# Categorizing Three-Dimensional Symmetry Using Reflection, Rotoinversion, and Translation Symmetry

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

**Definition**

**1.**

**tensor product**$b\otimes a$ is the linear map:${\mathbb{R}}^{m}\to {\mathbb{R}}^{n}$ defined such that $\forall x\in {\mathbb{R}}^{m}$:

**Definition**

**2.**

**orthogonal tensor**is a linear map $Q:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ satisfying ${Q}^{T}={Q}^{-1}$. Orthogonal tensors have a determinant equal to 1 or −1. Orthogonal tensors form a group, i.e., the composition of two orthogonal tensors is again an orthogonal tensor.

**Definition**

**3.**

**proper orthogonal tensors**. Proper orthogonal tensors represent a counterclockwise rotation of an angle $0\le \theta \le 2\pi $ around a vector $p\in {\mathbb{R}}^{3}$.

**Definition**

**4.**

**improper orthogonal tensors**. Improper orthogonal tensors represent a counterclockwise rotation of an angle $0\le \theta \le 2\pi $ around a vector $p\in {\mathbb{R}}^{3}$ in addition to a reflection around the plane perpendicular to $p$.

**Remark**

**1.**

- -
- Given $p\in {\mathbb{R}}^{3}$ and $0\le \theta \le 2\pi $, an orthogonal tensor $Q\left(p,\theta \right)=Q\left(-p,-\theta \right)=Q\left(-p,2\pi -\theta \right)$. Therefore, every orthogonal tensor can be associated with an angle $0\le \theta \le \pi $. Additionally, the transpose of an orthogonal tensor is given by: ${Q}^{T}=Q\left(-p,\theta \right)=Q\left(p,-\theta \right)$.
- -
- Given an orthogonal tensor $Q:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$, the cosine of the associated angle $\theta $ can be calculated using the trace of $Q$:$$\mathrm{tr}\left(Q\right)=\mathrm{det}\left(Q\right)+2\mathrm{cos}\left(\theta \right)=\{\begin{array}{c}1+2\mathrm{cos}\left(\theta \right),Q\mathrm{is}\mathrm{proper}\\ -1+2\mathrm{cos}\left(\theta \right),Q\mathrm{is}\mathrm{improper}\end{array}$$Assuming $0\le \theta \le \pi $ the sine of the associated angle $\theta $ can be calculated as$$\mathrm{sin}\left(\theta \right)=\sqrt{1-{\mathrm{cos}}^{2}\left(\theta \right)}$$The components of the vector $p$ can then be calculated when $\mathrm{sin}\left(\theta \right)\ne 0$ using the off diagonal components:$${p}_{i}={\displaystyle \sum}_{j,k=1}^{3}\frac{{\epsilon}_{ijk}{Q}_{jk}}{2\mathrm{sin}\left(\theta \right)}$$
- -
- The composition of two orthogonal tensors is again orthogonal. A reflection followed by a proper orthogonal transformation gives an improper orthogonal tensor.

**Remark**

**2.**

**Definition**

**5.**

**hyperplane**. A

**hyperplane**in ${\mathbb{R}}^{3}$ is the analytical representation of a geometric plane. Without loss of generality, $\Vert p\Vert =1$. In that case, the vector ${x}_{0}=-bp\in H$ and $H$ can also be represented as

**Definition**

**6.**

**Definition**

**7.**

**reflection around a hyperplane**$H\subset {\mathbb{R}}^{3}$ is the linear operation ${R}_{H}:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ such that $\forall x\in {\mathbb{R}}^{3}:{R}_{H}\left(x\right)=x-2\left(p\cdot \left(x-{x}_{0}\right)\right)p$.

**Remark**

**3.**

**Definition**

**8.**

**geometric rotation around**$L$ with an angle $\theta $ is given by the linear function ${R}_{L}$ defined such that $\forall x\in {\mathbb{R}}^{3}$:

**Theorem**

**1.**

**Proof of Theorem**

**1.**

**Remark**

**4.**

**Approximate Symmetry Characterization:**

- -
- a reflection around the hyperplane $H$ defined by ${p}_{rp}\in {\mathbb{R}}^{3}$ and $b=-0.5{c}_{p}$, and
- -
- a rotation of angle ${\theta}_{rp}$ around the axis defined by the geometric points represented by the two vectors ${x}_{f}$ and ${x}_{f}+p$

- -
- a reflection around the hyperplane defined by ${p}_{rp}\in {\mathbb{R}}^{3}$ and $b=-0.5{c}_{p}$.
- -
- A translation parallel to the hyperplane along the vector $\left(\begin{array}{c}0\\ {c}_{q}\\ {c}_{r}\end{array}\right)$.

