ylfink@water.UUCP (06/09/87)
DEPARTMENT OF COMPUTER SCIENCE
UNIVERSITY OF WATERLOO
SEMINAR ACTIVITIES
COMPUTER GRAPHICS/NUMERICAL ANALYSIS SEMINAR
- Thursday, June 11, 1987
Dr. Barry Joe of the University of Alberta will speak
on ``Cubic, Quartic, Rational, and Discrete Beta-
Splines''.
TIME: 3:30 PM
ROOM: MC 6091A
ABSTRACT
Cubic Beta-splines were introduced by Barsky in 1981 as
a generalization of cubic B-splines for use in
computer-aided geometric design and computer graphics.
They provide a means of constructing parametric spline
curves with geometric continuity (i.e. continuity of
the curve, unit tangent vector, and curvature vector),
in which the shape of the curve is controlled by shape
parameters. Extensions to cubic Beta-splines were
obtained by Beatty, Bartels, and Goodman in the next
few years. Goodman gave a general explicit formula for
cubic Beta-splines defined on uniform knot sequences.
The theoretical results for Beta-splines with higher
degrees of arc-length and geometric continuity were
given by Goodman, Dyn, and Micchelli in 1985.
In this talk, we present an alternative explicit
formula for cubic Beta- splines defined on nonuniform
knot sequences, which is valid for multiple knots and
is used for computing discrete Beta-splines. We
introduce a general explicit formula for quartic Beta-
splines, and illustrate the effects of the two extra
shape parameters available for quartic Beta-spline
curves. We also introduce rational Beta-spline curves
and surfaces which are generalizations of rational
B-spline curves and surfaces and contain weight
parameters for each control vertex for further shape
control. Finally, we solve the subdivision problem for
Beta-spline curves which is to express the curve in
terms of a larger number of control vertices and Beta-
splines. Discrete Beta-splines, which are analogous to
discrete B-splines, arise in this subdivision. We
present two algorithms for computing discrete Beta-
splines, one of which is also a new algorithm for
computing discrete B-splines.