## 1 Introduction
[6] explores coverage path planning for structural inspections using a drone, and various heuristic methods to solve the optimization problem (e.g., genetic algorithms ) are implemented here. An alternative approach to these optimization methods and solving NP-hard problems is to use geometric algorithms; however, care must be taken to use the appropriate algorithm for the problem at hand. The main contribution of this paper is to demonstrate the usage of a geometric method instead of complex numerical optimization methods and show that we get good results for the 3D virtual models of real-world examples. Ref. [7] describes a geometric algorithm using slicing planes for 'circling' around an underwater structure using an underwater robot for inspection (see Figure 3a), and Ref. [1] also describes a similar  
Figure 1: Viewpoints on surfaces. (**a**) Generic viewpoint on surface; (**b**) Viewpoints corresponding to surface patch. circling strategy for UAVs in urban environment. Ref. [8] describes a basic geometric algorithm for coverage path planning for spraying using drones. Geometric algorithms based on slicing the surface based on planes are commonly used in 3D printing applications; however, if the distance between planes is kept constant then we will immediately run into problems for even the simplest curved surfaces (see Figure 3b for how a gap in between viewpoints has occurred), and in general such a technique cannot be used efficiently without making some assumptions about the surface mesh. Ref. [7] have implemented the circling algorithm for their underwater inspection and note that it is 'inflexible' and 'not optimal'. Slicing algorithms have also been implemented in [9] for the purpose of inspecting large structures like wind turbines using UAVs (Figure 2B). Although geometric algorithms do not guarantee optimality or complete coverage, it is possible to use the geometric methods for applications such as panel inspection for aerospace systems, and obtain efficient algorithms that give near-optimal solutions by selecting the appropriate input parameters carefully. In addition, geometric algorithms are well known for their speed, simplicity, and ease of use. We proposed a simple geometric approach based on using the geodesics of the surface, in [10]. In this approach, the surface is first partitioned into near-developable patches. On each patch, the near rectangular camera image sensor projections are placed, side by side, with overlaps to ensure that all points on the surface are covered. What is done is to place one rectangle, and then construct adjacent rectangles by moving in the length and width  
Figure 3: Viewpoints on surfaces: (**a**) Circling path for underwater inspections; (**b**) Failure of a basic slicing algorithm. Figure 2: Coverage path planning of structures using UAVs. (**A**) UAV inspection of church using optimization methods [5]; (**B**) UAV inspection of wind turbine using slicing-based geometric method [9]. directions along the geodesics of the surface in those directions. The advantages of using geodesics are that (a) on near planar surfaces geodesics can form a near rectangular grid, and (b) geodesics are well-defined curves that can be constructed using simple algorithms. For finding geodesics, we use the approaches proposed by [11; 12]. Ref. [11] gives an algorithm to compute a discrete geodesic path on a surface mesh, and Ref. [12] uses this algorithm to fit woven cloth (resembling a 2D grid) onto the surface. We proceed in a similar fashion as [12] but dispense with the notions of warp, weft, and other terminologies related to fabric draping and replace the node mapping algorithms in [12] with our own node insertion algorithm. We show that the algorithm achieves good results for real-world scenarios. In [10], we have described the underlying process as follows. Figure 4a shows a generic shape that can be decomposed into several sections, each of which is near developable, i.e., a 2D grid can be mapped on the surface without much distortion.