**Property**

**1.**

**Theorem 2 and Property 2.**

**Proof of Theorem**

**2.**

## 3. Materials and Methods

#### 3.1. Symmetry Analysis Procedure

#### 3.2. Symmetry Analysis Example Models

#### 3.2.1. Model 1: Reflection Symmetry

#### 3.2.2. Model 2: Rotoinversion Symmetry

#### 3.2.3. Model 3: Translation Symmetry

## 4. Results

## 5. Discussions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Dialog Window for Translation Symmetry

## Appendix B. Subroutine on Wolfram Mathematica to Extract Object Characteristics

## References

- Wagemans, J. Characteristics and models of human symmetry detection. Trends Cogn. Sci.
**1997**, 1, 346–352. [Google Scholar] [CrossRef] - Tomkinson, G.R.; Olds, T.S. Physiological correlates of bilateral symmetry in humans. Int. J. Sports Med.
**2000**, 21, 545–550. [Google Scholar] [CrossRef] [PubMed] - Santoro, D.; Tantavisut, S.; Aloj, D.; Karam, M.D. Diaphyseal osteotomy after post-traumatic malalignment. Curr. Rev. Musculoskelet Med.
**2014**, 7, 312–322. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ten Berg, P.W.L.; Dobbe, J.G.G.; van Wolfswinkel, G.; Strackee, S.M.; Streekstra, G.J. Validation of the contralateral side as reference for selecting radial head implant sizes. Surg. Radiol. Anat.
**2016**, 38, 801–807. [Google Scholar] [CrossRef] [PubMed] - Tümer, N.; Blankevoort, L.; van de Giessen, M.; Terra, M.P.; de Jong, P.A.; Weinans, H.; Tuijthof, G.J.; Zadpoor, A.A. Bone shape difference between control and osteochondral defect groups of the ankle joint. Osteoarthr. Cartil.
**2016**, 24, 2108–2115. [Google Scholar] [CrossRef] [PubMed] - Sun, S.-P.; Hsu, K.-W.; Chen, J.-S. The evaluation of female breast symmetry by calculating related physics parameters. J. Med. Biol. Eng.
**2007**, 27, 87–90. [Google Scholar] - Jaremko, J.L.; Poncet, P.; Ronsky, J.; Harder, J.; Dansereau, J.; Labelle, H.; Zernicke, R.F. Indices of torso asymmetry related to spinal deformity in scoliosis. Clin. Biomech.
**2002**, 17, 559–568. [Google Scholar] [CrossRef] - Oxborrow, N.J. Assessing the child with scoliosis: the role of surface topography. Arch. Dis. Child
**2000**, 83, 453–455. [Google Scholar] [CrossRef] [PubMed] - Komeili, A.; Westover, L.; Parent, E.; ElRich, M.; Adeeb, S. Monitoring for Idiopathic Scoliosis Curve Progression Using Surface Topography Asymmetry Analysis of the Torso in Adolescents. Spine J.
**2015**, 15, 743–751. [Google Scholar] [CrossRef] [PubMed] - Ho, C.; Parent, E.; Watkins, E.; Moreau, M.J.; Hedden, D.; ElRich, M.; Adeeb, S. Asymmetry Assessment Using Surface Topography in Healthy Adolescents. Symmetry
**2015**, 7, 1436–1454. [Google Scholar] [CrossRef] [Green Version] - Komeili, A.; Westover, L.; Parent, E.; El-Rich, M.; Adeeb, S. Correlation between a Novel Surface Topography Asymmetry Analysis and Radiographic Data in Scoliosis. Spine Deform.
**2015**, 3, 303–311. [Google Scholar] [CrossRef] [PubMed] - Hill, S.; Franco-Sepulveda, E.; Komeili, A.; Trovato, A.; Parent, E.; Hill, D.; Lou, E.; Adeeb, S. Assessing Asymmetry Using Reflection and Rotoinversion in Biomedical Engineering Applications. J. Eng. Med.
**2014**, 228, 523–529. [Google Scholar] [CrossRef] [PubMed] - Hong, A.; Jaswal, N.; Westover, L.; Parent, E.C.; Moreau, M.; Hedden, D.; Adeeb, S. Surface topography classification trees for assessing severity and monitoring progression in adolescent idiopathic scoliosis. Spine
**2017**, 42, E781–E787. [Google Scholar] [CrossRef] [PubMed] - Ghaneei, M.; Komeili, A.; Li, Y.; Parent, E.C.; Adeeb, S. 3D Markerless Asymmetry Analysis in the Management of Adolescent Idiopathic Scoliosis. BMC Musculoskelet. Disord.
**2018**, 19, 385–395. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**(

