# Shape Optimum Design by Basis Vector Method Considering Partial Shape Dependence
## 4 Basis Vector Method and Partial Shape Dependency Relation
### Corresponding Relation of a Line Segment Composed of Any Two Points  
Considering two arbitrary points \(a\) and \(b\) inside the structure, and taking direction vectors from point \(a\) to point \(b\) in the original shape, the basic shape and transformed new shape are as follows:  
\[a_{0}^{\rightarrow}b_{0} = \left[x_{b0}-x_{a0},\ y_{b0}-y_{a0}\right]^{T} \tag{9}\] \[a_{1}^{\rightarrow}b_{1} = \left[x_{b1}-x_{a1},\ y_{b1}-y_{a1}\right]^{T}\] (10) \[\overset{\rightarrow}{ab} = \left[x_{b}-x_{a},\ y_{b}-y_{a}\right]^{T} \tag{11}\]  
Considering if the shape transformation on the weighting factor is \(\omega_{0}\) and \(\omega_{1}\), the direction vector after transformation according to Equation (8) is represented by Equation (12),  
\[\overset{\rightarrow}{ab} = \left\{\left(w_{0}x_{b0}+w_{1}x_{b1}\right)-\left(w_{0}x_{a0}+w_{ 1}x_{a1}\right),\ \left(w_{0}y_{b0}+w_{1}y_{b1}\right)-\left(w_{0}y_{a0}+w_{1}y_{a1}\right) \right\}^{T} \tag{12}\]  
Taking Equations (9) and (10) into consideration, the above equation becomes Equation (13)  
\[\overset{\rightarrow}{ab} = w_{0}a_{0}^{\rightarrow}b_{0}+w_{1}a_{1}^{\rightarrow}b_{1} \tag{13}\]  
Considering Equation (13), it is found that the direction vector consisting of two arbitrary points inside the structure has a linear combination relation with the same weighting coefficient as the shape transformation formula.  
Furthermore, to investigate the change of the line segment \(\overset{\rightarrow}{ab}\), Equation (14) is obtained by squaring both ends of Equation (13).  
\[\left|\overset{\rightarrow}{ab}\right|^{2} = w_{0}\left|a_{0}^{\rightarrow}b_{0}\right|^{2}+w_{1}\left|a_{1} ^{\rightarrow}b_{1}\right|^{2}+2w_{0}w_{1}\left|a_{0}^{\rightarrow}b_{0} \right|\left|a_{1}^{\rightarrow}b_{1}\right|\cos\theta \tag{14}\]  
Here, \(\theta\) is the angle between the vector \(a_{0}^{\rightarrow}b_{0}\) and \(a_{1}^{\rightarrow}b_{1}\). Using this formula, it can calculate the length of the line segment after shape transformation. As a special case, when the basis vector becomes parallel, the above equation becomes Equation (15).  
\[\left|\overset{\rightarrow}{ab}\right|^{2} = w_{0}\left|a_{0}^{\rightarrow}b_{0}\right|^{2}+w_{1}\left|a_{ 1}^{\rightarrow}b_{1}\right|^{2}+2w_{0}w_{1}\left|a_{0}^{\rightarrow}b_{0} \right|\left|a_{1}^{\rightarrow}b_{1}\right| \tag{15}\]  
To express the length of the line segment, Equation (16) is obtained.  
\[\left|\overset{\rightarrow}{ab}\right| = w_{0}\left|a_{0}^{\rightarrow}b_{0}\right|+w_{1}\left|a_{1}^{ \rightarrow}b_{1}\right| \tag{16}\]Also, when \(w_{0}+w_{1}=1\) is substituted, Equation (18) is obtained.  
\[\left|\overset{\rightarrow}{ab}\right|=\left|a_{0}b_{0}\right| \tag{17}\]  
Here, the length of the line segment whose shape has been converted indicates that the original line segment length is always maintained, regardless of the weighting factor.  
\[\left|\overset{\rightarrow}{ab}\right|=\left|a_{0}b_{0}\right| \tag{18}\]  
When using the Equation (18) basis vector method for shape optimization, if the complex shape is decomposed into multiple line segments and holes, and convex-concave are set to the same shape, then in the process of shape optimization--even if the combination of any weight coefficients is converted--the shape will remain unchanged.