Accretive Darboux growth in Lorentz-Minkowski spacetime

Abstract

It is known that the geometric methods are used in most fields in natural sciences. Kinematics on curves and surfaces as one of these methods is an essential tool for investigating the growth of some biological objects. In this study, the time-dependent and quaternionic models of accretive growth are considered. For this, first, it is defined as a growth velocity in the direction of the Darboux vector at every point on a spatial non-null curve in three-dimensional Lorentz-Minkowski spacetime. Second, accretive surface growth is presented through quaternions. With the help of the quaternionic form of the growth model, we reach that a point (cell tract) on the accretive surface makes a homothetic motion. Also, several examples and visualizations are given to support the theory.