## 2 Orthogonal Projection onto a Spatial Parametric Curve
In addition, it is not necessary to compute the osculating circle \(\tilde{c}\) which is the intersection of the osculating plane and the osculating sphere \(\tilde{s}\) of the spatial parametric curve \(c(t)\) at \(t=t_{0}\). Hence, our proposed method improves the efficiency and stability for computing \(q\). Figure 2: Illustration of the second-order geometric iteration for computing the minimum distance between a point and a spatial parametric curve. In this paper, we prove the case in which \(R(t)=\left|\dfrac{1}{k(t)}\right|=\dfrac{1}{k(t)}\). In addition, the proof method is also valid for the case \(R(t)=\left|\dfrac{1}{k(t)}\right|=\dfrac{-1}{k(t)}\) only with an opposite sign. The center of the osculating sphere of the spatial parametric curve \(c(t)\) is defined by  
\[\mathrm{m}(t)=c(t)+\beta(t)/k(t) \tag{10}\]  
Now substituting Equations (7) and (8) into Equation (10), we obtain  
\[\mathrm{m}(t)= c(t)+\dfrac{A^{3}(t)}{B(t)}[\dfrac{A(t)}{B(t)}c^{\prime\prime}(t )-\dfrac{\left\langle c^{\prime}(t),c^{\prime\prime}(t)\right\rangle}{A(t)B(t )}c^{\prime}(t)] \tag{11}\] \[= c(t)+\dfrac{A(t)^{2}}{B(t)^{2}}[c^{\prime\prime}(t)-A(t)\left\langle c ^{\prime}(t),c^{\prime\prime}(t)\right\rangle c^{\prime}(t)]\]  
The equation of the osculating plane of the spatial parametric curve \(c(t)\) can be expressed as following  
\[\left\langle\mathrm{x}-c(t),\gamma(t)\right\rangle=0 \tag{12}\]  
where \(\mathrm{x}=(x,y,z)\). Assume that the orthogonal projection of the test point \(p\) onto the osculating plane of the spatial parametric curve \(c(t)\) yields a point \(\mathrm{v}\). Thus, the equation of line segment \(\overrightarrow{p\mathrm{v}}\) is the following  
\[\Delta t=\dfrac{\left\langle c^{\prime}(t_{0}),q-c(t_{0})\right\rangle}{ \left\langle c^{\prime}(t_{0}),c^{\prime}(t_{0})\right\rangle} \tag{2}\]0  
where \(w\) is the parameter. Substituting Equation (13) into (12), we can obtain the parameter \(w\) as the following,  
\[\Delta t=\dfrac{\left\langle c^{\prime}(t_{0}),q-c(t_{0})\right\rangle}{ \left\langle c^{\prime}(t_{0}),c^{\prime}(t_{0})\right\rangle} \tag{2}\]1  
hence,  
\[\Delta t=\dfrac{\left\langle c^{\prime}(t_{0}),q-c(t_{0})\right\rangle}{ \left\langle c^{\prime}(t_{0}),c^{\prime}(t_{0})\right\rangle} \tag{2}\]2  
Substituting Equation (14) into Equation (13), we can get the coordinate value of the projection point \(\mathrm{v}\) as the following  
\[\Delta t=\dfrac{\left\langle c^{\prime}(t_{0}),q-c(t_{0})\right\rangle}{ \left\langle c^{\prime}(t_{0}),c^{\prime}(t_{0})\right\rangle} \tag{2}\]3  
where parameter \(w\) is the same as that of Equation (14). The corresponding parametric equation of the line segment that connects the projecting point \(\mathrm{v}\) and the center \(\mathrm{m}(t)\) of the osculating sphere can be expressed as  
\[\Delta t=\dfrac{\left\langle c^{\prime}(t_{0}),q-c(t_{0})\right\rangle}{ \left\langle c^{\prime}(t_{0}),c^{\prime}(t_{0})\right\rangle} \tag{2}\]4  
where \(u(t)\) is a parameter for the line segment between the center \(\mathrm{m}\) of the osculating sphere and the projection point \(\mathrm{v}\). We know that the equation of the osculating sphere is  
\[\Delta t=\dfrac{\left\langle c^{\prime}(t_{0}),q-c(t_{0})\right\rangle}{ \left\langle c^{\prime}(t_{0}),c^{\prime}(t_{0})\right\rangle} \tag{2}\]5  
Because the footpoint \(q\) is the intersection of the line segment Equation (16) and the osculating sphere Equation (17), we get  
\[\Delta t=\dfrac{\left\langle c^{\prime}(t_{0}),q-c(t_{0})\right\rangle}{ \left\langle c^{\prime}(t_{0}),c^{\prime}(t_{0})\right\rangle} \tag{2}\]6_i.e.,_\(|1-u(t)|\left\|\mathrm{m}(t)-\mathrm{v}\right\|=R(t)\).