1. **Problem Statement:** You have several points each with an attached vector of length $r$, and you want to find the surface defined by the ends of these vectors. 2. **Understanding the Problem:** Each vector starts at a point and extends a distance $r$ in some direction. The ends of these vectors form a set of points in space. 3. **Method to Find the Surface:** - If the vectors are all of the same length $r$ and originate from points on a known surface or shape, the ends of the vectors will lie on a surface offset by $r$ in the direction of each vector. - To find the surface defined by the vector ends, you can use the parametric form: $$\mathbf{P}_{end} = \mathbf{P}_{start} + r \hat{v}$$ where $\mathbf{P}_{start}$ is the starting point, $r$ is the vector length, and $\hat{v}$ is the unit vector direction. 4. **If you have multiple points and vectors:** - Collect all $\mathbf{P}_{end}$ points. - Use surface fitting techniques (e.g., polynomial fitting, spline surfaces, or mesh generation) to approximate the surface passing through these points. 5. **Summary:** - Calculate each vector end point using $\mathbf{P}_{end} = \mathbf{P}_{start} + r \hat{v}$. - Use these points to fit or interpolate a surface. This method works generally for any set of points and attached vectors of length $r$.