# Quadtree

> Mathematical theory and implementation details for Quadtree

## Mathematical Theory

**Mathematical Foundation**:

- **Recursive Subdivision**: Each node represents a rectangular region that can be subdivided into four equal quadrants
- **Subdivision Condition**: Subdivide when number of objects exceeds capacity threshold
- **Tree Height**: $h = \lceil \log_4(n) \rceil$ where $n$ is number of objects
- **Query Optimization**: $O(\log n + k)$ where $k$ is number of objects in query region

**Key Properties**:

- **Spatial Partitioning**: Efficiently organizes 2D space
- **Adaptive Structure**: Subdivides only where needed
- **Range Queries**: Fast spatial queries and collision detection

## Implementation

For implementation details and usage examples, see the source code in the `src/` directory.

## API Reference

```typescript
// Import the algorithm
import { Quadtree } from '@entropy-tamer/reynard-algorithms';
```

## Performance

See the [Performance Analysis](../performance/) section for detailed benchmarks and optimization strategies.