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741 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 twentieth century (Dierckx, 1993) As applied to functions y = y(x), for x [x1, x2], 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 variations of y(x) in the interpolation interval become more pronounced, the order of the interpolating 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 equations that must be solved for the polynomial coef cients The larger the condition number, the larger is the roundoff-error ampli cation and, hence, the smaller the accuracy of the computed coef cients As a matter of fact, the use of a nonorthogonal polynomial readily leads to unacceptably high roundoff-error ampli cations Orthogonal polynomials offer a remedy to these ampli cations but only to some extent As an alternative to polynomial interpolation, spline functions were developed by numerical analysts to allow for lower-degree interpolating polynomials Lower-degree interpolating polynomials are possible if they are de ned 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 suf cient Here is where parametric splines x = x(p), y = y(p) come into the picture Notice that geometric curves, as opposed to curves representing 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 representing functions While we limit ourselves to cubic parametric splines, other spline curves are available that go by names such as B zier curves, rational B zier curves (Srinivasan and Ge, 1997), and NURBS, which stands for non-uniform rational B zier splines (Rogers, 2001), as required for more advanced applications The cam pro les of interest are assumed to be closed smooth curves of the G2 type, ie, with uniquely de ned tangent and curvature everywhere, except for, possibly, some isolated points Moreover, functions describing the Cartesian coordinates of a cam pro le, ie, 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 de ned, and regard x and y as functions x(p) and y(p) Let (xi, yi), where xi x(yi) and yi y(yi), for i = 1, , n, be a discrete set of points generated on the cam pro le Since we need a closed smooth curve1 the pertinent periodic boundary conditions must be satis ed, namely, x1 = xn , x1 = xn , x1 = xn , where x and x are de ned as i i xi dx dp , xi.

Description Returns true if k is a key in the invoking SortedList Returns false otherwise Returns true if v is a value in the invoking SortedList Returns false otherwise Returns the value at the index specified by idx Returns an IDictionaryEnumerator for the invoking SortedList Returns the value of the key at the index specified by idx Returns an IList collection of the keys in the invoking SortedList Returns an IList collection of the values in the invoking SortedList Returns the index of the key specified by k Returns 1 if the key is not in the list Returns the index of the first occurrence of the value specified by v Returns 1 if the value is not in the list Sets the value at the index specified by idx to the value passed in v Returns a synchronized version of the SortedList passed in sl Sets Capacity to Count

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