## 2 An Overview of Line Simplification Algorithms
### Line Simplification Algorithms and Error Measures  
A general proposed division of simplification algorithms is local algorithms, which consider several consecutive original points; global algorithms, which process the whole domain of a function; and independent algorithms, which do not search for any mathematical relationship between neighboring points. An example of a simple independent algorithm is the \(n\)th point routine, which selects every fixed number of consecutive points, also called the equal intervals algorithm (EI) [29]. Since the goal is to optimally distribute and select the points to define the multilinear curve, more attention will be given to local and global algorithms. Local algorithms consider relationships between several neighboring points of a function and decide on the removal of a specific point depending on the distance from other points or the area enclosed by points in its proximity. One such algorithm is the Reumann-Witkam routine [30], which uses user-defined distance tolerance to decide if the point is to be kept or removed. First, a line is defined through the first two consecutive points, \(x_{\text{i}}\) and \(x_{\text{i+1}}\). The shortest distance from this line to each next point of the original dataset is calculated, and the points whose distance exceeds the predefined tolerance are kept. The process continues until reaching the last point. The simplification algorithm developed by [PERSON] [29] searches the space of a fixed number of consecutive original points. In the search space, the first and last original points are connected, and the points between them that have a perpendicular distance to that line segment smaller than the tolerance are removed. The Visvalingam Whyatt (VW) algorithm [31] is another local simplification algorithm that works by removing points that cause less change in the shape of the function. Successive triplets of points are formed along the piece-wise function. The algorithm removes point \(x_{\text{i}}\) if the area of the triangle with points \(x_{\text{i-1}}\), \(x_{\text{i}}\), and \(x_{\text{i+1}}\) is smaller than the user-defined value. By doing so, the points that give the smallest change in overall function shape are removed, and the iteration starts again [31]. Global line-simplification algorithms consider a relationship between all points of a given dataset in one iteration. Similar to the local Visvalingam Whyatt algorithm, the global VW algorithm [31] also forms successive triplets of points, but computes the effective area of all triangles formed. The algorithm eliminates the point with the minimum effective area in each iteration. This process is repeated, forming successive triplets and removing points until all areas are within the specified tolerance [29]. Another example of global simplification algorithm is the Ramer-Douglas-Peucker (RDP) algorithm, shown in Figure 3, which recursively divides the entire line and calculates the distance of points [32]. The algorithm works with an already discretized initial curve by connecting the first original point, \(x_{1}\), to the last original point of the dataset, \(x_{\text{n}}\), with a line segment and calculating the shortest distances of all original points to that line. The farthest distance is labeled \(d_{\text{max}}\), and the point corresponding to that distance, \(x_{\text{i}}\), is kept in a new simplified dataset if it is outside of the predefined tolerance \(\varepsilon\). The iterative process continues with two new line segments, with the point \(x_{\text{i}}\) being the new end point for the left line segment and the new start point for the right line segment. The process continues until all points are within the predefined tolerance. In assessing the accuracy of various line-simplification algorithms [32], the deviation of the discretized polyline from the original shape can be quantified.