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A curve is considered fair if it consists of continuous and few monotonic curvature segments. Polynomial curves such as Bezier and B-spline curves have complex curvature function, hence the curvature profile may oscillate easily with a little tweak of control points. Thus, bending energy and shear deformation energy are common fairness metrics used to produce curves with monotonic curvature profiles. The fairness metrics are used not just to evaluate the quality of curves, but it also aids in reaching to the final design. In this paper, we propose two types of fairness metric functionals to fair plane curves defined by the similarity geometry invariants, i.e. similarity curvature and its reciprocal to extend a variety of aesthetic fairing metrics. We illustrate numerical examples to show how log-aesthetic curves change depending on and G1 constraints. We extend LAC by modifying the integrand of the functionals and obtain quasi aesthetic curves. We also propose -curve to introduce symmetry concept ...

Fairness metric of plane curves defined with similarity geometry invariants