Usually, the animation curve is specified by key frames associated with animation time. The animation is then projected by advancing the time factor and interpolating along the curve with a monotonously increasing parameter extracted from the animation time factor. Interpolation of a uniform scale factor is the most simple one.
Translation is just a bit more complicated.
As for rotation - this is an entirely different story. The rotation along the axis in R 3 is highly dependent. This is why the the order of axes in the rotation sequence is very important. But even if you maintain the right order of primitive rotation actions, an effect called Gimbal Lock creeps in and prevents the system from being truly axis independent. The Gimbal lock effect is described in detail in the Image Synthesis course.