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### C1 Smooth Triangular Surface Patch Constructed by C-curves

Wei LI, Ichiro HAGIWARA, Zhuoqi WU
2005 JSME international journal. Series C, Mechanical systems, machine elements and manufacturing
The constructed triangular patch interpolates the positions and the first derivatives of the boundary composite C-Bézier curves.  ...  So, how to develop C-curve into surface becomes a very important problem. In Ref. (3), tensor product quadrilateral patches have been constructed using Ccurves.  ...  into C-Bézier representation (1) .  ...

### Bezier Crust on Quad Subdivision Surface [article]

Jianzhong Wang, Fuhua Cheng
2013 Pacific Conference on Computer Graphics and Applications
Although widely used, one issue remains with the approximating schemes, i.e., the process of interpolating a set of data points is a global process so it is difficult to interpolate large data sets.  ...  In this paper, we present a local interpolation scheme for quad subdivision surfaces through appending a G2 Bezier crust to the underlying surface, and show that this local interpolation scheme does not  ...  Nevertheless, with our interpolating scheme, since the Bezier crust is added to the underlying CCS patch, the twisting effect can be offset by the underlying CCS patch, such that the new surface patch  ...

### Interpolation Using Cubic Bѐzier Triangular Patches

Samsul Ariffin Abdul Karim, Azizan Saaban, Mohammad Khatim Hasan, Jumat Sulaiman, Ishak Hashim
2018 International Journal on Advanced Science, Engineering and Information Technology
In this study, we use a triangulation based method consists of positivity preserving cubic Bézier triangular patches to estimate the unknown value of rainfall amount at spatial localization.  ...  To preserve the positivity of the surface, the appropriate conditions for the Bézier ordinates are implemented.  ...  Consider the cubic Bézier triangular patch P(u, v, w) in (1) with c 300 = A, c 030 = B, c 003 = C where A, B, C > 0.  ...

### Page 6329 of Mathematical Reviews Vol. , Issue 97J [page]

1997 Mathematical Reviews
con- tinuity to connect two Bézier patches have been studied and meth- ods have been developed to ensure it.  ...  The technique has recently been extended to generate interpolating surfaces with given normal vectors at the interpolated vertices.  ...

### A sinusoidal polynomial spline and its Bezier blended interpolant

K.F. Loe
1996 Journal of Computational and Applied Mathematics
By blending with a Bezier curve using the data points as the control points, the blended curve is a proper smooth interpolant.  ...  Smooth interpolating surface patches can also be defined by blending sinusoidal polynomial tensor surfaces and Bezier tensor surfaces.  ...  Tensor product interpolating patches We can also construct tensor product interpolating patches in a similar way as we construct Bezier tensor surface patches.  ...

### A Geometric Algorithm for Ray/Bézier Surfaces Intersection Using Quasi-Interpolating Control Net

Yohan D. Fougerolle, Sandrine Lanquetin, Marc Neveu, Thierry Lauthelier
2008 2008 IEEE International Conference on Signal Image Technology and Internet Based Systems
The novelty of our approach resides in the use of bounds of the difference between a Bézier patch and its quasi-interpolating control net.  ...  In this paper, we present a new geometric algorithm to compute the intersection between a ray and a rectangular Bézier patch.  ...  For instance, in the case of a bicubic Bézier surface, i.e. d 1 = d 2 = 3 and Z ∞ (3) = 1/12, the bound of the maximum deviation is defined as: Figure 2 shows a bicubic Bézier patch with its control  ...

### Visualizing the Energy of Scattered Data by Using Cubic Timmer Triangular Patches

F A M Ali, S A A Karim, S C Dass, V Skala, A Saaban, M K Hasan, I Hashim
2019 Journal of Physics, Conference Series
This paper discusses the application of the new cubic Timmer triangular patches constructed by Ali et al. [1] to interpolate the irregularly scattered data with 1 continuity.  ...  The results indicate that the surface produced by cubic Timmer triangular patch is better than surface produced using cubic Ball and cubic Bezier triangular patches with overall coefficient of determination  ...  Acknowledgments This research is fully supported by Universiti Teknologi PETRONAS (UTP) through a research grant YUTP: 0153AA-H24 (Spline Triangulation for Spatial Interpolation of Geophysical Data).  ...

