Constructing Invariant Fairness Measures for Surfaces

This paper presents a general method which from an invariant curve fairness measure constructs an invariant surface fairness measure. Besides the curve fairness measure one only needs a class of curves on the surface for which one wants to apply the curve measure. The surface measure at a point is found by integrating the curve measure at the point over all curves in the class which passes through the point.The method is applied to the cases where the class of curves consists of all plane intersections, and the curve measure is the square of the curvature respectively the square of the curvature variation.The method is extended to the case where one considers, not the fairness of one curve, but the fairness of a one parameter family of curves. Such a family is generated by the flow of a vector field, orthogonal to the curves. The first, respectively the second order derivative along the curve of the size of this vector field is used as the fairness measure on the family.Six basic 3rd order invariants satisfying two quadratic equations are defined. They form a complete set in the sense that any invariant 3rd order function can be written as a function of the six basic invariants together with the mean and Gaussian curvature. Furthermore the geometry of a plane intersection curve is studied, and the derivative with respect to the arc length is determined for the total curvature, the normal curvature, the geodesic curvature and the geodesic torsion.