Real-time rendering under distant illumination with conformal geometric algebra

Precomputed radiance transfer (PRT) methods established for handling global illumination (GI) of objects from area lights in real time and many techniques proposed for rotating the light using linear algebra rotation matrices. Rotating area lights efficiently are crucial part for computer graphics since it is one of the main components of real-time rendering. Matrices commonly used for handling such rotations are not quite efficient and require high memory consumption; as a result, the need for proposing new more efficient rotation algorithms has been established. In this work, we use the conformal geometric algebra (CGA) as the mathematical background for GI in real-time under distant Image-based lighting (IBL) illumination, for diffuse surfaces with self-shadowing by efficiently rotating the environment light using CGA entities. Our work is based on spherical harmonics (SH), which are used for approximating natural, area-light illumination as irradiance maps. Our main novelty is that we extend the PRT algorithm by representing SH for the first time with CGA.The main intuition is that SH of band index 1 are represented using CGA entities and SH with band index larger than 1 are represented in terms of CGA-SH of band 1. Specifically, we propose a new method for representing SH with CGA entities and rotating SH by rotating CGA entities. In this way, we can visualize the SH rotations, rotate them faster than rotation matrices, and we provide a unique visual representation and intuition regarding their rotation, in stark contrast to usual rotation matrices, and we achieve consistently better visual results from Ivanic rotation matrices during light rotation. Via our CGA expressed SH, we provide a significant boost on the PRT algorithm since we represent SH rotations by CGA rotors (4 numbers) as opposed to 9x9 sparse matrices that are usually used. With our algorithm, we pave the way for including scaling (dilation) and translation of light coefficients using CGA motors.