The quasi-frame of the rational and polynomial Bezier curve by algorithm method in Euclidean space

PurposeFrames play an important role in determining the geometric properties of the curves such as curvature and torsion. In particular, the determination of the point types of the curve, convexity or concavity is also possible with the frames. The Serret-Frenet frames are generally used in curve theory. However, the Serret-Frenet frame does not work when the second derivative is zero. In order to eliminate this problem, the quasi-frame was obtained. In this study, the quasi frames of the polynomial and rational Bezier curves are calculated by an algorithmic method. Thus, it will be possible to construct the frame even at singular points due to the second derivative of the curve. In this respect, the contribution of this study to computer-aided geometric design studies is quite high.Design/methodology/approachIn this study, the quasi frame which is an alternative for all intermediate points of the rational Bezier curves was generated by the algorithm method, and some variants of this frame were analyzed. Even at the points where the second derivative of such rational Bezier curves is zero, there is a curve frame.FindingsSeveral examples presented at the end of the paper regarding the quasi-frame of the rational Bezier curve, polynomial Bezier curve, linear, quadratic and cubic Bezier curves emphasize the efficacy and preciseness.Originality/valueThe quasi-frame of a rational Bezier curve is first computed. Owing to the quasi frame, it will have been found a solution for the nonsense rotation of the curve around the tangent.