# Basic Functions for Representing Shape

This theorical field has been largely exploited in our research years. To explain the main ideas behind this topic, may be natural starting from the polynomial case and remember that the Bernstein polynomial basis is today recognized as the best basis to represent a polynomial from all viewpoints:
• the polynomial coefficients have a geometric meaning and control the polynomial shape;
• the basis is numerically well conditioned and its numerical evaluation is stable thanks to a recurrence relation;
• applied for representing curves and surfaces, this basis allow us the convex-hull, variation diminishing approximation and affine invariance properties.
Degree 3 Bernstein basis.

Unfortunately, the polynomial space is not recommend overall in the high degree case. Other classes of functions, tipically piecewise polynomials or piecewise rational polynomials, respectively called NUBS (Non Uniform B-Splines) and NURBS (Non Uniform Rational B-Splines) are more useful.
Degree 3 uniform B-spline basis.
Degree 3 uniform rational B-spline basis.

But these are not the entire spectrum and a lot of classes of functions were, and are everyday, proposed in literature within a useful basis representation.

Our research in this area addressed initially to L-spline functions and to their basis named LB-spline, for data fitting problems in one and two variables; then succesively to polynomial splines (NUBS) and rational splines (NURBS) in the modeling field and to p-Bezier and p-spline which are rational trigonometric function very useful to fit data and modeling in polar/spherical coordinates. Recently our interest was about ICC and ICC-spline which are irrational functions to model in polar coordinates and ISS patches for modeling in a spherical environment.

Actually our research is on subdivision surfaces and their powerful both for reconstruction and modeling.

#### References

• L.Bacchelli Montefusco, G.Casciola, Analysis of methods for the numerical evaluation of Basic L-splines, Calcolo, vol.20, n.2 (1983).
• G.Casciola, B-splines via recurrence relations, Calcolo, vol.26 n.2-3-4 (1989).
• G.Casciola, G.Valori, An inductive proof of the derivative B-spline recursion formula, Department of Mathematics, University of Bologna, (1989).
• G.Casciola, A recurrence relation for rational B-splines, Computer Aided Geometric Design, vol.14 (1997).
• G. Casciola, S. Morigi, Spline curves in polar and Cartesian coordinates, Curves and Surfaces with applications in CAGD, A. LeMeahute`, C. Rabut ed L.L. Schumaker (Eds.) (1997)
• G.Casciola, S.Morigi, Inverse Circular Curves, International Journal of Shape Modeling, Vol.8, no.1 (2002)