Engine Architecture and Math Foundations

Describe the overall architecture of a real-time physics engine and its main subsystems.

Implement a fixed-timestep simulation loop decoupled from the rendering loop.

Set up a basic OpenGL debug-drawing system to render lines, points, and transforms for visualization.

Implement a 3D vector and matrix math library with essential operations for physics simulation.

Implement conversions between axis-angle, rotation matrix, and quaternion representations of orientation.

Implement quaternion multiplication, normalization, and the rotation of a vector by a quaternion.

Compute the time derivative of a quaternion from an angular velocity vector for integration.

Explain the gimbal lock problem and why quaternions are preferred for representing orientation in 3D physics.