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OneSpace Designer: ME10 Dynamic Drafting MI Interfacing


Parametric B-Splines

The Spline entity (BSPL) in the Geometry Elements section contains values for the description of a rational parametric curve. A B-spline is uniquely defined by:

If you connect Bi, Bi+1 with a line for iE {1, ..., n-1} you get the control polygon of the spline.

Let t denote a parameter value, where: tE[X1,Xn+k]. Then a B-spline at curve P is given by:


For the 'ith' normalized B-spline basis function of order k, the basis functions Nik(t) are defined by the Cox-de Boor recursion formulas:

  Ni,1t) = (1, if t...[Xi,Xi+1]
  Ni,1(t) = (0, else
and


A B-spine curve is defined as a polynomial spline function of order k (degree k-1) which satisfies the following conditions:

It can also be shown that for any parameter value tE [X1, ..., Xn+k]


Each basis function is positive or zero for all parameter values, that is Ni,k>=0. Except for k=1, each basis function has precisely one maximum value. The maximum order of the spline is equal to the number of defining polygon vertices. The curve does not oscillate about any straight line more often than its defining control polygon, that is, the curve generally follows the shape of the defining polygon.

Any affine transformation can be applied to the curve by applying polygon vertices, that is, the curve is transformed by transforming the defining polygon vertices.

The curve lies within the convex hull of its defining polygon.

ME10 uses cordal parametization, that is:

Xi+1 - Xi = | Bi+1 - Bi |

iE{k,...,n}

where | | is the distance between Bi+1 and Bi

This means that knot values are determined by the distance between the defining control points.

The derivatives of a B-spline curve at any point on the curve are obtained by formal differentiation. The first derivative is:


while the second derivative is:


For further details about mathematical descriptions of B-splines, refer to Elements for Computer Graphics by D F Rogers, J A Adams (McGraw Hill).


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