# Shape Optimum Design by Basis Vector Method Considering Partial Shape Dependence
## 4 Basis Vector Method and Partial Shape Dependency Relation
### Holding Relationship of Arbitrary Three Points  
During the case of shape optimization accompanying the rotation of the structure, to investigate the change in the angle of the structure, consider the angle formed by the three nodes \(a\), \(b\), \(c\) inside the structure and make the angle between the original shape and the basic shape coincide, as shown in Equation (24). \[\theta_{0}=<a_{0}^{\rightarrow}b_{0},a_{0}^{\rightarrow}c_{0}>=<a_{1}^{ \rightarrow}b_{1},a_{1}^{\rightarrow}c_{1}> \tag{24}\]  
Here, \(<a_{0}^{\rightarrow}b_{0},a_{0}^{\rightarrow}c_{0}>\) is the angle between the vector \(a_{0}^{\rightarrow}b_{0}\) and \(a_{0}^{\rightarrow}c_{0}\). Similarly, \(<a_{1}^{\rightarrow}b_{1},a_{1}^{\rightarrow}c_{1}>\) is the angle between the vector \(a_{1}^{\rightarrow}b_{1}\) and \(a_{1}^{\rightarrow}c_{1}\). Since the aim is to investigate the influence on the angle due to structural rotation, and it is assumed that the side length constituting the angle does not change:  
\[\left|a_{0}^{\rightarrow}b_{0}\right|=\left|a_{1}^{\rightarrow}b_{1}\right|,\ \left|a_{0}^{\rightarrow}c_{0}\right|=\left|a_{1}^{\rightarrow}c_{1}\right| \tag{25}\]  
Also, the rotation angle \(\Delta\theta\) from the original shape to the basic shape is taken. The vector that constitutes the new angle \(\theta\) in the transformed shape is as shown in Equation (26). \[\overrightarrow{ab}=w_{0}a_{0}^{\rightarrow}b_{0}+w_{1}a_{1}b_{1}^{ \rightarrow}a\overleftarrow{c}=w_{0}a_{0}^{\rightarrow}c_{0}+w_{1}a_{1}^{ \rightarrow}c_{1} \tag{26}\]  
Then, take an inner product on each side of Equation (26):  
\[\overrightarrow{ab}\cdot\overrightarrow{ac}=w_{0}^{2}a_{0}^{\rightarrow}b_{ 0}\cdot a_{0}^{\rightarrow}c_{0}+w_{1}^{2}a_{1}^{\rightarrow}b_{1}\cdot a_{ 1}^{\rightarrow}c_{1}+w_{0}w_{1}\left(a_{0}^{\rightarrow}b_{0}\cdot a_{1}^{ \rightarrow}c_{1}+a_{1}^{\rightarrow}b_{1}\cdot a_{0}^{\rightarrow}c_{0}\right) \tag{27}\]  
Considering Equation (25) and rearranging it, Equation (28) can be obtained:  
\[\overrightarrow{ab}\cdot\overrightarrow{ac}=\left|a_{0}^{\rightarrow}b_{0} \right|\left|a_{0}^{\rightarrow}c_{0}\right|\cos\theta_{0}[1-2w_{0}w_{1}(1- \cos\Delta\theta)] \tag{28}\]  
Conversely, Equation (29) is obtained using the inner product calculation formula of the vector:  
\[\overrightarrow{ab}\cdot\overrightarrow{ac}=\left|\overrightarrow{ab}\right| \left|\overrightarrow{ac}\right|\cos\theta \tag{29}\]  
Making the first equation of Equation (26) square on both sides, Equation (30) can be obtained:  
\[\left|\overrightarrow{ab}\right|^{2}=w_{0}^{2}\left|a_{0}^{\rightarrow}b_{0} \right|^{2}+w_{1}^{2}\left|a_{1}^{\rightarrow}b_{1}\right|^{2}+2w_{0}w_{1}a _{0}^{\rightarrow}b_{0}\cdot a_{1}^{\rightarrow}b_{1} \tag{30}\]  
Considering Equation (29) and rearranging it, Equation (31) is obtained:  
\[\left|\overrightarrow{ab}\right|^{2}=\left|a_{0}^{\rightarrow}b_{0}\right|^{2 }[1-2w_{0}w_{1}(1-\cos\Delta\theta)] \tag{31}\]  
Likewise, Equation (32) is obtained from the second equation of Equation (26):  
\[\left|\overrightarrow{ac}\right|^{2}=\left|a_{0}^{\rightarrow}c_{0}\right|^{2 }[1-2w_{0}w_{1}(1-\cos\Delta\theta)] \tag{32}\]  
By substituting Equations (31) and (32) into Equation (29), Equation (33) is obtained:  
\[\overrightarrow{ab}\cdot\overrightarrow{ac}=\left|a_{0}^{\rightarrow}b_{0} \right|\left|a_{0}^{\rightarrow}c_{0}\right|\cos\theta[1-2w_{0}w_{1}(1-\cos \Delta\theta)] \tag{33}\]By comparing Equations (28) and (33), Equation (34) is obtained:  
\[\left|a_{0}^{\rightarrow}b_{0}\right|=\left|a_{1}^{\rightarrow}b_{1}\right|,\ \left|a_{0}^{\rightarrow}c_{0}\right|=\left|a_{1}^{\rightarrow}c_{1}\right| \tag{25}\]0  
The new angle \(\theta\) in the shape transformed by Equation (34) always coincides with the angle \(\theta_{0}\) in the original shape, irrespective of the weighting coefficient. When actual shape optimization is performed, it is possible to control the change of the angular value composed of three arbitrary points of the structure using Equation (34).