Abstract:

The traditional free curve modeling system can be described as “a (discrete) weighted average of a (discrete) sequence of control points with respect to a (discrete) sequence of basis functions.” This discrete property has been transformed into a continuous property, which could be described as “a (continuous) integral average of a (continuous) curve with respect to a (continuous) function family.” The corresponding change was similar to the transformation from the mathematical expectation of a discrete random variable defined by probability distribution law to the mathematical expectation of a continuous random variable defined by probability density functions in probability theory. Hence, the modeling method with continuous property was referred to as the density modeling method, where the continuous curve was known as a control curve, and the continuous function family was referred to as a basis density function. To preliminarily explore the density modeling method, we presented its model, constructed a basic density function of degree 1 and 2 satisfying non-negativity, normalization, and symmetry properties, and examined the derivatives of the basic density functions and the moment functions of the corresponding random variable. During density modeling, an arbitrary polynomial or even non-polynomial parametric curves could be used as the input, and the output curve was a polynomial curve of degree 1 or 2, respectively. The density modeling curve possesses properties such as convex hull, affine invariance, and symmetry properties.

Key words: free-form curve modeling, Bézier curve, basis function, mathematical expectation, probability density function