Rotations as a Special Case of Vector Transformations
Rotations as a Special Case of Vector Transformations
... with Applications to Game World Modeling
Rotations, as a class of affine vector transformations, represent rigid body transformations that change the direction of a vector within a given plane.
Efficient and accurate handling of such transformations poses challenges not only for game and animation engines, but also for simulations and real-time systems, where the limitations of floating-point arithmetic and operation throughput can negatively impact the computational outcomes.
My article introduces vector rotation with a focus on the relations between its two- and three-dimensional formulations.
We begin in the two-dimensional plane by deriving classical rotation matrices from linear combinations in orthogonal bases.
We then generalize this approach by introducing Rodrigues' rotation formula, which provides a compact expression for rotation around arbitrary axes in three-dimensional space. We verify that the standard rotation matrices emerge as special cases from this general form.
Finally, we illustrate the application of vector rotation in the context of embedded coordinate systems within large scene graphs, demonstrating how transformations propagate across hierarchical structures in real-time environments such as video games.