ComputerAided_Design_Engineering_amp_Manufactur.pdf

Similarly, if P i is in the sector P 2O 2Q, the distance D i can be obtained. The only exception is when P i lies
outside either of these sectors, in which case D i is defined as the minimum distance between P i and either
of the end points P 1 and P 2.
Having the deviation formula available, we can therefore find the optimal biarc which passes through
two end points P 1 and P 2 by formulating it as a least square problem:
© 2001 by CRC Press LLC
�
Minimize Di( �)
i
( ) 2
(2.8a)
subject to ( �min �� ��max)
(2.8b)
where �min and �max are constrained by the Equation (2.6)
This is a nonlinear optimization problem of one variable with simple-bounded constraints and can
be easily solved by any of the commonly available optimization algorithms such as the sequential quadratic
programming method.
Biarc Curve Fitting to a Large Set of Discrete Data Points
Having developed the formulation of the biarc curve and the optimal biarc sub-problem, we are now in
the position to tackle the approximation of a large set of data points so that the resulting biarc curves
are C continuous and within a specified tolerance of the data points. We assume the data points are
given in the sequence that traces out the desired profile. The approach combines both a search process
and an optimization process. In the search process, the algorithm starts from an initial point and searches
forward to find the largest number of consecutive data points that can be approximated by a single biarc
and still not violate the tolerance constraint. The search process uses a binary search approach, whereas
finding the optimal biarc within the search steps solves an optimal biarc sub-problem as formulated by
the model in the previous section.
The algorithm can be summarized as follows:
if (max_deviation_opt_biarc(1,n) � tolerance)
add optimal_biarc(start,end);
else
start � 1
while (start � n)
end � longest_end(start, start�1, n);
add optimal_biarc(start,end);
start � end;
endwhile
endif
where optimal_biarc returns the optimal biarc between two end points and longest_end is the binary
search to find the furthest point that does not violate the tolerance constraint.
The routine longest_end (Integer Routine longest_end(start, low, high)) can be summarized as
follows:
deviation � max_deviation_opt_biarc(start, high)
while (deviation � tolerance)
high � (low � high)/2
if (high � low) return high;
else deviation � max_deviation_opt_biarc(start, high)
endif
endwhile
return high
where max_deviation_opt_biarc solves the optimal biarc subproblem and returns the maximum deviation
of the optimal biarc.
1