Input of log-aesthetic curve segments with inflection end points and generation of log-aesthetic curves with G² continuity

The log-aesthetic curves include the logarithmic (equiangular) spiral, clothoid, and involute curves. Although most of them are expressed only by an integral form of the tangent vector, it is possible to interactively generate and deform them and they are expected to be utilized for practical use of industrial and graphical design. However, their input method proposed so far by use of three so-called control points can generate only an aesthetic curve segment with monotonic curvature variation and can not create a curve with the curvature-extremal point or the inflection point. Hence at first we propose a technique to input a log-aesthetic curve segment with an inflection point at its end. Then we propose a method to generate an log-aesthetic curve with G² continuity from a sequence of 2D points input with, for example, a liquid crystal pen tablet.