Non-stationarization of the Typical Curves and its Extension to Surfaces

Abstract

It is known that if the degree of the typical plane Bézier curve is increased infinitely, the curve will converge to the logarithmic (equiangular) spiral. The logarithmic spiral is one of log-aesthetic curves and they are formulated by : the slope of the logarithmic curvature graph. In this paper we define the non-stationary typical Bézier curve by making the transition matrix of the typical Bézier curve non-stationary and dependent on each side of the control polyline and defining the transition matrix in the Frenet frame. We propose a method that generates a curve such that if its degree is increased infinitely it will converge to a log-aesthetic curve with arbitrary  and : the slope of the logarithmic torsion graph in case of the space curve, by controlling the relationship between the rotation angle and the scaling factor. Furthermore we extend the non-stationarization for free-form surfaces and propose the non-stationary typical surface with the unit scaling factor.

Quasi-Log-Aesthetic Curves in Polynomial Bézier Form

“…This paper proposed quasi-log-aesthetic curves in polynomial Bézier form using Taylor polynomials. We have shown that quasi-log-aesthetic curves are better approximation than the curves generated by discretization [6]. Because of the limitation of polynomial curves, quasi-log-aesthetic curves are not good approximations when log-aesthetic curves are close to circular arcs.…”

“…We compare quasi-log-aesthetic curves with Bézier curves generated by Miura's method [6] using logarithmic curvature graphs. Miura's method discretizes log-aesthetic curves, and then using the discretized equation, Bézier control points are placed on the curve either by a constant arc length or by a constant tangential angle.…”

“…This paper proposes quasi-log-aesthetic curves in polynomial Bézier form and compares them with the curves generated by Miura's method [6] and quasi-log-aesthetic curves in rational cubic Bézier form [10]. We show that quasi-log-aesthetic curves are better approximation than the curves generated by Miura's method.…”

“…Miura et al have proposed a method for approximating log-aesthetic curves with polynomial curves by discritizing the equations of log-aesthetic curves [6]. Miura's work was motivated by class A Bézier curves [2] proposed by Farin.…”