# Mapping Based Quality Metrics for Mesh Deformation Algorithms Using Radial Basis Functions
## 3 Mapping-Based Metrics
### Mapping Relationships
\begin{array}{ccc}1+H_{11}&H_{12}&H_{13}\\ H_{21}&1+H_{22}&H_{23}\\ H_{31}&H_{32}&1+H_{33}\end{array} \right]\left[ \begin{array}{c}V_{1}\\ V_{2}\\ V_{3}\end{array} \right]=\mathbf{F}\left[ \begin{array}{c}V_{1}\\ V_{2}\\ V_{3}\end{array} \right] \tag{12}\]  
Therefore, the image of the vector can be written as:  
\[\mathbf{v}=\left[ \begin{array}{ccc}V_{1}&V_{2}&V_{3}\end{array} \right]\mathbf{F}^{ T}\left[ \begin{array}{c}\mathbf{E}^{1}\\ \mathbf{E}^{2}\\ \mathbf{E}^{3}\end{array} \right]=\left[ \begin{array}{ccc}V_{1}&V_{2}&V_{3}\end{array}  \right]\left[ \begin{array}{c}\mathbf{e}^{1}\\ \mathbf{e}^{2}\\ \mathbf{e}^{3}\end{array} \right] \tag{13}\]  
Therefore, the mapping \(M\) for the vectors is:  
\[\mathbf{v}=M(V)=\left[ \begin{array}{ccc}V_{1}&V_{2}&V_{3}\end{array} \right] \left[ \begin{array}{c}\mathbf{e}^{1}\\ \mathbf{e}^{2}\\ \mathbf{e}^{3}\end{array} \right] \tag{14}\]  
Based on Equation (14), the coordinates of image vector \(\mathbf{v}\) according to image base vectors is the same as the coordinates of the initial vector \(\mathbf{V}\) according to initial base vectors  
Figure 1: Configurations in deforming the element process. of the reference configuration. Therefore, the mapping can be measured according to the change of unit base vectors of initial configuration over the computational domain. Given \(l_{V}\), the length of vector \(\mathbf{V}\) can be formulated as:  
\[l_{\mathbf{V}}=\sqrt{\mathbf{V}\cdot\mathbf{V}}=\sqrt{V_{i}V_{i}} \tag{15}\]  
The length of vector \(\mathbf{v}\) can obtained by:  
\[{l_{\mathbf{v}}}^{2}=\mathbf{v}\cdot\mathbf{v}=\left[ \begin{array}{cc}V_{1}&V_{2}&V_{3} \end{array} \right]\mathbf{F}^{T}\mathbf{F}\left[ \begin{array}{c}\mathbf{V}_{1}\\ \mathbf{V}_{2}\\ \mathbf{V}_{3}\end{array} \right] \tag{16}\]  
According to Equation (11), it easily follows that:  
\[{l_{\mathbf{v}}}^{2}=\left[ \begin{array}{cc}V_{1}&V_{2}&V_{3}\end{array} \right] (\mathbf{I}+\mathbf{H}^{T})(\mathbf{I}+\mathbf{H})\left[ \begin{array}{c}\mathbf{V}_{1}\\ \mathbf{V}_{2}\\ \mathbf{V}_{3}\end{array} \right]=l_{\mathbf{v}}^{2}+2\left[ \begin{array}{cc}V_{1}&V _{2}&V_{3}\end{array} \right]\mathbf{E}\left[ \begin{array}{c}\mathbf{V}_{1}\\ \mathbf{V}_{2}\\ \mathbf{V}_{3}\end{array} \right] \tag{17}\]  
where \(\mathbf{E}\) is the green strain tensor. Equation (17) can be given in a components format by:  
\[v_{i}\cdot v_{i}=V_{i}\cdot V_{i}+2E_{ij}V_{i}V_{j} \tag{18}\]  
For a 2-D case, consider specified vectors \(\mathbf{Y},\mathbf{W}\) and their image vectors \(\mathbf{y},\mathbf{w}\). The area of the parallelogram formed by the two vectors is computed as follows:  
\[S=|\mathbf{Y}\times\mathbf{W}|=|Y_{1}W_{2}-Y_{2}W_{1}| \tag{19}\]  
And its image is given by:  
\[s=|\mathbf{y}\times\mathbf{w}|=|y_{1}w_{2}-y_{2}w_{1}|=\det(\mathbf{F})\cdot S \tag{20}\]  
The angle between two vectors is related to the dot product. Let \(\theta\) be the angle between \(\mathbf{y},\mathbf{w}\) and \(\theta_{0}\) be the angle between \(\mathbf{Y},\mathbf{W}\), then:  
\[\cos\theta_{0}=\frac{\mathbf{Y}\cdot\mathbf{W}}{|\mathbf{Y}||\mathbf{W}|}=\frac{Y_{i}W_{i}}{ \sqrt{Y_{i}Y_{i}}\sqrt{W_{i}W_{i}}} \tag{21}\]  
\[\cos\theta=\frac{\mathbf{y}\cdot\mathbf{w}}{|\mathbf{y}||\mathbf{w}|}=\frac{Y_{i}W_{i}+2E_{ij }Y_{i}W_{j}}{\sqrt{Y_{i}Y_{i}+2E_{ij}Y_{i}Y_{j}}\sqrt{W_{i}W_{i}+2E_{ij}W_{i} W_{j}}} \tag{22}\]  
The mapping quality of the deformation algorithm is related to the variety of the unit base vectors in the reference configuration as mentioned before. Therefore, the mapping-based size metrics \(f_{size}\) and skew metrics \(f_{skew}\) of the mesh deformation algorithm of a 2-D mesh are established to measure the area change and angle change based on the mapping relationship of the unit base vectors.