Accretive Darboux growth along a space curve

In this article, we give a mathematical framework to model the kinematics of the surface growth of objects such as some crustacean creatures. For this, a growth velocity in the direction of the Darboux vector field is defined at each point on a spatial generating curve. A local orthonormal frame (alternative moving frame) is added to each point of the generating curve and a velocity is given in terms of local coordinate directions to obtain a system of differential equations. Using the analytical solutions of this system, various surface examples, including some seashells are provided and the shapes of these surfaces are illustrated.