Short answers, pulled from the story.
Rendering in computer graphics is the process of generating an image from input data such as 3D models. A software component that performs this task is called a rendering engine, render engine, graphics engine, or renderer. The field draws from optics, mathematics, and software engineering, with applications in video games, film visual effects, medical diagnosis, and design visualization.
Real-time rendering generates and displays images immediately, fast enough to create the impression of motion or animation, as required by video games. Offline rendering, also called pre-rendering, generates frames for later viewing and can use slower, higher-quality methods. Interactive applications must primarily use real-time rendering, though they may incorporate pre-rendered content.
Path tracing is a rendering technique that uses Monte Carlo integration to compute the average brightness of sampled light paths traveling from a light source to the camera. It was proposed and named in 1986 by Jim Kajiya in the same paper that introduced the rendering equation. Monster House, the first feature film rendered entirely using path tracing, was released about 20 years after Kajiya's original proposal.
Radiosity, first proposed in 1984, splits surfaces and lights in a scene into patches and computes form factors representing the fractions of light transferred between each pair of patches. These fractions are assembled into a matrix equation solved using linear algebra. The resulting irradiance values can be precomputed and stored in textures called irradiance maps for use in real-time rendering.
The rendering equation was introduced in 1986 by Jim Kajiya in the same paper that proposed path tracing. It provided ray tracing with a rigorous mathematical foundation by expressing how light is emitted, reflected, and propagated in an environment. Path tracing was introduced as a statistically unbiased method for solving this equation.
Neural networks are widely used for denoising path-traced images, reducing the number of sample paths needed to achieve acceptable quality. They are also used to encode bidirectional reflectance distribution functions and to represent neural radiance fields, which encode the color and intensity of light at all points in a volume. As of 2023, 3D Gaussian splatting emerged as a neural-network-adjacent approach representing scenes as fuzzy, partially-transparent blobs rather than discrete points.