### Hierarchical triangular splines

Alex Yvart, Stefanie Hahmann, Georges-Pierre Bonneau
2005 ACM Transactions on Graphics
Several interpolants based on this kind of surfaces have been developed over the last fifteen years. However, with current 3D acquisition equipments, models are becoming more and more complex.  ...  This model enables designers to create a complex smooth surface of arbitarry topology composed of a small number of patches, to which details can be added by locally refining the patches until an arbitrary  ...  Similar to H-splines but dealing with arbitrary topology, we have developed an interpolation scheme with level of details.  ...

### Shape aware normal interpolation for curved surface shading from polyhedral approximation

Xunnian Yang, Jianmin Zheng
2012 The Visual Computer
In this paper we propose two novel normal interpolation schemes along with interpolation of cubic Bézier triangles for rendering curved surfaces from rough triangular meshes.  ...  Firstly, the interpolating normal is computed by a Gregory normal patch to each Bézier triangle by a new definition of quadratic normal functions along cubic space curves.  ...  This type of interpolating normals can also be blended with the analytical normals of the cubic Bézier patch.  ...

### Page 7053 of Mathematical Reviews Vol. , Issue 97K [page]

1997 Mathematical Reviews
Summary: “We develop a method for interpolating scattered data on sphere-like surfaces based on a local triangular patch which is constructed from a blend of certain spherical Bernstein-Bézier polynomials  ...  In Section 5, the compatibility conditions for two adjacent Bézier patches along a common boundary curve are developed.  ...

### Scattered Data Interpolation Using Quartic Triangular Patch for Shape-Preserving Interpolation and Comparison with Mesh-Free Methods

Samsul Ariffin Abdul Abdul Karim, Azizan Saaban, Van Thien Nguyen
2020 Symmetry
Quartic Bézier triangular patches with 15 control points (ordinates) can also be used for scattered data interpolation.  ...  Thus, we cannot obtain the exact Bézier ordinates, and the quartic scheme is global and not local. Therefore, the quartic Bézier triangular has received less attention.  ...  The sufficient condition for the positivity of the quartic triangular patch developed in this study is the same as the sufficient condition for the quartic Bézier triangular patch developed in Saaban et  ...

### Adaptive patch-based mesh fitting for reverse engineering

Hongwei Lin, Wei Chen, Hujun Bao
2007 Computer-Aided Design
By interpolating the normal and boundary-fitting curves of each quadrilateral patch with a Bézier patch, an initial G 1 smoothly stitching Bézier patches is generated.  ...  In this paper, we propose a novel adaptive mesh fitting algorithm that fits a triangular model with G 1 smoothly stitching bi-quintic Bézier patches.  ...  Fig. 6 . 6 With our patch-by-patch approach, Bézier patches can be constructed individually, each of which interpolates a shared boundary-fitting curve and normal curve.  ...

### Comparing faceted and smoothed tool surface descriptions in sheet metal forming simulation

D. M. Neto, M. C. Oliveira, J. L. Alves, L. F. Menezes
2014 International Journal of Material Forming
The description of arbitrarily-shaped tool surfaces using the traditional linear finite elements is compared with two distinct smooth surface description approaches: (i) Bézier patches obtained from the  ...  Smoothing the contact surfaces with Nagata patches allows creating more accurate tool models, both in terms of shape and normal vectors, when compared with the conventional linear finite element mesh.  ...  This behaviour can be associated with the high interpolation degree used in some Bézier patches, which can create instabilities in the local search.  ...

### Construction of Bézier surface patches with Bézier curves as geodesic boundaries

R.T. Farouki, N. Szafran, L. Biard
2009 Computer-Aided Design
The existence conditions and interpolation scheme, developed in a general context in earlier studies, are adapted herein to ensure that the geodesic-bounded surface patches are compatible with the usual  ...  Precise conditions for four Bézier curves to constitute geodesic boundaries of a polynomial or rational surface patch are identified, and an interpolation scheme for the construction of such surfaces is  ...  Bézier patch that interpolates these curves as geodesic boundaries.  ...

### An iterative Clough-Tocher interpolant

G. Farin, P. Kashyap
1992 Mathematical Modelling and Numerical Analysis
This paper has benefitted from discussions with members of  ...  It is assumed that the reader is familiar with the theory of Bézier triangular patches, as outlined in [8] , [6] .  ...  An w-th degree Bernstein-Bézier triangular patch is of the form = n the b { are Bézier ordinates which form the control net of the triangular patch ; u = (u, v,w ) are the barycentric coordinates of the  ...
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