## 2 An Overview of Line Simplification Algorithms
### Discussion of the Mentioned Algorithms  
The local and global simplification algorithms mentioned earlier work through either point addition or point removal. Initially, a continuous curve is discretized, with points being either removed (VW) or added successively (RDP). Due to this discretization process, the accuracy and final simplified polyline are influenced by the position and density of the original points rather than solely by the complexity of the line [29]. A more effective approach involves directly selecting points from continuous curves, bypassing the need to discretize the initial curve and, thereby, minimizing information loss.  
Moreover, the mentioned algorithms rely on user-defined tolerance as a criterion for point removal. For example, the VisualizingWyatt algorithm requires the input of minimal triangle size, which is difficult to estimate since the optimal triangle size is not unique and can vary for various material curves to achieve consistent accuracy.  
Figure 4: Example of a method for determining the areal difference measure (\(A_{\text{diff}}\)) calculated by substracting the final simplified area (\(A_{\text{simplified}}\)) from the original area (\(A_{\text{original}}\)).  
Figure 3: Example of the Ramer–Douglas–Peucker method for polyline simplification with initial curve discretized by 9 points.  
Additionally, even slight variations in the tolerance value can lead to different numbers and positions of simplified points on the final polyline. Given that the provided tolerance does not directly indicate the deviation of stresses or strains of the simplified polyline from the original curve and lacks intuitiveness, an alternative stopping criterion must be considered for this purpose.  
While the algorithms can be adjusted to stop when a predefined total number of points in a simplified polyline is reached, the desired number of points alone also does not provide meaningful information about the deviation of the simplified polyline from the original curve. That is why incorporating the previously mentioned error measures into the stopping criterion proves to be a more effective approach. The positional error sum depends on the discretization of the original curve, as it sums errors from the original points to a simplified line. Thus, it greatly depends on the number of points in the original curve and the scale of the stresses and strains and cannot be used to compare curves with a different number of points or different material constants. Again, if the positional error sum is a user-defined value, it is difficult to estimate its value, as it does not contain specific information that can be known to the user beforehand. The positional error sum could potentially be useful only in determining the goodness of algorithms when comparing simplified curves with the same number of points obtained from the original curve discretized with an identical number of points.  
A better quantifying error measure, which could also potentially be used as a stopping criterion, is the areal difference measure. This measure can be converted to a percentage and used to quantify the percentage that is not explained by the simplified curve. An areal difference measure could be used as a stopping criterion, but it fails to provide information on the maximum deviation of stress values to the original curve, which is information of importance when optimally discretizing a material stress-strain curve. For this reason, another measuring error that considers the deviation of stresses between the simplified and original curves is proposed.