Summer of Math Exposition

Billboarding in 3D graphics is a technique that has evolved significantly from its early use in video games to modern applications. Originally used in games like Super Mario 64 to conserve computational resources, billboarding involves displaying a 2D image that always faces the camera, creating the illusion of a 3D object. This technique has two main types: viewpoint-oriented, which directly faces the camera, and screen-aligned, which remains parallel to the camera’s viewing plane. The mathematics behind billboarding involves calculating vectors, cross products, and rotation matrices to ensure the billboard maintains its orientation relative to the camera. As scenes grew more complex, performance optimization techniques like culling and occlusion culling were developed to render only visible billboards, reducing computational load. Despite its effectiveness, traditional billboarding can struggle with complex scenes or objects viewed from multiple angles. Modern approaches, such as Neural Radiance Fields (NeRFs) and implicit neural representations, address these limitations by using neural networks to reconstruct 3D scenes from multiple images, allowing for more realistic renderings and interpolation between captured images. This evolution from simple 2D sprites to advanced neural network techniques highlights the significant progress in graphics rendering and the ongoing innovation in creating immersive 3D environments.