# Algebraic Point Projection for Immersed Boundary Analysis on Low Degree NURBS Curves and Surfaces
## 4 Methodology of Algebraic Point Projection
### Algebraic Point Projection for a NURBS Curve
#### 4.1.5 Inversion to Parametric Space  
Given a footpoint \(\mathbf{x}_{f}\) in physical space, finding a corresponding parameter \(u_{f}\) such that \(\mathbf{C}(u_{f})=\mathbf{x}_{f}\) is the point inversion problem. The direct approach to carrying out point inversion is by solving a system of polynomial equations, which may not be easy to do since there is no analytical solution for high-order polynomials of \(deg>4\), and since \(\mathbf{x}_{f}\) may not lie exactly on curve \(\mathbf{C}(u)\)[22].  
Figure 11: Second-order algebraic point projection with test points at a far distance: (**a**) using non-corrected normal vector; and (**b**) using corrected normal vector.  
Figure 10: Recursive first-order algebraic point projection at test distance \(\bar{d}=0.3\).  
This drawback can be overcome by using the Bezout matrix [25], as shown in the following. Evaluate \(\mathbf{M}^{B}(\mathbf{x}_{f})\) with \(\mathbf{x}_{f}=(x_{f},y_{f})\):  
\[\mathbf{M}^{B}(\mathbf{x}_{f})=\mathbf{M}_{x}^{B}x_{f}+\mathbf{M}_{y}^{B}y_{f}+ \mathbf{M}_{w}^{B}=[m_{ij}]_{p\times p} \tag{29}\]  
Then, Equation (6) can be rewritten as the following over-constrained linear system:  
\[A\bar{\mathbf{u}}\equiv\begin{bmatrix}m_{11}&m_{12}&\cdots&m_{1(p-1)}\\ m_{21}&m_{22}&\cdots&m_{2(p-1)}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ m_{p1}&m_{p2}&\cdots&m_{p(p-1)}\end{bmatrix}\begin{pmatrix}u^{p-1}\\ u^{p-2}\\ \vdots\\ u\end{pmatrix}=-\begin{pmatrix}m_{1p}\\ m_{2p}\\ \vdots\\ m_{pp}\end{pmatrix} \tag{30}\]  
Matrix \(A\) is full ranked if Equation (5) has only one common root, i.e., if \(\mathbf{x}_{f}\) is not a double point [30]. Thus, \(u_{f}\) can be obtained by solving a linear least square problem resulting from Equation (30), which requires bounded computational cost unlike numerical iterations using the Newton-Raphson method. In addition, if a physical test point \(\mathbf{x}\) is initially on the curve, then \(\mathbf{x}_{f}=\mathbf{x}\), and the point inversion can be directly applied, without needing to find the footpoint. Alternative to solving the over-constrained system, one can discard any one row of \(\mathbf{A}\) in Equation (30), which will make it full rank and the point inversion solution can be obtained by solving linear system of equations.  
To compute \(u_{f}\) on a NURBS curve, which is piece-wise polynomial, projection onto the appropriate Bezier segment is required. For this purpose, one can identify the closest Bezier segment using individual algebraic level sets, and apply algebraic point projection on the closest Bezier segment. Denoting the computed parameter on the closest Bezier curve as \(u_{f}^{B}\), the corresponding parameter on the original NURBS curve \(u_{f}^{N}\) can be obtained by purely linear scaling and offset. Unlike iterative or recursive schemes, the algebraic method guarantees the existence of a definite footpoint without needing to manipulate user-controlled parameters such as stop criterion or recursion limit in an effort to coax a solution. If a test point is close to the connection node of two adjacent Bezier segments, a result of \(u_{f}^{B}<0\) or \(u_{f}^{B}>1\) may be obtained. In this case, higher projection precision can be achieved when a second point projection to the adjacent Bezier segment is attempted (see Figure 12).