“Where there is matter, there is geometry.”
The triangle is a basic atomic rendering primitive. It is what graphics hardware is tuned to rapidly turn into shaded fragments and put into the framebuffer. However, objects and animation paths that are created in modeling systems can have many different underlying geometric descriptions. Curves and curved surfaces can be described precisely by equations. These equations are evaluated and sets of triangles are then created and sent down the pipeline to be rendered.
The beauty of using curves and curved surfaces is at least fourfold: (1) They have a more compact representation than a set of triangles, (2) they provide scalable geometric primitives, ...