To extend the proposed framework for non-spherical objects like an American football, several modifications would be necessary to accommodate the unique dynamics and interactions of such shapes. First, the concept of roto-translational invariance may need to be redefined, as non-spherical objects do not exhibit uniform behavior under rotation and translation. Instead, a more complex representation that accounts for the object's orientation, shape, and aerodynamic properties would be essential.
One approach could involve developing a multi-faceted representation that captures the object's geometry and its interaction with the environment. This could include using computational fluid dynamics (CFD) simulations to model the aerodynamic forces acting on the object, which would vary significantly based on its orientation and spin. Additionally, incorporating machine learning techniques that can learn from high-dimensional data, such as convolutional neural networks (CNNs), could help in extracting features relevant to the object's dynamics.
Furthermore, the Gram-Schmidt process may need to be adapted or replaced with a method that can handle the complexities of non-spherical shapes. For instance, using a combination of principal component analysis (PCA) and custom feature extraction techniques could help in deriving representations that are sensitive to the object's unique dynamics. Finally, real-world data collection would need to be expanded to include various launch angles, spins, and velocities specific to non-spherical objects to ensure the model is trained effectively.