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https://github.com/lohedges/aabbcc

Dynamic AABB trees in C++ with support for periodic systems.
https://github.com/lohedges/aabbcc

aabb aabb-trees axis-aligned-bounding-box collision-detection spatial-decomposition spatial-partitioning

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Dynamic AABB trees in C++ with support for periodic systems.

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README 

        
# AABB.cc



Copyright © 2016-2018 [Lester Hedges](http://lesterhedges.net)



[![Build Status](https://travis-ci.org/lohedges/aabbcc.svg?branch=master)](https://travis-ci.org/lohedges/aabbcc)



Released under the [Zlib](http://zlib.net/zlib_license.html) license.



## About

A C++ implementation of a dynamic bounding volume hierarchy

([BVH](https://en.wikipedia.org/wiki/Bounding_volume_hierarchy)) using

axis-aligned bounding boxes ([AABBs](https://en.wikipedia.org/wiki/Minimum_bounding_box)).

The data structure provides an efficient way of detecting potential overlap

between objects of arbitrary shape and size and is commonly used in

computer game engines for collision detection and ray tracing.



Because of their speed and flexibility, AABB trees are also well suited

to overlap detection in physics applications, such as molecular simulation.

They are particularly helpful for systems where there is a large size disparity

between particle species, or whenever the particle density is extremely

inhomogeneous. In such situations, traditional neighbour finding tools, such

as [cell lists](https://en.wikipedia.org/wiki/Cell_lists), can become extremely

inefficient (both in terms of memory footprint, and search speed). A good

overview of the pros and cons of various neighbour finding algorithms

can be found [here](http://hoomd-blue.readthedocs.io/en/stable/nlist.html).

(Note that this only discusses the cost of _querying_ different data

structures, not the additional overhead of building them, or maintaining

them as objects move around.)



In statistical physics, a common means of approximating a bulk (infinite)

system is through the use of [periodic boundary conditions](https://en.wikipedia.org/wiki/Periodic_boundary_conditions).

Here, particles that are on opposite sides of the unit box can interact through

its periodic image. This library supports periodic and non-periodic systems

in an arbitrary number of dimensions (>= 2). Support is also provided for simulation

boxes that are partially periodic, i.e. periodic along specific axes. At present,

only orthorhombic simulation boxes are supported.



The code in this library was adapted from parts of the [Box2D](http://www.box2d.org)

physics engine.



## Installation

A `Makefile` is included for building and installing the AABB library.



To compile and install the library, documentation, python wrapper, and demos:



```bash

make build

make install

```



By default, the library installs to `/usr/local`. Therefore, you may need admin

privileges for the final `make install` step above. An alternative is to change

the install location:



```bash

make PREFIX=MY_INSTALL_DIR install

```

If you would rather use a header-only version of the library in your application,

simply run:



```bash

make header-only

```



The resulting library header, `header-only/AABB.hpp`, can be directly included

in your source code without the need for compiling and linking.



Further details on using the Makefile can be found by running make without

a target, i.e.



```bash

make

```



## Compiling and linking

To use the library with a C/C++ code first include the library header file

in the code.



```cpp

#include 

```



Then to compile, we can use something like the following:



```bash

g++ example.cc -laabb

```



This assumes that we have used the default install location `/usr/local`. If

we specify an install location, we would use a command more like the following:



```bash

g++ example.cc -I/my/path/include -L/my/path/lib -laabb

```



## Python wrapper

A python wrapper can be built using:



```bash

make python

```



You will require [python2.7](https://www.python.org/download/releases/2.7)

(and the development files if your package manager separates them) and

[SWIG](http://www.swig.org). To use the module you will need the python file

`aabb.py` and the shared object `_aabb.so` from the `python` directory.

If you wish to use a different version of python, simply override the

`PYTHON` make variable on the command line, e.g.



```bash

make PYTHON=3.5 python

```



(Note that you'll also need to update the shebang at the top of

[hard_disc.py](python/hard_disc.py) to reflect your changes in order for the

python demo to work.)



## Example

Let's consider a two-component system of hard discs in two dimensions, where

one species is much larger than the other. Making use of AABB trees, we can

efficiently search for potential overlaps between discs by decomposing the

system into its two constituent species and constructing a tree for each one.

To test overlaps for any given disc, we simply query the two trees

independently in order to find candidates. This decomposition ensures that

each AABB tree has a well defined length scale, making it simple to construct

and quick to query.



The image below shows the example hard disc system (left) and the AABB tree

structures for each species (middle and right). Each leaf node in a tree is

the AABB of an individual disc. Moving up the tree, AABBs are grouped together

into larger bounding volumes in a recursive fashion, leading to a single AABB

enclosing all of the discs at the root. The box outline in the left-hand image

shows the periodic boundary of the system.



![AABBs for a binary hard disc system.](https://raw.githubusercontent.com/lohedges/assets/master/aabbcc/images/aabb.png)



To query overlaps between discs we start at the root node and traverse the

tree. At each node we test whether the current AABB overlaps the AABB of the

chosen disc. If so, we add the two children of the node to the stack of nodes

to test. Whenever we reach a leaf node with which an overlap is found we record

a potential overlap with that disc (we know that the AABBs of the discs overlap,

but we need still need to check that discs themselves actually overlap). The

animation below shows an example of such a query. The disc of interest is

highlighted in green and the boundary of the periodic simulation box is shown

in orange. At each stage of the search the AABB of the current node in the tree

is shown in white. Leaf nodes that overlap with the trial disc are highlighted

green. Note that the green leaf node on the right-hand edge of the simulation

box overlaps through the periodic boundary.



