ComputerAided_Design_Engineering_amp_Manufactur.pdf

FIGURE 2.14(b) Circular approximation of Bezier curve B. Tolerance : 0.9, convergence criteria : 0.09, number of
circular arcs: 3, processor time: 0.33.
biarcs. A C-shaped biarc has both the centers of the circular arcs on the same side of the curve while an
S-shaped biarc has them on opposite sides.
A few methods have been proposed to fit biarcs to approximate or interpolate a set of two-dimensional
data points. Some of these methods assume that the biarcs have to pass through all the data points while
the rest only require that the data points are within a specified tolerance of the fitted biarcs. Because
defining a biarc given only the end-point positions and tangent directions is an under-constrained
problem, researchers have proposed methods to introduce the additional constraint required to uniquely
define these biarcs by considering minimal total curve length, 23,33 minimal spline strain energy, 17,34
minimal radii difference, 14,15 minimal radii ratio, 15 or minimal curvature difference. 17,21,23 Ong et al. 35 had
proposed minimization of total undercut area as an optimizing criterion, but the method was applied
to approximating B-spline curves—though it can be easily modified to a set of discrete data points as
well. In all of the methods described above, except for those of Meek and Walton21 and Ong et al., 35 the
number of biarcs has to be predetermined beforehand and is usually overspecified.
Formulation of Biarc Curve
Before we describe our approach, 36 we will develop a formulation of the biarc curve that is necessary for
the rest of the analysis. For practical purposes, we can assume that the spanned angles of the two circular
arcs of a biarc do not exceed 2�. Without loss of generality, we can transform the two end points P1 and
P2 to lie on the horizontal line with P1 as the origin and P2 on the positive axis. The two end tangents t1 and t2 form angles � and �, respectively, to the vector from P1 to P2, the angle being positive if it is in
the counterclockwise direction and negative in the clockwise direction. An additional variable � which
denotes the angle from P1 to Q, the joining point of the two circular arcs, is introduced to assist in the
formulation (see Figure 2.15) From the figure, it can be seen that to maintain C continuity at Q, the two
arc centers O1 and O2 must be collinear with Q. This condition, together with the continuity requirement,
implies that the two following equality constraints must be satisfied:
1
C 0
where L is the distance from P 1 to P 2.
© 2001 by CRC Press LLC
R1 sin ��
R1 cos ��
( ( ) � sin( ���) ) � R2( �sin(
���) � sin( �)
) � L
( ( ) � cos(
���) ) � R2( cos(
� � �)
� cos(
�)
) �
0
(2.4)