Collision Detection

Collision detection is typically a two-stage process for efficiency. The broad phase quickly identifies pairs of objects that might be colliding using spatial partitioning data structures, culling obviously separate objects. Common broad-phase algorithms include:

• Sweep and Prune: Sorts object bounding boxes along axes.
• Spatial Hashing: Places objects into a grid of cells.
• Bounding Volume Hierarchies (BVH): A tree structure organizing objects.

The narrow phase then performs precise geometric intersection tests on the candidate pairs from the broad phase, using algorithms like the Separating Axis Theorem (SAT) for convex shapes or Gilbert–Johnson–Keerthi (GJK) for distance/intersection computation.

Exact geometric intersection tests are computationally expensive. Bounding volumes are simplified shapes that encapsulate complex geometry for fast preliminary checks. Common types include:

• Axis-Aligned Bounding Boxes (AABB): Simple rectangles/cubes aligned to world axes; intersection test is a cheap range check.
• Bounding Spheres: Defined by a center and radius; intersection test checks distance between centers.
• Oriented Bounding Boxes (OBB): Boxes aligned to the object's rotation, tighter fit than AABBs but more complex math.

A Bounding Volume Hierarchy (BVH) is a tree structure where each node has a bounding volume containing its children. This allows for logarithmic-time search to reject entire groups of objects that cannot be colliding, making it the gold-standard acceleration structure for complex scenes in ray tracing and physics.

Standard discrete collision detection checks for intersections at discrete time steps (e.g., every frame). For fast-moving objects, this can cause tunneling, where an object passes through another between checks. Continuous Collision Detection (CCD) solves this by modeling the continuous motion of objects between time steps.

CCD algorithms compute the time of impact (TOI) by solving for when the swept volumes of moving shapes intersect. Techniques include:

• Linear CCD: Assumes linear motion between frames; common for spheres and capsules.
• Conservative Advancement: Iteratively advances time until contact is found.

CCD is critical for simulation stability in robotics, bullet physics, and games with fast projectiles, but is significantly more computationally intensive than discrete detection.

These data structures organize objects in space to minimize the number of pairwise collision checks. Key types include:

• Uniform Grid: Space is divided into fixed-size cells. Objects are assigned to cells they overlap. Checks are limited to objects in the same or adjacent cells. Simple but inefficient for uneven object distributions.
• Quadtree (2D) / Octree (3D): Recursively subdivides space into quadrants/octants. Subdivision stops when a node contains fewer than a threshold number of objects. Efficient for sparse, clustered scenes.
• k-d Tree: A binary tree that recursively splits space along alternating axes. Excellent for nearest-neighbor searches but can be costly to update for dynamic scenes.
• Bounding Volume Hierarchy (BVH): As mentioned, organizes objects hierarchically, not space. Often the best choice for dynamic scenes as it can be refitted or rebuilt efficiently.

The complexity of collision detection depends heavily on shape geometry. A convex shape has no interior angle greater than 180 degrees; a line segment between any two points inside the shape lies entirely within it (e.g., spheres, boxes, capsules).

Convex collision algorithms like GJK and EPA (Expanding Polytope Algorithm) are efficient and robust. Most physics engines require convex shapes for primary collision geometry.

A concave (or non-convex) shape has indentations or holes (e.g., a bowl, a torus, a complex game asset). Detecting collisions directly with concave shapes is complex. The standard solution is convex decomposition, where the concave mesh is broken down into a collection of convex hulls. Collision detection is then performed on these convex parts, and results are aggregated.

A Signed Distance Field is a powerful volumetric representation for collision and rendering. It is a scalar field where the value at any point in 3D space is the shortest signed distance to the surface of an object. The sign indicates if the point is inside (negative) or outside (positive) the object.

For collision detection, SDFs enable:

• Proximity Queries: Finding the distance to a surface is trivial—just sample the field.
• Collision Normal: The gradient of the SDF at a point gives the surface normal, crucial for collision response.
• Complex Shapes: Can represent any shape, including smooth, organic, or Boolean combinations, with a uniform query cost.

SDFs are memory-intensive but are central to sphere tracing in rendering and are increasingly used in physics for soft bodies and particle interactions. They are a key component of Neural Radiance Fields (NeRFs) for implicit 3D scene representation.

Collision response is the computational process that follows collision detection, calculating and applying the appropriate forces, impulses, or positional corrections to simulate realistic physical interaction. It determines whether objects bounce, slide, stick, or break apart.

• Key methods include penalty forces (spring-like repulsion) and impulse-based methods (instantaneous velocity changes).
• Coefficients of restitution and friction are applied to model material properties like bounciness and sliding resistance.
• Essential for creating believable behavior in games, robotics, and visual effects.

A Bounding Volume Hierarchy is a tree data structure used to accelerate spatial queries like collision detection and ray tracing. It works by recursively partitioning a set of objects into nested volumes.

• Common bounding volumes include Axis-Aligned Bounding Boxes (AABB), Spheres, and Oriented Bounding Boxes (OBB).
• The hierarchy allows the simulation to quickly reject large groups of objects that are far apart (broad phase), before performing expensive exact intersection tests on nearby pairs (narrow phase).
• Critical for achieving real-time performance in complex scenes with thousands of objects.

Constraint solving is the process of finding a configuration for a system of simulated objects that satisfies a set of defined relationships or limits. While collision detection finds intersections, constraints maintain desired separations or connections.

• Joint constraints (e.g., hinge, ball-and-socket) restrict the relative motion between connected rigid bodies.
• Contact constraints are generated from collision detection to prevent interpenetration and model friction.
• Solvers iteratively adjust positions and velocities to satisfy all constraints simultaneously, often using methods like Sequential Impulse or Projected Gauss-Seidel.

A Signed Distance Field is a volumetric representation where the value at any 3D point is the shortest distance to an object's surface, with sign indicating inside (negative) or outside (positive).

• Collision detection with SDFs is efficient: testing a point involves a simple lookup and sign check to determine containment.
• Precise contact normals can be derived from the SDF's gradient (direction of greatest increase).
• Widely used in ray marching for rendering, but also in robotics for efficient proximity queries and soft body simulations.

Numerical stability in simulation refers to the property where small computational errors (e.g., floating-point rounding) do not accumulate and cause the solution to diverge to unrealistic states or "explode."

• Collision detection is highly sensitive to numerical precision; tolerances (epsilons) must be carefully tuned to avoid missed collisions or jitter.
• Integration methods like Implicit Euler are used for stiff systems (e.g., springs) because they remain stable with larger time steps, unlike Explicit Euler.
• A core concern for deterministic simulation, where identical inputs must produce identical outputs across different hardware.