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.

Number of pages | 19 |
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Publication status | Published - 1998 |
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