基于Groebner基方法的射影不变曲线的构造

Construction of projective invariant curve based on Groebner basis method

提出利用Groebner基方法对射影不变曲线的构造方程进行求解。首先构造出二次曲线的一般方程且其系数用参数表示;然后,利用拉格朗日乘子法得到满足最优拟合曲线时的条件,该问题由7个三次方程构成,其一般形式的解最多可以达到2187个。利用多项式环字典序下的Groebner基具有消元的性质将原问题转化为三角型方程组,进而求解。讨论了两组点集通过该类方法拟合出的不变曲线,并用实例分析了曲线在射影变换时具有拓扑结构和次数不变性。

In this paper,the Groebner basis method is proposed to solve the construction equations of projective invariant curves.First,the general equation of conic is constructed and its coefficients are expressed by parameters.Then,using the Lagrange multiplier method,the conditions for satisfying the optimal fitting curve are obtained.The problem consists of seven cubic equations,and the general form of the solution can reach 2187 at most.The original problem is transformed into a trigonometric system of equations by using the elimination property of Groebner basis under the dictionary order of polynomial rings.The invariant curves fitted by two point sets using this method are discussed,and the topological structure and degree invariance of curves in projective transformation are analyzed with examples.