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THB7 8/15/03 1:58 PM Page 187

GEOMETRY OF PLANAR CAM PROFILES 187

7.4.1 Cubic Splines

We focus here on the representation of the closed contours of planar cam mechanisms
using the simplest geometric splines, namely, cubic splines.
A spline curve is an interpolation tool fully developed in the second half of the twen-
tieth century (Dierckx, 1993). As applied to functions y = y(x), for x Π[x 1, x 2], classical
interpolation tools include orthogonal polynomials of various kinds that go by names such
as Legendre, Chebyshev, etc. (Kahaner et al., 1989). It is well known that, as the varia-
tions of y(x) in the interpolation interval become more pronounced, the order of the inter-
polating polynomial must be increased. However, the larger this order, the larger will be
the condition number (Golub and Van Loan, 1983) of the underlying linear system of equa-
tions that must be solved for the polynomial coefficients. The larger the condition number,
the larger is the roundoff-error amplification and, hence, the smaller the accuracy of the
computed coefficients.
As a matter of fact, the use of a nonorthogonal polynomial readily leads to unaccept-
ably high roundoff-error amplifications. Orthogonal polynomials offer a remedy to these
amplifications but only to some extent. As an alternative to polynomial interpolation,
spline functions were developed by numerical analysts to allow for lower-degree interpo-
lating polynomials. Lower-degree interpolating polynomials are possible if they are
defined piecewise. Thus, a cubic spline function is a piecewise cubic polynomial. Quintic
spline functions are piecewise quintic polynomials.
As applied to geometric curves of the form F(x, y) = 0, spline functions, also called
nonparametric splines, are not sufficient. Here is where parametric splines x = x(p), y =
y(p) come into the picture. Notice that geometric curves, as opposed to curves represent-
ing functions, can have slopes making arbitrary angles with the coordinate axes, can cut
themselves, thus giving rise to double points, and can have cusps. Not so curves repre-
senting functions. While we limit ourselves to cubic parametric splines, other spline curves
are available that go by names such as Bézier curves, rational Bézier curves (Srinivasan
and Ge, 1997), and NURBS, which stands for non-uniform rational Bézier splines (Rogers,
2001), as required for more advanced applications.
2
The cam profiles of interest are assumed to be closed smooth curves of the G type,
i.e., with uniquely defined tangent and curvature everywhere, except for, possibly, some
isolated points. Moreover, functions describing the Cartesian coordinates of a cam profile,
i.e., x = x(y) and y = y(y), are also periodic functions of y. In the following discussion
we introduce an alternative parameter, p, as yet to be defined, and regard x and y as func-
tions x(p) and y(p).
Let (x i, y i), where x i ∫ x(y i) and y i ∫ y(y i), for i = 1,..., n, be a discrete set of points
1
generated on the cam profile. Since we need a closed smooth curve the pertinent periodic
boundary conditions must be satisfied, namely,
x = x , x¢ = x¢, x¢¢= x¢¢, y = y , y¢ = y¢, y¢¢= y¢¢ (7.24)
1 n 1 n 1 n 1 n 1 n 1 n
where x¢ i and x≤ i are defined as
dx dx
2
x ¢ ∫ , x ¢¢∫ (7.25)
i i
dp dp 2
xx xx
= i
= i
with similar definitions for y¢ i and y≤ i. Conditions in Eq. (7.24) can be satisfied by resorting
to a periodic parametric cubic spline, as discussed in Rogers (2001) and outlined below.
1 If cusps are allowed at the endpoints P 1(x 1, y 1) and P 2(x 2, y 2), then the slope and curvature conditions of Eq. (7.24)
must be relaxed.