[ont.events] Cubic, Quartic, Rational, and Discrete Beta-Splines.

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.