■Y. Muraki, K. Konno, and Y. Tokuyama,
A Smooth Interpolation Method for Generating Non-Distorted
Surfaces of Three and Five Sided Faces Based on Regular Polygon
The Journal of Art and Science, Vol.9, No.2, pp.49-57, (2010).
N-Sided hole filling is being studied by a lot of people from of old.
However, in the case of the shape which
extremely different length of edge generated by fillet operation,
the distorted and discontinuous shape is generated,
and the other method requires a manual input by the user.
Each of methods have its merits and demerits.
Therefore it is still difficult to solve these three problems at same time.
In this paper, we have studied a method
for solving these three problems simultaneously. First, divided edges and
its parameters are determined, and
generate the edges that the area is divided by generating a regular shape.
And then, each area is interpolated
by a curved surface.
■Tokuyama, Y., Yoshii, Y., Konno, K., and Sone, J.,
Curved Mesh Generation Based on Limit Subdivision Surface and Gregory Patch Interpolation,
The Journal of IIEEJ, Vol.35, No.5, pp.878-887, (2006).
In CG animation, the method for modeling of CG characters
such as humans or animals is important. If these shapes are
represented by parametric surfaces, such as Bezier, B-spline, or
NURBS surfaces, complex procedures are required to make adjacent
surfaces connect smoothly. To solve this problem, instead of
parametric surfaces, subdivision surfaces are often used to
represent shapes. However, how best to fulfill efficient modeling
intended by designers is still problematic.
On the other hand, smooth surfaces can be generated by interpolating
a curved mesh with Gregory Patches.
This method has the advantage that the G1 continuity of adjacent
surfaces can be maintained even if the curved mesh is modified.
Here, we propose a modeling method that involves interpolating a
curved mesh with Gregory Patches after generating a curved mesh of
a Catmull-Clark subdivision surface.
The shape difference between the generated surface and the
subdivision surface is very small.
One of the characteristics of our method is that if some vertices or
edges are modified, only the surfaces connected to them are affected.
Also, the surfaces wthin a specified area can be modified freely by
keeping the continuity between them and the adjacent surfaces
outside the specified area.
The cutting and Boolean operations can be executed on the
generated surfaces easily and our method can be applied to the
polygon meshes generated by Boolean operations and that contain
holes or bosses.
In the modeling of free-form surfaces, a method that first
generates a rough shape with a polygon mesh and then modifies
its curved mesh to obtain the final shape is effective.
■Konno, K., Tokuyama, Y. and Chiyokura, H.,
A G1 connection around complicated curve meshes
using C1 NURBS Boundary Gregory Patches, Computer Aided Design,
Vol. 33, No.4, pp. 293-306, (2001).
A method to design complicated free-form surfaces
using a curve mesh
is a very effective method to intuitively design a shape.
When a user designs a curve mesh,
it is important that the user can concentrate on shape modeling
without considering the type of curves and
the equations of surfaces interpolate an area of the curve mesh.
One of the merits of using a NURBS curve is that various shapes
such as arcs and free-form curves,
can be represented by the same equation.
Therefore a user can design curve meshes without taking
consideration of type of curves and/or special features.
However, it is difficult to interpolate the area of curve
meshes smoothly with NURBS surfaces.
In this paper, we propose a method to interpolate and connect
complicated curve meshes with G1 continuity by using
C1 continuous NURBS Boundary Gregory patch representation.
■Tokuyama, Y. and Konno, K.,
Reparameterization of piecewise rational Bezier curves and its applications,
The Visual Computer, Springer-Verlag, Heidelberg, Vol. 17, No.6, pp. 329-336, (2001).
A piecewise rational Bezier curve is constituted by several
rational Bezier curve segments.
It can be represented by
a rational B-spline curve that the multiplicity of each interior knot
When using these curves to generate a ruled surface or skinned surface,
it is necessary to merge the knot vector of all curves to form the knot
vector of the generated surface.
