#
Bézier Triangles with G^{2} Continuity across Boundaries

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## Abstract

**:**

## 1. Introduction

- A simple technique for generating a smooth surface with ${G}^{2}$ continuity over a triangular mesh.
- Surface construction using a simple linear blending of two Bézier triangles rather than a manifold.
- Interactive control of the blending region on each Bézier triangle allows sharp features such as darts, corners and creases to be created in a controlled manners.

## 2. Related Work

## 3. Cubic Bézier Triangles

## 4. Smooth Blending of Bézier Triangles

#### 4.1. Barycentric Coordinates with Respect to Different Triangular Domains

#### 4.2. Defining Blending Regions

#### 4.3. Blending Bézier Triangles

## 5. Experimental Results

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Blending of Bézier triangles: (

**a**) A given triangular mesh; (

**b**) Cubic Bézier triangles with ${G}^{0}$ continuity; (

**c**) Blending regions (in yellow) between adjacent triangles; (

**d**) A smoothly blended surface; (

**e**) A blended surface with sharp features.

**Figure 4.**Four domain triangles ${T}_{0},{T}_{1},{T}_{2}$ and ${T}_{3}$, and the three subdomains ${T}_{01},{T}_{02}$ and ${T}_{03}$.

**Figure 9.**Blended triangles with (

**b**) $h=.106$ and (

**d**) $h=.212$. (blending regions are shown in yellow in (

**a**),(

**c**)).

**Figure 10.**(

**a**) A stellated dodecahedron; (

**b**) Surfaced by PN triangles; (

**c**),(

**d**) Using our method (blend regions are shown in yellow in (

**c**)).

**Figure 11.**Planar Bézier triangles (defined using the scheme shown in (

**a**)) produce (

**b**) on object with flat faces ($h=0$), which can be (

**c**) blended by setting $h=.106$. (

**d**) Different values of h produces a variety of darts, crease and corners.

**Figure 12.**(

**a**) A different geometry surfaced by Bézier triangles (

**b**), and (

**c**) Blending regions with different extents, and (

**d**) Sharp features generated by blending different types of Bézier triangles.

**Figure 13.**Comparison with PN triangles: (a), (d), (g), (h): control meshes; (b), (e), (h), (k): PN triangles with ${G}^{0}$ continuity; (c), (f), (i), (l): smooth surfaces with ${G}^{2}$ continuity.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lee, C.-K.; Hwang, H.-D.; Yoon, S.-H.
Bézier Triangles with *G*^{2} Continuity across Boundaries. *Symmetry* **2016**, *8*, 13.
https://doi.org/10.3390/sym8030013

**AMA Style**

Lee C-K, Hwang H-D, Yoon S-H.
Bézier Triangles with *G*^{2} Continuity across Boundaries. *Symmetry*. 2016; 8(3):13.
https://doi.org/10.3390/sym8030013

**Chicago/Turabian Style**

Lee, Chang-Ki, Hae-Do Hwang, and Seung-Hyun Yoon.
2016. "Bézier Triangles with *G*^{2} Continuity across Boundaries" *Symmetry* 8, no. 3: 13.
https://doi.org/10.3390/sym8030013