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**Proceedings of CAD'17, 2017, 311-316 **

**Fairness Metric of Plane Curves Defined with Similarity Geometry Invariants **

**Abstract.** A curve is considered fair if it consists of continuous and few monotonic curvature segments. Polynomial curves such as Bézier 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. Curve synthesis is a process of generating curves with a well-defined Cesáro equation, which describes the curvature κ of a curve as a function of its arc length s. Log-aesthetic curves (LAC in short) are generated with a Cesáro equation derived by letting the Logarithmic curvature graph (LCG) as a linear function with the gradient as a. This curve has gained its momentum in design environment and now is it used for automobile and architecture design.

**Keywords.** Fairness metric, Plane curve, Similarity geometry, Invariants, Log-aesthetic curve

**DOI:** 10.14733/cadconfP.2017.311-316