Ribs and fans are interesting geometric entities that are derived from an ordinary $B'{e}zier$ curve or surface. A rib itself is a $B'{e}zier$ curve or surface with a lower degree than the given curve or surface. A fan is a vector field whose degree is also lower than its origin. First, we present methods to transform the control points of a $B'{e}zier$ curve or surface into the control points and vectors of its ribs and fans. Then, we show that a $B'{e}zier$ curve of degree n is decomposed into a rib of degree (n-1), a fan of degree (n-2), and a scalar function of degree 2. We also show that a $B'{e}zier$ surface of degree (m, n) is decomposed into a rib of degree (m-1, n-1) and three fans of degrees (m-1, n-2), (m-2, n-1), and (m-2, n-2), respectively. In addition, the lengths of the fans are further controlled by scalar functions of degree 2 and (2, 2). We present relevant notations and definitions, introduce theories, and present some of design examples.