# A Novel Breast-Volume Self-Measurement Method with Improved Convenient and Accuracy
## 3 Results  
### Breast-Volume Mathematical Model  
#### 3.1.1 Breast Shape Formula Fitting  
First, input the necessary parameters for breast-volume measurement, which include the following:  
1. x-axis parameter Distance \(Ai-Ai^{\prime}\), denoted as \(H_{Ai-Ai^{\prime}}\). 2. y-axis parameters Distance \(Bi-Ci\), denoted as \(W_{Bi-Ci}\), Distance \(Ai^{\prime}-Ci\), denoted as \(W_{Ai^{\prime}-Ci}\), Distance \(Ai^{\prime}-Bi\), denoted as \(W_{Ai^{\prime}-Bi}\). 3. z-axis parameters Distance \(Di-Ai\), denoted as \(L_{Di-Ai}\); Distance \(Ei-Ai\), denoted as \(L_{Ei-Ai}\). 4. Anchor points Ai, Bi, Ci, Gi, and Fi The coordinate of \(Ai\) in the breast shape measurement coordinate system \(o-xyz\) is  
Figure 5: 3D scanning method. denoted as \((Aix,Aiy,Aiz)\), where \(Aix,Aiy,\) and \(Aiz\) are the \(x,y,\) and \(z\)-coordinates of \(Ai\), respectively. Similarly, the coordinate of \(Bi\) is \((Bix,Biy,Biz)\), the coordinate of \(Ci\) is \((Cix,Ciy,Ciz)\), the coordinate of \(Gi\) is \((Gix,Giy,Giz)\), and the coordinate of \(Fi\) is \((Fix,Fiy,Fiz)\). 5. \(\angle GiCiAi\) and \(\angle FiBiAi\) Calculate \(\angle GiCiAi\) and \(\angle FiBiAi\) in the projection plane \(yoz\). Through practical breast shape analysis, it was found that using a block ellipsoid to denote the lower breast can better reflect a realistic breast shape. Hence, with the right breast as an example, the shape equation was derived. The left and right regions are divided by the plane that is perpendicular to the y-axis and passes the line \(A1D1\) projected on the \(yoz\) plane, and both regions can be described by a 1/8 ellipsoid. For the complete ellipsoid, as shown in Figure 6, the local coordinate system \(o-xyz\) is built with the center of the ellipsoid as the origin, which can be expressed by the following mathematical equations:  
\[\text{Complete ellipsoid in the left region}:\ \frac{x^{2}}{{H_{Ai-A^{\prime}}}^{2}}+ \frac{y^{2}}{{W_{Ai^{\prime}-Bi}}^{2}}+\frac{z^{2}}{{L_{Di-A^{\prime}}}^{2}}=1 \tag{1}\]  
\[\text{Complete ellipsoid in the right region}:\ \frac{x^{2}}{{H_{Ai-A^{\prime}}}^{2}}+ \frac{y^{2}}{{W_{Ai^{\prime}-Ci}}^{2}}+\frac{z^{2}}{{L_{Di-A^{\prime}}}^{2}}=1 \tag{2}\]  
Since the practical breasts only account for 1/8 of the ellipsoids in the left and right regions respectively, only the 1/8 parts of Equations (1) and (2) are taken, as shown below: 1/8 ellipsoid in the left region:  
\[\frac{x^{2}}{{H_{Ai-A^{\prime}}}^{2}}+\frac{y^{2}}{{W_{Ai^{\prime}-Bi}}^{2}}+ \frac{z^{2}}{{L_{Di-A^{\prime}}}^{2}}=1,\ x\geq 0,\ y\geq 0,\ z\geq 0 \tag{3}\]  
1/8 ellipsoid in the right region:  
\[\frac{x^{2}}{{H_{Ai-A^{\prime}}}^{2}}+\frac{y^{2}}{{W_{Ai^{\prime}-Ci}}^{2}}+ \frac{z^{2}}{{L_{Di-A^{\prime}}}^{2}}=1,\ x\geq 0,\ y\leq 0,\ z\geq 0 \tag{4}\]  
The anchor point \(Ai\) is projected onto the \(YOZ\) plane and the projected point \(A11\) is obtained. The coordinate of \(A11\) can be expressed as follows:  
\[A11=Ai\times l(n,\ x)=(0,\ Aiy,\ Aiz) \tag{5}\]  
By moving the origin \(o\) of the local coordinate systems of 1/8 ellipsoids in the left and right regions to the point \(A11\) of the global coordinate system, we can rewrite Equations (3) and (4) as  
Figure 6: Schematic of the complete ellipsoid in the right and left regions. \(1/8\) ellipsoid in the left region:  
\[\frac{\left(x\right)^{2}}{H_{Ai-Ai^{\prime}}^{2}}+\frac{\left(y+Aiy_{y}\right)^{ 2}}{W_{Ai^{\prime}-Bi}}+\frac{\left(z-Aiz_{z}\right)^{2}}{L_{Di-Ai^{\prime}}^{2 }}=1,\ x\geq 0,\ y\geq 0,\ z\geq 0 \tag{6}\]  
\(1/8\) ellipsoid in the right region:  
\[\frac{\left(x\right)^{2}}{H_{Ai-Ai^{\prime}}^{2}}+\frac{\left(y+Aiy_{y}\right) ^{2}}{W_{Ai^{\prime}-Ci}}+\frac{\left(z-Aiz_{z}\right)^{2}}{L_{Di-Ai^{\prime} }^{2}}=1,\ x\geq 0,\ y\leq 0,\ z\geq 0 \tag{7}\]  
When the ellipsoid rotates around the z axis, it can better fit the practical breast shape (Figure 7).