Adjustable Speed Subdivision Surfaces
K. Karˇciauskas and J. Peters
Abstract
—We introduce a nonuniform subdivision algorithm
that partitions the neighborhood of an extraordinary point in the
ratio
σ
: 1

σ
, where
σ
∈
(0
,
1)
. We call
σ
the
speed
of the non
uniform subdivision and verify
C
1
continuity of the limit surface.
For
σ
= 1
/
2
, the CatmullClark algorithm is recovered. Other
speeds are useful to vary the contraction near extraordinary
points.
I. INTRODUCTION
With few exceptions, polynomial subdivision algorithms
generalize uniform (box)spline subdivision [BHR93]. In the
univariate
case, standard Bspline knot insertion yields a
stable, local
nonuniform
refinement of the control polygon,
one knot at a time. Cashman et al. [CDS07], [CDS08] recently
proposed factoring nonuniform Bspline subdivision into
small stencils suitable for
simultaneous
knot insertion. Earlier,
Goldman and Warren [GW93] extended uniform subdivision
of curves to knot intervals in globally geometric progression.
When the univariate refinement is tensored and applied to
tensorproduct splines, the knot intervals must be carried over
to parallel isocurves. In practice, the least superset of all
isoknot intervals is enforced by inserting knots. By con
trast, Nonuniform Recursive Subdivision Surfaces, proposed
by Sederberg et al. [SZSS98], allow freely assigning knot
intervals to every edge of the control net. Subdivision in the
regular part surrounding an extraordinary node, i.e. a control
point associating more or fewer than four direct neighbors, is
constructed by halving each knot interval.
blend with hole
Fig. 1.
Shape completion. (
top
) Regular speed
σ
= 1
/
2
. (
bottom
) High
speed
σ
= 3
/
4
. (
left
) Blend with hole. (
middle
) Two steps of subdivision.
(
right
) Highlight lines when filling the remaining hole according to [SS05].
Note the strong fault in the highlight line in the center area of the zoomed in
detail (
top
,
right
).
We came across a different generalization, not covered by
[GW93], [SZSS98], [CDS07], when looking for reparameter
izations for highquality surface constructions. Our approach
is philosophically different. We take the point of view that
subdivision surfaces are splines with singularities and that
subdivision is a process for generating a sequence of nested
surface rings. Once the BernsteinB´ezier (BB) form of a
surface ring is welldefined, e.g. as in Appendix VC, we
do not refine the control net further, but work with the
polynomial representation. In this scenario, the nonuniformity
is motivated by the the desire to adjust, possibly repeatedly,
the relative width of polynomial surface rings. Figure 1 gives
an example of such an adjustment in practice. The resulting
family of algorithms has properties akin to and specializes to
CatmullClark subdivision. In Section II, we derive the rules
and in Section III, we verify the properties. Besides adjustable
control net refinement for computer graphics and adjustable