The Functional Gradient Description Method of Space Coordinate Transformation

Abstract

The 3-dimension coordinate datum transformation is very important in geomatics. Generally, there are small angle coordinate transformation model (i.e., Bursa model, et al) and large angle coordinate transformation model (i.e., Rodrigo matrix model, et al). However, these papers only focus on the fixed and static situation regardless of the small or large angle. This paper proposes the conception of coordinate transformation gradient field,which can realize the space coordinate transformation from small angle to arbitrary angle, and from static to dynamical. Based on the equivalent characteristics of the unit quaternion rotation matrix and the Rodrigo matrix, through studying the mathematical relationship between the spatial coordinate transformation and the functional gradient, we derive a functional gradient expression of the arbitrary transformation formula in space. The study shows that the essence of spatial coordinate transformation is a kind of potential field mathematically. This potential field conception can unify all the space coordinate transformations. It is the theoretical foundation for the further study of time continuous space coordinate transformation,and this study gives a new solution for the attitude determination of motion carriers.

Keywords

• |spatial coordinate transformation|functional gradient field|complete equivalence