	Querying an AABB tree for overlaps.



You may be wondering why the AABBs shown in the previous animation are not

the minimum enclosing bounding box for each disc. This is a trick that is

used to avoid frequent updates of the AABB tree during dynamics (movement

of the discs). Whenever an AABB changes position we need to delete it from

the tree then reinsert the new one (at the updated position). This can be a

costly operation. By "fattening" the AABBs a small amount it is possible to

make many displacements of the objects before an update is triggered, i.e.

when one of the discs moves outside of its fattened AABB. During dynamics it

is also possible for the tree to become unbalanced, leading to increasingly

inefficient queries. Here trees are balanced using a surface area heuristic

and active balancing is handled via tree rotations. The animation below shows

an example of a hard disc simulation. Dynamic AABB trees were used to maintain

a configuration of non-overlapping discs throughout the trajectory.



	Dynamics using AABB trees for overlap tests.



The code used to create the above animation can be found at

`demos/hard_disc.cc`. When the library is built, you can run the demo

and use [VMD](http://www.ks.uiuc.edu/Research/vmd/) to view the trajectory

as follows:



```bash

./demos/hard_disc

vmd trajectory.xyz -e demos/vmd.tcl

```



A python version of the demo can be found at `python/hard_disc.py`. This

provides an example of how to use the python wrapper module.



## Usage

There are several steps that go into building and using an AABB tree. Below

are some examples showing how to use the various objects within the library.



### AABB

This should be the minimum enclosing axis-aligned bounding box for an object

in your simulation box. There is no need to fatten the AABB; this will be done

when an object is inserted into the AABB tree. For example, to create an AABB

for a two-dimensional disc we could do the following:



```cpp

// Particle radius.

double radius = 1.0;



// Set the particle position.

std::vector position({10, 10});



// Compute lower and upper AABB bounds.

std::vector lowerBound({position[0] - radius, position[1] - radius});

std::vector upperBound({position[0] + radius, position[1] + radius});



// Create the AABB.

aabb::AABB aabb(lowerBound, upperBound);

```



(While we refer to _particles_ in this example, in practice a particle could

be any object, e.g. a sprite in a computer game.)



### Tree

#### Initialising a tree

To instantiate dynamic AABB trees for a periodic two-component system in

two dimensions:



```cpp

// Fattening factor.

double fatten = 0.1;



// Periodicity of the simulation box.

std::vector periodicity({true, true});



// Size of the simulation box.

std::vector boxSize({100, 100});



// Number of small discs.

unsigned int nSmall = 100;



// Number of large discs.

unsigned int nLarge = 10;



// Create the AABB trees.

aabb::Tree treeSmall(2, fatten, periodicity, boxSize, nSmall);

aabb::Tree treeLarge(2, fatten, periodicity, boxSize, nLarge);

```



Many of the arguments to the constructor of `Tree` are optional, see the

[Doxygen](http://www.doxygen.nl) documentation for details.



Note that both the periodicity and box size can be changed on-the-fly, e.g.

for changing the box volume during a constant pressure simulation. See the

`setPeriodicity` and `setBoxSize` methods for details.



#### Inserting a particle

To insert a particle (object) into the tree:



```cpp

// Particle radius.

double radius = 1.0;



// Particle index (key).

unsigned int index = 1;



// Set the particle position.

std::vector position({10.0, 10.0});



// Compute lower and upper AABB bounds.

std::vector lowerBound({position[0] - radius, position[1] - radius});

std::vector upperBound({position[0] + radius, position[1] + radius});



// Insert particle into the tree.

tree.insertParticle(index, position, lowerBound, upperBound);

```



Here `index` is a key that is used to create a map between particles and nodes

in the AABB tree. The key should be unique to the particle and can take any value

between 0 and `std::numeric_limits::max() - 1`.



For spherical objects, the insertion can be simplified to:



```cpp

tree.insertParticle(index, position, radius);

```



#### Removing a particle

If you are performing simulations using the [grand canonical ensemble](https://en.wikipedia.org/wiki/Grand_canonical_ensemble)

you may wish to remove particles from the tree. To do so:



```cpp

tree.removeParticle(index);

```



where `index` is the key for the particle to be removed. (You'll need to

keep track of the keys).



#### Querying the tree

You can query the tree for overlaps with a specific particle, or for overlaps

with an arbitrary AABB object. The `query` method returns a vector containing

the indices of the AABBs that overlap. You'll then need to test the objects

enclosed by these AABBs for actual overlap with the particle of interest.

(using your own overlap code).



For a particle already in the tree:



```cpp

// Query AABB overlaps for particle with key 10.

std::vector particles = tree.query(10);

```



For an arbitrary AABB:



```cpp

// Set the AABB bounds.

std::vector lowerBound({5, 5}};

std::vector upperBound({10, 10}};



// Create the AABB.

aabb::AABB aabb(lowerBound, upperBound);



// Query the tree for overlap with the AABB.

std::vector particles = tree.query(aabb);

```



## Tests

The AABB tree is self-testing if the library is compiled in development mode, i.e.



```bash

make devel

```



## Disclaimer

Please be aware that this a working repository so the code should be used at

your own risk.



It would be great to hear from you if this library was of use in your research.



Email bugs, comments, and suggestions to [email protected].