**a**) Model 1 is a purely bilateral symmetric object modelled after an ant and is (

**b**) imported into Geomagic as a mesh, (

**c**) duplicated as a point cloud (grey) and reflected over an arbitrary plane (Step 1), where (

**d**) the reflected model (grey) is aligned to the original model (yellow) using the Best Fit function (Step 2).

**Figure 2.**(

**a**) Model 2 is a bilaterally and rotationally symmetric object reminiscent of two people holding a ball and is (

**b**) imported into Geomagic as a mesh, (

**c**) duplicated as a point cloud and reflected over an arbitrary plane (Step 1), where (

**d**) the reflected model (grey) is aligned to the original model (yellow) using the Best Fit function (Step 2).

**Figure 3.**(

**a**) Model 3 is an object representing an alternating rung ladder and is (

**b**) imported into Geomagic as a mesh, (

**c**) duplicated as a point cloud and reflected over an arbitrary plane (Step 1), where (

**d**) the reflected model (grey) is aligned to the original model (yellow) using the Best Fit function (Step 2).

**Table 1.**Transformation matrices exported from Geomagic after the alignment of the mirrored test models. These matrices are used to extract the characteristics of object’s symmetry in Table 2. Equation (13) is the general form of the matrices which contain components of the Psym normal and rotation.

Symmetry/Model | Transformation Matrix Exported from Geomagic |
---|---|

Model 1 | $T=\left[\begin{array}{cccc}-0.500& 0.750& -0.433& 124.102\\ 0.750& 0.625& 0.217& -62.052\\ -0.433& 0.217& 0.875& 35.826\\ 0.00& 0.00& 0.00& 1.00\end{array}\right]$ |

Model 2 | $T=\left[\begin{array}{cccc}-0.750& -0.125& 0.650& 63.841\\ -0.625& -0.188& -0.758& 392.180\\ -0.217& 0.974& -0.063& 88.015\\ 0.00& 0.00& 0.00& 1.00\end{array}\right]$ |

Model 3 | $T=\left[\begin{array}{cccc}-0.5014& -0.7494& 0.4324& 176.1938\\ -0.7501& 0.6256& 0.2146& 73.4342\\ 0.4313& 0.2167& 0.8758& -76.7989\\ 0.00& 0.00& 0.00& 1.00\end{array}\right]$ |

Results | Model 1 | Model 2 | Model 3 |
---|---|---|---|

Psym Normal | $p=\left[\begin{array}{c}-0.866002\\ 0.433063\\ -0.250095\end{array}\right]$ | $p=\left[\begin{array}{c}0.865975\\ 0.433315\\ -0.249994\end{array}\right]$ | $p=\left[\begin{array}{c}0.86643\\ 0.432667\\ -0.24921\end{array}\right]$ |

Fixed Point | $x=\left[\begin{array}{c}80.2827\\ -3.74933\\ 2\end{array}\right]$ | $x=\left[\begin{array}{c}99.9904\\ 150.009\\ 199.836\end{array}\right]$ | $x=\left[\begin{array}{c}-12055.3\\ 18492\\ -10217.7\end{array}\right]$ |

Rotation | 4.21943 × 10^{−18} rad0.00° | −1.57126 rad 90.026° | −0.00123492 rad 0.006° |

Translation Magnitude | 0.0160606 mm | 352.614 mm | 29.8993 mm |

**Table 3.**Validation of approach. Displaying the measurements outlined above that compare the method’s results to the expected results.

Deviation | RMS Error | Characteristic | Absolute Difference | With Respect to Model Height/Model Max Length | |
---|---|---|---|---|---|

Model 1 | 0.002 mm (standard) | 0.007 mm | Translation magnitude | 0.016 mm | 0.012%/0.005% |

Model 2 | 0.003 mm (standard) | 0.026 mm | Fixed Point | 0.037 mm | 0.038%/0.028% |

Model 3 | 12.1988 mm/−12.1894 mm (max/min) | 0.000 mm | Translation magnitude | 0.101 mm | 0.050%/0.044% |

© 2019 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**

Baclig, M.M.; Westover, L.; Adeeb, S.
Categorizing Three-Dimensional Symmetry Using Reflection, Rotoinversion, and Translation Symmetry. *Symmetry* **2019**, *11*, 1132.
https://doi.org/10.3390/sym11091132

**AMA Style**

Baclig MM, Westover L, Adeeb S.
Categorizing Three-Dimensional Symmetry Using Reflection, Rotoinversion, and Translation Symmetry. *Symmetry*. 2019; 11(9):1132.
https://doi.org/10.3390/sym11091132

**Chicago/Turabian Style**

Baclig, Maria Martine, Lindsey Westover, and Samer Adeeb.
2019. "Categorizing Three-Dimensional Symmetry Using Reflection, Rotoinversion, and Translation Symmetry" *Symmetry* 11, no. 9: 1132.
https://doi.org/10.3390/sym11091132