Varying the Betas in Beta-splines

http://www2.eecs.berkeley.edu/Pubs/TechRpts/1983/CSD-83-112.pdf

The Beta-spline introduced recently by Barsky is a generalization of the uniform cubic B-spline: parametric discontinuities are introduced in such a way as to preserve continuity of the unit tangent and curvature vectors at joints (geometric continuity) while providing bias and tension parameters, independent of the position of control vertices, by which the shape of a curve or surface can be manipulated. We introduce a practical method by which different values of the bias and tension at each point along a curve, the actual position being determined by substituting these values into the equations for a uniformly-shaped Beta-spline. We explore the properties of the resulting piecewise polynomial curves and surfaces. An important characteristic is their local response when either the position of a control vertex or the value of a shape parameter is altered.

There is also a conceptually simple and obvious way to directly generalize the equations defining the uniformly-shaped Beta-splines so that each shape parameter may have a distinct value at every joint. Unfortunately, the curves which result lack many desirable properties.

BibTeX citation:

```@techreport{Barsky:CSD-83-112,
Author = {Barsky, Brian A. and Beatty, John C.},
Title = {Varying the Betas in Beta-splines},
Institution = {EECS Department, University of California, Berkeley},
Year = {1982},
Month = {Dec},
URL = {http://www2.eecs.berkeley.edu/Pubs/TechRpts/1982/6326.html},
Number = {UCB/CSD-83-112},
Abstract = {The Beta-spline introduced recently by Barsky is a generalization of the uniform cubic B-spline: parametric discontinuities are introduced in such a way as to preserve continuity of the unit tangent and curvature vectors at joints (geometric continuity) while providing bias and tension parameters, independent of the position of control vertices, by which the shape of a curve or surface can be manipulated.  We introduce a practical method by which different values of the bias and tension at each point along a curve, the actual position being determined by substituting these values into the equations for a uniformly-shaped Beta-spline. We explore the properties of the resulting piecewise polynomial curves and surfaces. An important characteristic is their local response when either the position of a control vertex or the value of a shape parameter is altered.  <p>  There is also a conceptually simple and obvious way to directly generalize the equations defining the uniformly-shaped Beta-splines so that each shape parameter may have a distinct value at every joint. Unfortunately, the curves which result lack many desirable properties.}
}
```

EndNote citation:

```%0 Report
%A Barsky, Brian A.
%A Beatty, John C.
%T Varying the Betas in Beta-splines
%I EECS Department, University of California, Berkeley
%D 1982
%@ UCB/CSD-83-112
%U http://www2.eecs.berkeley.edu/Pubs/TechRpts/1982/6326.html
%F Barsky:CSD-83-112
```