# Computing the Closest Approach Distance of Two Ellipsoids

## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Problem

#### 2.2. Solution Method

#### 2.3. Initial Guess and Stopping Criteria

## 3. Experimental Results

## 4. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Two contacting ellipsoids ${e}_{1}\left(\mathbf{x}\right)=1$ and ${e}_{2}(\mathbf{x}-d\mathbf{n})=1$ with the closest approach distance d in the inter-center direction $\mathbf{n}$ and their concentric ellipsoids (with the same aspect ratios) contacting along the curve $\mathbf{x}\left(u\right)$ where the gradient vectors $\nabla {e}_{1}\left(\mathbf{x}\right)$ and $\nabla {e}_{2}(\mathbf{x}-d\mathbf{n})$ are parallel.

**Figure 2.**Distributions of the number of solver iterations with ${\u03f5}_{u}={10}^{-8}$ for 10 million pairs of ellipsoids generated randomly with $\gamma =3$ and $\Gamma =3$.

**Figure 3.**Maximum and average number of solver iterations for 1 million samples generated for each $\gamma $ value from 1 to 200.

**Figure 4.**Maximum and average number of solver iterations for 1 million samples generated for each $\Gamma $ value from 1 to 200.

**Figure 5.**Average number of solver iterations for 1 million samples when ${\u03f5}_{u}$ decreases from 1 to ${10}^{-8}$. The solid lines are for $\gamma =3$, and the dashed lines are for $\gamma =6$.

**Figure 6.**Real-time simulation of 180 deformable models (top) with 4404 ellipsoidal particles (bottom) using the as-rigid-as-possible solid simulation technique presented in [3].

**Table 1.**Maximum and average number of solver iterations and average computation time in ns. ${t}_{1}$ and ${t}_{18}$ are times in nanoseconds using a single core with a single thread and 18 cores with 36 threads, respectively.

Figure 2 | Figure 6 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$|{\mathbf{u}}_{\mathbf{i}}-{\mathbf{u}}_{\mathbf{i}-\mathbf{1}}|<{\mathbf{10}}^{-\mathbf{8}}$ | $|{\mathbf{u}}_{\mathbf{i}}-{\mathbf{u}}_{\mathbf{i}-\mathbf{1}}|<{\mathbf{10}}^{-\mathbf{8}}$ | $\parallel {\mathbf{x}}_{\mathbf{1}}^{\left(\mathbf{i}\right)}-{\mathbf{x}}_{\mathbf{2}}^{\left(\mathbf{i}\right)}\parallel <{\mathbf{10}}^{-\mathbf{2}}{\mathit{r}}_{\mathit{m}\mathit{i}\mathit{n}}$ | |||||||||||||

# Solver Iters. | Time | # Solver Iters. | Time | # Solver Iters. | Time | ||||||||||

Max. | Avg. | ${\mathit{t}}_{\mathbf{1}}$ | Max. | Avg. | ${\mathit{t}}_{\mathbf{1}}$ | ${\mathit{t}}_{\mathbf{18}}$ | Max. | Avg. | ${\mathit{t}}_{\mathbf{1}}$ | ${\mathit{t}}_{\mathbf{18}}$ | |||||

Hybrid Newton’s Method | 14 | 4.30 | 310 | 14 | 3.87 | 285 | 12.7 | 13 | 2.46 | 205 | 9.4 | ||||

Hybrid Fixed-Point Iteration Method | 28 | 7.37 | 491 | 28 | 6.07 | 433 | 16.9 | 10 | 2.43 | 189 | 8.7 | ||||

Brent’s Method | 28 | 11.62 | 578 | 29 | 11.21 | 553 | 22.0 | 12 | 4.99 | 487 | 20.8 | ||||

Cross-Section Search | 41 | 40.34 | 16,914 | - | - | - | - | - | - | - | - |

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**MDPI and ACS Style**

Choi, M.G.
Computing the Closest Approach Distance of Two Ellipsoids. *Symmetry* **2020**, *12*, 1302.
https://doi.org/10.3390/sym12081302

**AMA Style**

Choi MG.
Computing the Closest Approach Distance of Two Ellipsoids. *Symmetry*. 2020; 12(8):1302.
https://doi.org/10.3390/sym12081302

**Chicago/Turabian Style**

Choi, Min Gyu.
2020. "Computing the Closest Approach Distance of Two Ellipsoids" *Symmetry* 12, no. 8: 1302.
https://doi.org/10.3390/sym12081302