Curve and surface fitting based on the nonhomogeneous linear differential system

Abstract

In this paper, we establish the connection between exponential representation of curve (surface) and the classical linear differential system. That is, a curve (surface) can be described by a vector-valued function X(t)=eAtX0(t) (X(s,t)=eAtX0(s,t)), which can be viewed as a solution of the ordinary (partial) linear differential system X˙(t)=AX(t)+f(t) (∂X(s,t)∂t=AX(s,t)+f(s,t)). Further, we propose two algorithms to fit the given discrete data points by the solution curve (surface) of the ordinary (partial) linear differential system with nonhomogeneous term. We also show that the solution of the linear differential system described by X˙(t)=AX(t)+f(t) (∂X(s,t)∂t=AX(s,t)+f(s,t)) is not unique, so we find a way to determine an optimal A based on the flow field information from the given data points. Numerical examples illustrate the effectiveness of the proposed algorithms.