# Parametric Blending of Triangular Meshes

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## Abstract

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## 1. Introduction

- We present a simple and effective method for generating a parametric blending surface that smoothly joins the local regions of two triangular meshes.
- Our method provides a user with several parameters for controlling the shape of a blending surface.
- We propose an intuitive control mechanism for the direct manipulation of a blending surface.

## 2. Related Work

## 3. Parametric Blending Surface

#### 3.1. Local Parameterization Based on Geodesic Polar Coordinates

#### 3.2. Constructing Blending Surface

#### 3.3. Local Trimming of Triangular Meshes

## 4. Shape Control of Blending Surface

## 5. Experimental Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Overview of our method: (

**a**) selection of local regions; (

**b**) 2D parameterizations of (

**a**); (

**c**) two base surfaces; (

**d**) blending surface connecting two regions.

**Figure 3.**Two configurations of a virtual source vertex ${\mathbf{s}}^{\prime}$ that (

**a**) exists and (

**b**) does not exist.

**Figure 8.**Trimming the local region of a triangular mesh in 2D parameter space: (

**a**) initial trimming; (

**b**) construction of smooth boundary.

**Figure 10.**Two blending surfaces generated by setting (

**a**) $\alpha =\beta =0$ and (

**b**) $\alpha =\beta =1$.

**Figure 11.**Direct manipulation of surface point $\mathbf{p}$ to $\mathbf{q}$ (deformed blending surface is rendered by wireframes).

**Figure 12.**Blending surfaces: (

**a**) blending surface generated by circular contact curves; (

**b**) blending surface generated by a cross-shaped contact curve and a rectangular contact one; (

**c**,

**d**): trimming results of the blending regions.

**Figure 13.**Blending surfaces controlled by blending function ${f}_{\mu}(t)$: (

**a**) $\mu =0.05$; (

**b**) $\mu =0.5$; (

**c**) $\mu \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.95$ and (

**d**) mean curvature plots of (b).

**Figure 15.**(

**a**) Two intersecting cylinders; (

**b**) blending surface connecting two cylinders; (

**c**) merging three surfaces; (

**d**) 3D printing of (

**c**); (

**e**,

**f**) blending surfaces generated by different boundary curves.

**Figure 16.**(

**a**) Blending surfaces connecting separate cylinders; (

**c**) blending surfaces connecting; (

**b**) intersecting cylinders of arbitrary positions and orientations.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shin, M.-C.; Hwang, H.-D.; Yoon, S.-H.; Lee, J.
Parametric Blending of Triangular Meshes. *Symmetry* **2018**, *10*, 620.
https://doi.org/10.3390/sym10110620

**AMA Style**

Shin M-C, Hwang H-D, Yoon S-H, Lee J.
Parametric Blending of Triangular Meshes. *Symmetry*. 2018; 10(11):620.
https://doi.org/10.3390/sym10110620

**Chicago/Turabian Style**

Shin, Min-Chul, Hae-Do Hwang, Seung-Hyun Yoon, and Jieun Lee.
2018. "Parametric Blending of Triangular Meshes" *Symmetry* 10, no. 11: 620.
https://doi.org/10.3390/sym10110620