Approximate Boxes

Approximate Boxes is a small C++ project that intends to approximate a volume with cubes using octrees. It leverages the power of the Computational Geometry Algorithms Library (CGAL) to perform this task.

Features

• Approximation of a volume provided by an OFF file with cubes using octrees
• Saving the boxes as hexahedral elements in the Gmsh file format

Prerequisites

• C++ compiler with C++17 support
• CMake (version 3.10 or higher)
• CGAL library (Download and install from here)

Building the Project

1. Clone the repository:

   git clone https://github.com/paulotto/approximate_boxes.git

2. Navigate to the project directory:

   cd approximate_boxes

3. Create a build directory and navigate to it:

   mkdir build && cd build

4. Run CMake and build the project:

   cmake ..
   make

Usage

After building the project, you can run the example program that uses an octree with the following command:

./bin/ApproximateBoxesExampleOctree "../resources/ellipsoid.off"

This command will generate a Gmsh file with the boxes approximating the volume of the ellipsoid, placed in the resources folder.

Note: The degree of approximation is primarily determined by the number of nodes used for the surface in the OFF file. The more nodes, the more accurate the approximation will be and the smaller the boxes. Furthermore, the approximation can be influenced by the ApproximateBoxes<>::SetOctreeMaxDepth(unsigned int depth) and ApproximateBoxes<>::SetOctreeMaxInliers(unsigned int inliers) functions.

To have a completely connected mesh, it is necessary to divide the larger boxes into smaller ones so that every box has the same dimensions. This can be done by setting the ApproximateBoxes<>::SetDivideBoxes(bool divide) function to true (default: true). Below you can see an example where the ellipsoid is approximated with and without dividing the boxes.

Note: If the boxes are divided, the number of boxes will increase significantly, leading to longer computation times.

Ellipsoid Approximation Without Dividing of the Boxes

Ellipsoid	Ellipsoid + Approximation	Approximation Cut

Ellipsoid Approximation With Dividing of the Boxes

Ellipsoid	Ellipsoid + Approximation	Approximation Cut

Simple Approximation Implementation

There is also a simpler implementation of the approximation without using an octree. Here, the bounding box of the geometry is divided into smaller boxes, which are then checked for intersection with the geometry.

To run the example for this implementation, use the following command:

./bin/ApproximateBoxesExampleSimple "../resources/ellipsoid.off"

Note: The dimensions of the approximating boxes are determined by the dividing factor that is used to divide the bounding box of the geometry. The smaller the dividing factor, the smaller the boxes and the more accurate the approximation. The dividing factor can be set for each dimension individually as parameter of the function ApproximateBoxes<>::ApproximateGeometrySimple(CGAL::Vector_3<>)

Ellipsoid	Ellipsoid + Approximation	Approximation Cut

License

This project is licensed under the LGPL-3.0 License. See the LICENSE file for more details.