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It is known that if the degree of the typical plane Bezier 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 α: the slope of the logarithmic curvature graph. In this paper we define the non-stationary typical Bezier curve by making the transition matrix of the typical Bezier 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 α and β: 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.

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