# Animation
📗 The movement of objects in games can be represented by vector-valued functions of time, in 2D \(p\left(t\right) = \begin{bmatrix} x\left(t\right) \\ y\left(t\right) \end{bmatrix}\), and in 3D \(p\left(t\right) = \begin{bmatrix} x\left(t\right) \\ y\left(t\right) \\ z\left(t\right) \end{bmatrix}\), represent the position of an object or agent at time \(t\). The velocity and acceleration can be represented the same way as \(p'\left(t\right)\) and \(p''\left(t\right)\).
📗 In computer graphics, the paths are usually represented by splines formed by cubic Bezier curves, and each piece is given by \(\left(1 - t\right)^{3} p_{0} + 3 \left(1 - t\right)^{2} t p_{1} + 3 \left(1 - t\right) t^{2} p_{2} + t^{3} p_{3}\) where \(p_{0}\) is the starting position, \(p_{3}\) is the end position, and \(p_{1}, p_{2}\) are two controls points that are not necessarily on the curve. Splines are used to compute \(p_{1}, p_{2}\) in a way that the velocity and acceleration functions are continuous at the points where multiple Bezier curves connect.