point-to-triangle distance formula, which calculates the distance between the query point and the plane of the triangle, and then determines the closest point on the triangle to the query point.
• The closest point on triangle algorithm, which calculates the closest point on the triangle to the query point by projecting the query point onto the plane of the triangle and then determining the closest point on the triangle's edges or vertices.

Implementation Details

To implement the efficient querying of triangles within a radius of a point in 3D space, we can follow the following steps:

1. Build a spatial partitioning data structure, such as an octree, k-d tree, or BVH, from the triangular mesh.
2. Calculate the bounding sphere for the triangular mesh and determine if the query point is within the bounding sphere.
3. If the query point is within the bounding sphere, traverse the spatial partitioning data structure and calculate the distance between the query point and each triangle within the relevant regions.
4. Use an efficient distance calculation algorithm, such as the point-to-triangle distance formula or the closest point on triangle algorithm, to accurately determine the distance between the query point and each triangle.
5. Filter out triangles that are outside the query radius and return the remaining triangles.

Efficient querying of triangles within a radius of a point in 3D space is a crucial task in many 3D graphics and geometric algorithms. By using spatial partitioning data structures, approximate distance calculations, and efficient distance calculation algorithms, we can significantly reduce the computational cost of querying triangles within a given radius of a point. The following are the key concepts and techniques covered in this article:

• Spatial partitioning data structures, such as octrees, k-d trees, and BVHs, to reduce the number of distance calculations required.
• Approximate distance calculations, such as bounding sphere distances, to quickly filter out triangles that are clearly outside the query radius.
• Efficient distance calculation algorithms, such as the point-to-triangle distance formula or the closest point on triangle algorithm, to accurately determine the distance between the query point and each triangle.

References

• Octree
• k-d tree
• Bounding Volume Hierarchy (BVH)
• Point-to-Triangle Distance Formula
• Closest Point on Triangle Algorithm

Learn how to implement radius point query algorithms for 3D triangle meshes in this informative article.

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