As we have seen in Example 4.6.1, we may introduce
numerical errors during the formulation of the governing equations
in a shape interrogation problem.
Formulation of the governing simultaneous nonlinear polynomial
equations in multivariate Bernstein form for shape interrogation
usually involves arithmetic operations in Bernstein form (see
Sect. 1.3.2) starting from the given input
Bézier curve or surface. Therefore to achieve an
accurate formulation
[255,254], we suggest:
Use of rational arithmetic (RA) or rounded
interval arithmetic (RIA)
[273] (see also Sect. 4.8), if the control points of
the given curve or surface are floating point numbers to maintain a
pristine or guaranteed precision
statement of the problem, respectively.
Use of RIA if the
control points of the given curve or surface are irrational numbers
to avoid any
numerical contamination by standard FPA. This
happens, for example, when the curve or surface is rotated, since the rotation
matrix involves
cosines and sines, which are generally irrational.
Conversion of the coefficients of the nonlinear equations in
Bernstein form into intervals with FP number
boundaries if rational arithmetic is used in the formulation.
Rational and rounded interval arithmetic operations can be implemented
effectively in object-oriented languages such as C++.
Computation time comparison for various combinations of
arithmetic for the formulation of the governing equations and their
solution is presented in [246].
