## 2 Orthogonal Projection onto a Spatial Parametric Curve  
Assume that \(c(t)\) is a \(C^{2}\) curve in three-dimensional Euclidean space \(R^{3}\), and \(p\) is a test point. The scalar product of vectors \(x,y\in R^{3}\) will be denoted by \(\left\langle x,y\right\rangle\), and the norm of a vector \(x\) by \(\left\|x\right\|\). The curve's curvature will be denoted by the symbol \(k\) which is computed by the formula \(k(t)=\dfrac{\left\|c^{\prime}(t)\times c^{\prime\prime}(t)\right\|}{\left\|c ^{\prime}(t)\right\|^{3}}\). Recall that the osculating sphere has a radius \(\left|1/k\right|\) and its center is determined by Equation (10). A first-order geometric iterative method which computes the footpoint \(q\) of the test point \(p\) is as follows: The intersection of the osculating sphere \(\bar{s}\) of the curve \(c(t)\) at \(t=t_{0}\) and the line segment which connects the test point \(p\) and the center of the osculating sphere is the footpoint \(q\). The footpoint \(q\) is expressed in terms of \(c(t_{0})\) and the derivative \(c^{\prime}(t_{0})\) (see Figure 1):  
\[q=c(t_{0})+\Delta tc^{\prime}(t_{0}) \tag{1}\]  
Then,  
\[\Delta t=\dfrac{\left\langle c^{\prime}(t_{0}),q-c(t_{0})\right\rangle}{ \left\langle c^{\prime}(t_{0}),c^{\prime}(t_{0})\right\rangle} \tag{2}\]  
We increase \(t_{0}\) by \(\Delta t\) and repeat the above procedure until \(\Delta t\) is less than a given tolerance. Thus, we can compute the projection of the test point \(p\) onto the curve in a simple way. In order to improve the efficiency of the geometric iteration, we present a second-order iterative method to compute the minimum distance between the test point \(p\) and a spatial parametric curve  
Figure 1: Illustration of the first-order geometric iteration for computing the minimum distance between a point and a spatial parametric curve. \(c(t)\). This geometric second-order iterative method which computes the footpoint \(q\) of the test point \(p\) is as follows: orthogonal projecting of the test point \(p\) onto an osculating plane of the spatial parametric curve \(c(t)\) at \(t=t_{0}\) yields a point \(\
u\). Then, the intersection of an osculating sphere \(\tilde{s}\) and a line segment which connects the point \(\
u\) and a center of the osculating sphere \(\tilde{s}\) is the footpoint \(q\). The intersection of the osculating plane of the spatial parametric curve \(c(t)\) at \(t=t_{0}\) and the osculating sphere \(\tilde{s}\) of the spatial parametric curve \(c(t)\) at \(t=t_{0}\) is a osculating circle \(\tilde{c}\) of the spatial parametric curve \(c(t)\) at \(t=t_{0}\). We assume that, for the moment, the osculating circle \(\tilde{c}\) is parameterized such that it has the same Taylor polynomial as the curve \(c(t)\). Similar to the first-order geometric iterative method, we get (see Figure 2):  
\[q=\tilde{c}(t_{0}+\Delta t)=c(t_{0})+\Delta tc^{\prime}(t_{0}) \tag{3}\]  
Then  
\[\Delta t=\frac{\langle c^{\prime}(t_{0}),q-c(t_{0})\rangle}{\langle c^{\prime }(t_{0}),c^{\prime}(t_{0})\rangle} \tag{4}\]  
We increase \(t_{0}\) by \(\Delta t\) and repeat the above procedure until \(\Delta t\) is less than a given tolerance. This equation now can be used to compute the parameter increment \(\Delta t\). Iteration yields a second-order algorithm for computing the footpoint \(q\) which is determined by the given parameter value. This algorithm also fits to solve the inversion problem, e.g., computing the parameter value \(t\) for a point which is known to lie on the curve. **Remark 1**. : During the process of calculating the footpoint \(q\), since the footpoint \(q\) only depends on the projection point \(\
u\), which is the projection of the test point \(p\) onto the osculating plane, while the osculating plane is given by the osculating sphere \(\tilde{s}\) of the spatial parametric curve \(c(t)\) at \(t=t_{0}\).