# Online Uniformly Inserting Points on the Sphere +
## 3 Results
### Analysis of the Lower Bound
#### 3.2.2 3-Point Sequence Insertion  
**Lemma 1**.: _For 3-point sequence insertions, the gap ratio is no less than 1.78._  
Proof of Lemma 1.: Insert points \(p_{1},p_{2},p_{3}\) one after another. After inserting \(p_{1}\) and \(p_{2}\), we have Equation (4) according to the previous case.  
After inserting point \(p_{3}\), these 3 points determine a plane which intersects the sphere. Let \(R\) denote the radius of the sphere, \(O^{\prime}\) denote the center of the circumcircle of \(\Delta p_{1}p_{2}p_{3}\), and \(r\) denote the radius of the circumcircle; and let \(\angle p_{1}O^{\prime}p_{2}=\beta,\angle p_{2}O^{\prime}p_{3}=\gamma\), \(\angle O^{\prime}Op_{1}=\delta\). Obviously, \(G_{3}=2R(\pi-\delta)\). W.l.o.g., assuming that \(0<\gamma\leq 2\pi-\beta-\gamma\leq\beta<\pi\) and \(0<\delta\leq\pi/2\), we have \(g_{3}=\stackrel{{\frown}}{{d}}(p_{2},p_{3})=R\cdot\angle p_{2} Op_{3}\). Thus,  
\[r_{3}=\frac{G_{3}}{g_{3}}=\frac{2(\pi-\delta)}{\angle p_{2}Op_{3}} \tag{5}\]  
By the law of cosines, we have \(|p_{2}p_{3}|^{2}=2R^{2}(1-\cos\angle p_{2}Op_{3})\) for \(\Delta Op_{2}p_{3}\), and we also have \(|p_{2}p_{3}|^{2}=2r^{2}(1-\cos\gamma)\) for \(\Delta O^{\prime}p_{2}p_{3}\), where \(r\) is the radius of the circumcircle of \(\Delta p_{1}p_{2}p_{3}\), and \(r=R\sin\delta\). Combining these two equations we have \(\angle p_{2}Op_{3}=\arccos(1-\sin^{2}\delta(1-\cos\gamma))\).  
Figure 6: Analysis of 2- and 3-point sequence insertion. (**a**) 2-point sequence insertion; (**b**) 3-point sequence insertion.  
In the same way, we can get the value of \(\alpha\), \(\alpha=\angle p_{1}Op_{2}=\arccos(1-\sin^{2}\delta(1-\cos\beta))\).  
So we have  
\[r_{2}=\frac{2\pi-\arccos(1-\sin^{2}\delta(1-\cos\beta))}{\arccos(1-\sin^{2} \delta(1-\cos\beta))}\]  
\[r_{3}=\frac{2(\pi-\delta)}{\arccos(1-\sin^{2}\delta(1-\cos\gamma))}\]  
In order to get to lower bound, we should find the minimum gap ratio for the worst case of \(\{r_{2},r_{3}\}\). In this way, the lower bound must exist in the case where \(r_{2}=r_{3}\). As \(\delta\) is independent to other parameters, we first compute the partial derivative to \(\delta\): \(\frac{\mathrm{d}r_{3}}{\mathrm{d}\delta}=-\frac{\sin 2\delta(1-\cos\beta)}{ \arccos^{2}\chi\sqrt{1-\chi^{2}}}<0\), where \(\chi=1-\sin^{2}\delta(1-\cos\beta)\). In the same way, \(\frac{\mathrm{d}r_{3}}{\mathrm{d}\delta}<0\). So the gap ratio of \(r_{3}\) and \(r_{2}\) is a decrement function w.r.t. \(\delta\). Thus, the minimal gap is achieved when \(\delta=\pi/2\) where all the three points are on the spherical large arc.  
So we set \(\delta=\pi/2\) and let \(r_{2}\)=\(r_{3}\), and we get \(\gamma=\frac{\pi\beta}{2\pi-\beta}\). By the assumption that \(0<\gamma\leq 2\pi-\beta-\gamma\leq\beta<\pi\), we can finally get:  
\[\begin{cases}\gamma=\frac{\pi\beta}{2\pi-\beta}\\ \gamma\leq\frac{2\pi-\beta}{2}\\ \gamma\geq 2(\pi-\beta)\end{cases} \tag{6}\]  
By solving (6), we can get \(0.719\pi\leq\beta\leq 0.764\pi\). Considering that with \(\delta\) fixed, the gap ratio is a decreasing function w.r.t. \(\beta\). In this case, the minimal gap ratio is \(1.78\).