When merging the knot vectors of the curves, however, two problems tend to
The first is that the merged knot vector may be very large. This gives rise to
a large number of control points, resulting in bad performance in other
applications. The second is that there may be knots with close
, but distinctly different values. Then, extremely narrow surface patches will
be generated, resulting in numerical difficulties in other applications.
To avoid these problems,
in this paper, among two curves,
the second one is reparameterized to have the same knot vector as that of
the first one.
The reparameterized rational B-spline curve has the same shape and degree as
■Konno, K., and Chiyokura, H.,
An Approach of Designing and Controlling Free-Form Surfaces by Using
NURBS Boundary Gregory Patches,
Computer Aided Geometric Design, Vol.13, No. 9, pp.825-849,(1996).
Designers require a means of designing complex free-form surfaces
easily and intuitively.
One general approach to designing such surfaces
is to first define a curve mesh consisting of
characteristic lines, such as cross sections and boundary curves,
then to interpolate the curve mesh using free-form surfaces.
NURBS surfaces are widely used but make the interpolation of
an irregular curve mesh difficult. This has been a major limiting
constraint on designers.
In this paper, we propose a new surface representation
that enables the smooth interpolation of an irregular curve mesh with
NURBS curves and surfaces.
■Konno, K., and Chiyokura, H.,
G1 and G2 Surface Interpolation over Curve Meshes and Its Shape Control,
International Journal of SHAPE MODELING, Vol. 2, No 1, pp. 1-20, (1996).
A popular method to represent complicated surfaces is to interpolate the
curve meshes that are defined by the boundary curves of the surface.
The curve meshes
consist of Bezier curves, rational
Bezier curves or composite curves depending on the shape we design.
To interpolate these curve meshes smoothly,
we use Gregory patches, rational boundary Gregory patches,
general boundary Gregory patches and G2 Gregory patches
as surface representations.
One of the advantages of these surface representations is
that cross boundary derivatives can be independently defined for each boundary.
This enables us to create smooth surface shapes even on irregular curve meshes.
In addition to smooth surface interpolation,
these Gregory patches enable intuitive modification of their shapes with
new control point layouts based on the cross boundary derivatives.
Designers can easily modify surface shapes
without losing G1/G2 continuity
between adjacent patches by moving control points.
■Tokuyama, Y. and Konno, K.,
Approximate conversion of a rational boundary Gregory patch to a nonuniform B-spline surface,
The Visual Computer, Springer-Verlag, Heidelberg, Vol.11, No.7, pp. 360-368, (1995).
A rational boundary Gregory patch is characterized by the fact that
any n-sided loop can be smoothly interpolated and that
it can be smoothly connected to an adjacent patch.
Thus, it is well-suited to interpolate complicated wire frames in shape
Although a rational boundary Gregory patch can be exactly converted to
a rational Bezier patch to enable the exchange of data,
problems of high degree and singularity tend to arise as a result of conversion.
The objective of this paper is to present an algorithm that can approximately
convert a rational boundary Gregory patch to a bicubic non-uniform B-spline
The approximating surface has C1 continuity between its inner patches.
■Konno, K., Takamura, T., and Chiyokura, H.,
A Control Method of Free-Form Surfaces with Curvature Continuity,
IPSJ Journal, Vol.33, No.9, pp.1133-1142,(1992).
A popular method to represent a free-form surface in CAD
is to interpolate curve meshes which describe the surface boundaries.
The boundary curves represent rough shape of an object
which a user want to design.
Then, it is important that the surfaces created from the boundary
curve meshes are valid for users.
Depending on the shape of curve meshes, however,
distorted surfaces may sometimes be generated.
Moreover, generated surfaces may have to specify curvature continuity as
a user requires.
We propose a new surface interpolation method to interpolate
G2 continuous curve meshes with G2 continuity.
In our method, the cross boundary derivatives and curvature vectors
can be controlled to modify a surface shape without