## 2 Orthogonal Projection onto a Spatial Parametric Curve
There, we have  
\[\Delta t=\dfrac{\left\langle c^{\prime}(t_{0}),q-c(t_{0})\right\rangle}{ \left\langle c^{\prime}(t_{0}),c^{\prime}(t_{0})\right\rangle} \tag{2}\]7  
Because the footpoint \(q\) is located between the center \(\mathrm{m}(t)\) of the osculating sphere and the projecting point \(\mathrm{v}\), so the parameter \(u(t)\) should be  
\[\Delta t=\dfrac{\left\langle c^{\prime}(t_{0}),q-c(t_{0})\right\rangle}{ \left\langle c^{\prime}(t_{0}),c^{\prime}(t_{0})\right\rangle} \tag{2}\]8  
Substituting Equation (20) into Equation (18), we get the footpoint coordinate  
\[\Delta t=\dfrac{\left\langle c^{\prime}(t_{0}),q-c(t_{0})\right\rangle}{ \left\langle c^{\prime}(t_{0}),c^{\prime}(t_{0})\right\rangle} \tag{2}\]9  
Secondly, we present the proof of the theorem, our proof method used in the following is distinguished from that literature [23]. We use the method of the numerical analysis similar to those in the literature [24; 25; 26; 27; 28]. **Theorem 1**: Let \(q(t)-c(t)\) be a three-dimensional real parametric curve function, assume that \(q(t)-c(t)\) has first, second derivatives in an interval \(I\). If \(q(t)-c(t)\) has a simple root \(\alpha\in I\), and \(t_{n}\) is close to \(\alpha\), then the iterative method Equation (4) is quadratically convergent. **Proof. ** Let \(e_{n}=t_{n}-\alpha\), and let \(\alpha\) be a simple root of \(q(t)-c(t)=0\), _i.e.,_\(q(\alpha)-c(\alpha)=0,q^{\prime}(\alpha)-c^{\prime}(\alpha)\
eq\ 0\). For \(c(t)\), using the [PERSON]'s expansion around \(\alpha\), we get  
Figure 1:0  
Figure 1:1  
Figure 1:2  
where \(\mathrm{B}_{i}=(1/i!)c^{(i)}(\alpha),i=0,1,2,...\). For convenience, we employ the symbolic computation of the Maple 18 package to compute the Taylor's expansion as follows. Combining the derivation Equation (7)-(21) with the condition of orthogonal projection Equation (6), as well as Equation (22)-(24), and after simplifying, we get a formula of the parameter increment \(\Delta t\),  
Figure 1:3  
since  
Figure 1:4  
Substituting Equation (26) into Equation (25), we can get the following error equation  
Figure 1:5  
This means that the method defined by Equation (4) is quadratically convergent. In the following, we prove that Equation (4) is globally convergent. **Theorem 2**: For the test point \(p\), if there only exists one orthogonal projection point, then the Equation (4) is globally convergent. Proof. : This proof is analogous to the proof in the literature [29; 30]. If we start from an initial point on the left-hand side of \(\alpha\), then the footpoint \(q\) must be located on the right-hand side of the initial point, so \(\Delta t\) is a positive real number. By Equation (4), we define the following iterative sequence:  
Figure 1:6  
where  
Figure 1:7  
For \(t_{n}<\alpha\), the sequence \(\{t_{n}\}\) is strictly monotonic increasing. If \(t_{n}>\alpha\), and then, by at most three iterations, the sequence \(\{t_{n}\}\) converges to \(\alpha\). We should note that this iterative sequence \(\{t_{n}\}\) is similar to an attenuated pendulum. Moreover, if the initial point is to the right-hand side of the root \(\alpha\), the convergence works in a similar way. On the other hand, the footpoint \(q\) is jointly determined by the center of the osculating sphere and the orthogonal projection point \(v\). Furthermore, the footpoint \(q\) is always associated with the test point \(p\), but it is not restricted by the initial iteration point. Therefore, Equation (4) is globally convergent. **Remark 5**: The global convergence of the method in the iterative Equation (4) only requires that \(c(t)\) is \(C^{2}\), however, the global convergence of the method in the literature [26; 27; 28] must satisfy some stronger sufficient conditions.