# Algebraic Point Projection for Immersed Boundary Analysis on Low Degree NURBS Curves and Surfaces
## 1 Introduction  
Given a test point and a parametric entity (curve or surface), the generalized point projection problem is to find the closest point (footpoint) on the entity as well as the corresponding parameter value. Since the footpoint is the closest point on the curve or surface, the line connecting the test point to the footpoint is normal to the curve or the surface [1]:  
\[g(u)=\mathbf{C}^{\prime}(u)\cdot(\mathbf{C}(u)-\mathbf{P})=0 \tag{1}\]  
Given a parametric curve or surface entity \(\mathbf{C}(u)\in\mathbb{R}^{n}  
where \(\mathbf{C}(u)\) is a physical point on the curve or surface of interest. The distance function \(d_{E}(\mathbf{x})\) is continuous for all \(\mathbf{x}\in\mathbb{R}^{n}\) and differentiable almost everywhere. The \"footpoint\" of projection in parametric space \(u_{f}\) is defined as:  
\[u_{f}(\mathbf{x})=\operatorname*{arg\,min}_{u\in[a,b]}\lVert\mathbf{x}- \mathbf{C}(u)\rVert \tag{3}\]  
where \([a,b]\) is the parameter range. In general, \(u_{f}(\mathbf{x})\) may be non-unique, discontinuous, or non-existent, as illustrated in Figure 1. The footpoint of a test point near the curve or surface segment with high local curvature can be non-unique, leading to discontinuity of point projection process as illustrated in Figure 1a. The non-existence is illustrated in Figure 1b, and occurs around points where \(\mathbf{C}(u)\) has only \(\mathcal{G}^{0}\) continuity. This problem is of importance in geometric modeling. For instance, while fitting a curve or surface to sampled data, one may need to compute corresponding parameter values and errors at data points since the error is the distance between the data point and the fitting curve or surface [2]. Point projection also plays an important role in computer aided engineering (CAE), especially when boundaries are immersed into the domain and evolved. Such immersed boundary analysis [3] uses a non-conforming mesh to significantly reduce computational cost required for mesh generation as the boundaries evolve. Often, the immersed boundaries are represented as parametric splines, and, more recently, in isogeometric analysis, the underlying domain is also approximated by parametric splines. Isogeometric analysis (IGA) [4; 5] is aimed at building behavioral approximations isparametrically on a spline geometry, most commonly on NURBS curves and surfaces. The goal is to eliminate the need to mesh the geometry for analysis and to ensure the exactness of geometry to the CAD model during analysis. [PERSON] and [PERSON] [6] developed an Enriched Isogeometric Analysis (EIGA) for immersed boundary problems in which both the domain as well as the enrichments are described by NURBS entities, which are then blended to describe the enriched approximation. In any immersed boundary problem solution, capturing the interaction of the field approximation defined on the immersed (explicitly defined) boundary with the approximations on the enriched domain requires one to determine the nearest point on the boundary from any given point in the underlying domain. This projection from the spatial point to the boundary is necessary to compute the influence of the domain approximation on those approximations defined on the boundary (see Figure 2). For example, in solutions to mechanical contact problems [7; 8], point projection is needed to define the normal gap and tangential slip between two bodies. In fluid-structure interaction (FSI) problems, point  
Figure 1: Special cases encountered during point projection: (**a**) Exact projection from points on a straight line to a bell-shaped curve. Discontinuity occurs at the circled central point due to non-uniqueness of point projection. (**b**) Projection from points on a straight line to a cone-shaped curve. The dashed line segment has no footpoint on the curve. projection is required to transfer kinematic and traction data between non-matching fluid-structure interface [9].