Computer-Aided Design 40 (2008) 616–624
www.elsevier.com/locate/cad
Functional splines with different degrees of smoothness and
their applications
Chun-Gang Zhu
a,∗
, Ren-Hong Wang
a
, Xiquan Shi
b
, Fengshan Liu
b
a
Institute of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
b
Department of Applied Mathematics and Theoretical Physics, Delaware State University, Dover, DE 19901, USA
Received 15 June 2006; accepted 21 February 2008
Abstract
Implicit curves and surfaces are extensively used in interpolation, approximation and blending. [Li J, Hoschek J, Hartmann E. G
n−1
-functional
splines for interpolation and approximation of curves, surfaces and solids. Computer Aided Geometric Design 1990;7:209–20] presented a
functional method for constructing G
n−1
curves and surfaces which are called functional splines. In this paper, functional splines with different
degrees of smoothness are presented and applied to some typical problems.
c
2008 Elsevier Ltd. All rights reserved.
Keywords: Geometric continuity; Functional splines; Interpolation; Blending
1. Introduction
Blending, a basic task in Computer Aided Geomet-
ric Design (CAGD), is the process of using additional
curves/surfaces to smooth the sharp edges and corners produced
by the intersection of curves/surfaces (together with the ad-
ditional curves/surfaces, those curves/surfaces are called orig-
inal) [6]. The resulting curves/ surfaces are called blending
curves/surfaces. A measure of the smoothness of the blend-
ing curves/surface is the geometric continuity. The concept of
geometry continuity is either borrowed from differential geom-
etry using reparametrization or from algebraic geometry based
on contact of order n. A blending curve/surface is called G
n
-
blending curve/surface if any original curve/surface has the
contact of order n along the intersection points/curves of the
original curves/surfaces. For example, G
1
-blending means that
the tangent planes at the curves of contact are the same, and
there is no cusp for blending surfaces. G
2
-blending guaran-
tees the continuity of the normal curvatures. We do not want
to go into details about the definition of geometric continuity
(see [4,9,24]).
∗
Corresponding author. Tel.: +86 0411 84708351x8017.
E-mail addresses: cgzhu@dlut.edu.cn (C.-G. Zhu), renhong@dlut.edu.cn
(R.-H. Wang), xshi@desu.edu (X. Shi), fliu@desu.edu (F. Liu).
Implicitly defined curves and surfaces have their advantages,
such as (1) the closure property under some geometric oper-
ations (intersection, union, offset etc.), (2) the implicit alge-
braic equation presentation captures all elements of that set, (3)
implicit algebraic curve segments have more degrees of free-
dom than parametric curves, and (4) the algorithms for curve
and surface fittings do not need the parametrization of the data.
Due to the above advantages, people began to pay much atten-
tion to the study of modeling, especially blending, with implicit
algebraic curves and surfaces [1,3,7,8,10–14,18,21,25,27].
There are several methods for blending two implicit alge-
braic surfaces. They depend essentially on the representations
of the implicit algebraic surfaces to be blended. Important
contributions for blending implicit surfaces are due to Hoff-
mann and Hopcroft [15–17], Rockwood and Owen [22], and
Li et al. [18]. The first two methods are special cases of the
elliptic functional splines. Hoffmann and Hopcrofts solutions
are G
1
, Rockwood and Owens are G
n
. The last method pre-
sented the G
n−1
functional spline which was an extension of
the G
1
-conic-section spline technique introduced by [19]. A
basic paper for G
n
-blending of algebraic curves and surfaces
was presented by Warren [24]. Hartmann gave pencils of im-
plicit G
n
-blending surfaces for blending suitcase-, house- and
3-beam-corners of (nearly) arbitrary triples of surfaces in pa-
pers [12,13]. For other methods to blend surfaces with smooth
piecewise algebraic surfaces refer to papers [1–3,5,7].
0010-4485/$ - see front matter
c
2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.cad.2008.02.006