In this paper, we investigate the surface-dependent growth model in
Euclidean 3-space. The surface-dependent model is developed to model the
kinematics of surface growth for objects that can be generated by the
curves on the surface, such as parasites and plants. This paper includes
two main purposes for this model. The first one is to parametrize this
model by using the quaternions and homothetic motions.Furthermore, we
express the matrix representations of the surface-dependent growth
model. The second one is to construct the surface-dependent growth model
by using the growth velocity components related to the Darboux frame at
each point of the generating curve. Moreover, to support the theory
studied in the paper various illustrated examples are presented.