Abstract
In this paper, similarity groups in the complex plane C, polynomial curves and complex Bezier curves in C are introduced. Global similarity invariants of polynomial curves and complex Bezier curves in C are given in terms of complex functions. The problem of similarity of two polynomial curves in C are solved. Moreover, in case two polynomial curve (complex Bezier curve) are similar for the similarity group, a general form of all similarity transformations, carrying one curve into the other curve, are obtained.
Highlights
The invariance is a very important tool in areas data registration, object recognition, computer aided design applications
The aim of registration or object recognition is to ...nd the corresponding relationship between two point sets(or two curves) and compute the transformation which aligns two point sets(or two curves)(see [1,2,3,4]) Generally, Euclidean invariant features are used in above mentioned methods and a representation of polynomial curve or Bézier curve in the complex plane C are a useful method to investigate of their global invariants. In [16], taking customary rational Bézier curves in complex plane, complex rational Bézier curves are investigated
For Bézier curves, rational curves and implicit algebraic curves, detecting whether two plane curves are similar by an orientation preserving similarity transformation is important
Summary
The invariance is a very important tool in areas data registration, object recognition, computer aided design applications. This paper presents the similarity conditions of two point sets and the similarity conditions of two polynomial paths(two complex Bézier curve) in the complex plane. Z(u); W (u) for the groups GM (C ) and GM +(C ) is reduced to the problem of similarity of two polynomial curves(or two complex Bézier curves) Z(u); W (u) for the groups M (C ) and M +(C ), resp. For the groups of Euclidean motions M (n) and Euclidean rigid motions M +(n) in the n-dimensional Euclidean space, the problems of equivalence two Bézier curves of degree m and its global invariants are investigated in [15]. The paper contains solutions of problems of global similarity of complex Bézier curves and polynomial curves for the above mentioned groups without using differential invariants of a complex Bézier curve and a polynomial curve.
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