Log-aesthetic curve (LAC) is a curve family composed of transcendental curves that includes logarithmic spiral, clothoid, circle involute and Nielsen's spiral. They have linear logarithmic curvature graphs (LCGs) and are highly aesthetic. In order to implement G~2 LAC in industrial design successfully, one needs guidance on the existence and uniqueness whether a LAC segment satisfy given G~2 Hermite data. This paper focuses shows the existence and uniqueness of solution for single segment G~2 LAC. A LAC equation that incorporates both start and end curvatures, and end tangential angle is first derived. Then, the end points of the LAC segments are calculated using the derived LAC equation, which is also a representation of the solution region of LAC given a set of G~2 Hennite data. The derived function is investigated for its existence and uniqueness. It is shown that the solution region is a curve that do not self-intersect anywhere, thus the solution of single segment G~2 LAC is always unique.
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