The use of interval Bézier curves for a complete description of approximation errors was proposed by Sederberg and Farouki[5]. The basic idea is to extend splines to polynomials whose coefficients are intervals with well defined arithmetic operations. Such splines define a region in space rather than a curve. This notion captures very nicely the semantics of a tolerance specification, especially when it is generalized in 3D: if the assumption is made that the sensing error is Gaussian, then it can be described it by an ellipsoid around each sensed point (using a step value). Thus, along a sensed toolpath, an offset surface is produced (see [3]). We have only assumed that the enclosing envelopes are described by ellipses in planes orthogonal to the toolpath. Hence our algorithm allows for representing volumetric error and can easily be extended to other shapes than ellipses - which means different offset surfaces. This approach will require the ability to answer the question: is one ellipse inside the other one ? as fast as possible - when they are in the same plane. The final test will be to check the reliability of the proposed algorithm on real sensed data, along manufacturing toolpaths on parts that are inspected.
The algorithm uses a property that is associated with curves of the same
degree, which is the basis of interval splines.
Since a Bézier curves of degree is deduced from the control point
by the recursive equation (see [4]):
For curves of same degree, if the corresponding control points
are on a line (resp. on a plane), then during this
recursive process each corresponding
will also be on a line (resp. on a plane), hence for all
the different evaluations (
,
...) will give points on
a line (resp. on a plane). An easy way to ensure that the control points are on
a line is to have initial points on a line too, since the control points
are deduced by a linear operator.