TSplines opens up many interesting avenues for academic research. The purpose of this webpage is to stimulate research on the topic by posting links to research papers on TSplines and to recommend important open questions related to TSplines theory. Researchers who wish to add a link from this webpage to their TSplines related work, please contact us. (NOTE: TSplines, Inc. owns the exclusive license to commercialize the patented TSplines technology.)
Publications List

TSplines and TNURCCS (SIGGRAPH'03) Thomas W. Sederberg, Jianmin Zheng, Almaz Bakenov, Ahmad Nasri
This paper introduces TSplines and TNURCCs (NonUniform Rational CatmullClark Surfaces with Tjunctions). TNURCCs enable true local refinement of a CatmullClarktype control grid: individual control points can be inserted only where they are needed to provide additional control, or to create a smoother tessellation, and such insertions do not alter the limit surface.
 TSpline
Simplification and Local Refinement (SIGGRAPH'04) Thomas W. Sederberg, Jianmin Zheng, Tom Lyche, David Cardon, G. Thomas Finnigan, Nicholas North
A typical NURBS surface model has a large percentage of superfluous control points that significantly interfere with the design process.
This paper presents an algorithm for eliminating such superfluous control points, producing a TSpline. The algorithm can remove substantially more control points than competing methods such as BSpline wavelet decomposition.

Adaptive TSpline
Surface Fitting to ZMap Models Jianmin Zheng, Yimin Wang, Hock Soon Seah
Surface fitting refers to the process of contructing a smooth representation for an object surface from a fairly large number of measured 3D data points. This
paper presents an automatic algorithm to construct smooth parametric surfaces using TSplines from zmap data. The algorithm begins with a rough surface
approximation and then progressively refines it in the regions where the approximation accuracy does not meet the requirement.

Conversion between TSplines and Hierarchical BSplines Yimin Wang, Jianmin Zheng, Hock Soon Seah
This paper presents algorithms for transformation between TSplines and hierarchical (rational) BSplines. With the transformation, a surface expressed in terms of TSplines can be converted into a hierarchical
(rational) BSpline representation and vice versa. The correspondence between TSpline representation and hierarchical (rational) BSpline representation
is generally not onetoone. The conversion algorithms can yield compact representations.

Improvement on the Dimensions
of Spline Spaces on TMesh ChongJun Li, RenHong Wang, Feng Zhang
This paper discusses the dimensions of spline spaces on a Tmesh by using the Smoothing CofactorConformality method. The result is
improved essentially with another relaxed constraint depending on the order of the smoothness, the degree of the spline functions and the
structure of the Tmesh.

Manifold TSpline Ying He, Kexiang Wang, Hongyu Wang, Xiafeng Gu, Hong Qin
This paper develops manifold TSplines. In our shape modeling framework, the manifold Tsplines are globally welldefined except at a finite number of extraordinary points, without the need of
any tedious trimming and patching work. We present an effecient algorithm to convert triangular meshes to manifold TSplines. Becuse of the natural, builtin hierarchy
of TSplines, we can easily reconstruct a manifold TSpline surface of high quality with LOD control and hierarchical structure.

Automatic and Interactive Mesh to TSpline Conversion WanChiu Li, Nicholas Ray, Bruno Levy
In Geometry Processing, and more specifically in surface approximation, one of the most important issues is the
automatic generation of a quaddominant control mesh from an arbitrary shape (e.g. a scanned mesh). However, in the industry, designers
still use manual tools (see e.g. cyslice). The main difference between a control mesh constructed by an automatic
method and the one designed by a human user is that in the second case, the control mesh follows the features of
the model. More precisely, it is well known from approximation theory that aligning the edges with the principal
directions of curvature improves the smoothness of the reconstructed surface, and this is what designers intuitively
do.
In this paper, our goal is to automatically construct a control mesh that follows the features of a model, mimicking
the mesh that a designer would create manually. The control mesh generated by our method can be used by a wide
variety of representations (splines, subdivision surfaces . . . ). We demonstrate our method applied to the automatic conversion from a mesh of
arbitrary topology into a TSpline surface.

TSpline Merging Heather Ipsom
Geometric models, such as for use in CAD/CAM or animation, are
often constructed in a piecewise fashion. Historically, these models have
been made of NURBS surfaces. For various reasons it is problematic and often
times mathematically impossible to combine several NURBS models into one
continuous surface. The recent invention of a surface type called Tsplines
has made the combining of NURBS surfaces into a single continuous surface
possible, but much of the mathematics has yet to be explored. This thesis
explores the mathematics and algorithms necessary to merge multiple NURBS,
Tspline, or TNURCC surfaces into a single continuous surface. This thesis
addresses two main problems. The first problem is merging surfaces with
different parameterizations. In order to merge surfaces with different
parameterizations, it is often necessary to modify the parameter values of
the surface, which can change the shape of the surface. This change can be
alleviated through shape control methods. The second problem is merging
surfaces that meet at extraordinary points, or points with a valence other
than four. Results show that the merging algorithm is able to successfully
convert models composed of multiple NURBS, Tspline, or TNURCCS surfaces
into models composed of a single Tspline or TNURCC surface. The resulting
models are gapfree and contain little distortion in the parameterization.

TSpline Simplification David Cardon
This work focuses on generating approximations of complex
Tspline surfaces with similar but less complex Tsplines. Two approaches to
simplifying Tsplines are proposed: a bottomup approach that iteratively
refines an oversimple Tspline to approximate a complex one, and a topdown
approach that evaluates existing control points for removal in producing an
approximations. This thesis develops and compares the two simplification
methods, determining the simplification tasks to which each is best suited.
In addition, this thesis documents supporting developments made to Tspline
research as simplification was developed.

Isogeometric Analysis using TSplines Y. Bazilevs, V.M. Calo, J.A. Cottrell, J.A. Evans, T.J.R Hughes, S. Lipton, M.A. Scott, T.W. Sederberg
We explore Tsplines, a generalization of NURBS enabling local refinement, as a basis for isogeometric analysis. We review Tsplines as a surface design methodology and then develop it for engineering analysis applications. We test Tsplines on some elementary twodimensional and threedimensional fluid and structural analysis problems and attain good results in all cases. We summarize current limitations and future opportunities.

Watertight Trimmed NURBS Thomas W. Sederberg, G. Thomas Finnigan, Xin Li, Hongwei Lin, Heather Ipson
This paper addresses the longstanding problem of the unavoidable
gaps that arise when expressing the intersection of two NURBS surfaces
using conventional trimmedNURBS representation. The solution
converts each trimmed NURBS into an untrimmed TSpline,
and then merges the untrimmed TSplines into a single, watertight
model. The solution enables watertight fillets of NURBS models,
as well as arbitrary feature curves that do not have to follow isoparameter
curves. The resulting TSpline representation can be exported
without error as a collection of NURBS surfaces.

TSplines: A Technology for Marine Design with Minimal Control Points Matthew T. Sederberg, Thomas W. Sederberg
A typical NURBS hull model involves far more control points than
are needed for the design process, which needlessly complicates
fairing and form finding. A relatively new surface representation
called Tsplines eliminates the superfluous rows and columns of
control points that are unavoidable in NURBS. This paper presents
methods for designing hulls using Tsplines.
