# Defining Feasible Joint and Geometric Workspaces Through Boundary Functions
## 4 Boundary Functions for Workspace Definition
The first represents a joint trajectory consisting of five points, while the second shows the corresponding geometric trajectory. These paths are opposite to those presented in Figure 8.  
The feasible geometric regions from Figures 8 and 9 are overlaid in Figure 10. Notably, these regions do not exhibit symmetry, which is a result of the scaling applied relative to the centroid and the nonlinear properties inherent to the trajectory. Figure 10 highlights the application of traditional numerical methods for solving this type of problem. Initially, a set of \(n\) random joint variables was generated. Following a conventional approach, each detected collision was marked with a red point, while  
Figure 10: Overlay of the feasible geometric regions from Figures 8 and 9, highlighting their asymmetry due to centroid-based scaling and trajectory nonlinearities. Figure 4:0 Alternative feasible trajectory. The first plot depicts a five-point joint space path, while the second shows its corresponding geometric trajectory. noncolliding configurations were represented by blue points. These methods, however, entail a high computational cost and do not produce a continuous boundary trajectory in the joint space. Their efficiency is directly dependent on the volume of analyzed data. Figure 11 presents two plots from this experiment: the first in the joint space and the second in the geometric space. In the latter, overlapping regions containing both collisions and noncollisions can be observed. This phenomenon is associated with the poses and trajectory patterns described in Figures 8 and 9. Additionally, the lateral shapes of the geometric workspace, as seen in Figure 7, are also visible. Although the methodology presented in this work is exemplified using rotational joints in planar mechanisms, the proposed boundary function formulation is not limited to revolute joints. In the case of translational (prismatic) joints, the joint variable \(q_{i}\) affects the position vectors linearly instead of trigonometrically. As a result, the same structure of boundary functions \(\beta_{i,j}(\mathbf{q})\) remains applicable, and the constraints reduce to affine relationships in the configuration space. Building on this foundation, the method is being extended to more complex manipulator configurations. Its applicability to spatial mechanisms, including those incorporating hybrid kinematic structures and parallelogram-based linkages, is currently under investigation. In such cases, boundary functions defined as \(\beta_{i,j}(\mathbf{q})=0\) represent implicit hypersurfaces in high-dimensional configuration spaces. These surfaces can be visualized using established computational techniques, such as isosurface extraction (e.g., marching cubes or adaptive grid subdivision), which are suitable for feasibility analysis in spatial domains. To maintain computational efficiency, the evaluation of boundary functions can be restricted to lower-dimensional joint subspaces or slices, avoiding the need for exhaustive random sampling. In this way, the deterministic and symbolic character of the formulation is preserved while enabling scalability to more complex robotic systems. Moreover, data-driven parameterization of boundary surfaces is currently being explored as part of a broader strategy to generalize the framework while retaining its analytical structure.