ComputerAided_Design_Engineering_amp_Manufactur.pdf

FIGURE 2.10
or best-fit, or the curve where the points are taken, is within the given tolerance. 20,28 Piecewise circulararc-fitting
algorithms suffer from a similar lack of smoothness in the machined surface, though to a lesser
extent. One way to overcome this is to use biarc-curve fitting, 14,17,21 which ensures tangential C continuity
between the circular segments.
The machining of objects with curved surfaces essentially reduces to machining a series of plane curves
defined in a two-dimensional plane. Hence, without loss of generality, we can discuss the approximation
of curves by linear or circular segments by considering only the case of curves defined in the xy plane.
The following section will introduce two approaches in the generation of CNC cutter path data using
linear or circular segments to approximate general curves that are used in the creation of curved objects
in CAD systems. In the first method, we will illustrate the approach using the cubic Bezier curve as the
general curve example and approximate it by a minimum number of linear or circular segment. In the
second method, the general curve is given in the form of a large number of points that may have been
digitized from a scaled model using CMM technology or laser-ranging technology. Biarcs are used to
approximate the desired shape to within specified tolerances.
1
( )
C 0
A Continuity Approach
Let b0
� ( x0,
y0),
b1
� ( x1,
y1),
b2
� ( x2,
y2)
and b3
� ( x3,
y3)
be the four control points in a twodimensional
space, and t be a parameter (from 0 to 1) that characterizes a cubic Bezier curve. The cubic
Bezier curve can then be written as follows:
© 2001 by CRC Press LLC
NC program listing for CNC rotary table.
3 t
x() t � xiB i ()
�
3
�
i�0
x0( 1 � t)
3
3 t
y() t � yiB i ()
�
3
�
i�0
y0( 1 � t)
3
�
�
3x1t ( 1 � t)
2
3y1t ( 1 � t)
2
3x2t 2
� ( 1 � t)
x3t 3
�
3y2t 2
� ( 1 � t)
y3t 3
�
(2.1a)
(2.1b)