Frechet distance between surfaces

[Dedalus]

I am wondering whether or not anybody has any ideas of how to visualize and calculate the Frechet distance between two surfaces, or the sets that they encompass.

Let M be an m-dimensional finitely triangulated manifold (with or without boundary). Let f1 and f2 be continuous maps M---->R^n, n>m≥0. The Frechet Distance between maps is defined as
σ_F (f1,f2)=〖inf〗_(α,β) 〖max〗_xϵM∥[f1(α(x)) -f2(β(x))∥(Euclidean norm) where α,β are all the possible homeomorphisms (injective, bi-continuous) from M to M.

I'm particularly interested in calculating the Frechet Distance between two convex sets, one contained in the other. I'm wondering if ther'es a way to consider instead the Hausdorff distance between the surfaces, in which case the maximum epsilon of th epsilon-neighborhood of the outer surface would be the maximum distance from the surface to the center of mass of the set it bounds. I know there's a bound on the Frechet distance in two-space using the hausdorff distance, but it hasn't yet been extended to 3 space.

Thanks for any ideas or direction.

 

[jvicens]

I have some ideas on how to visualize and calculate the Frechet distance between two surfaces. The first step would be to represent the two surfaces as a set of points in a three-dimensional space. This can be done by using techniques such as point clouds or meshes.

Once the surfaces are represented as points, we can use the definition of the Frechet distance to calculate it. This involves finding the minimum distance between the two surfaces for all possible homeomorphisms. This can be a computationally intensive task, so it may be helpful to use algorithms such as the Dijkstra algorithm or the A* algorithm to efficiently search for the minimum distance.

Additionally, as mentioned in the forum post, the Hausdorff distance can also be used as an approximation for the Frechet distance in three-dimensional space. This can be calculated by finding the maximum distance between the points of one surface and the closest point on the other surface. However, it is important to note that the Hausdorff distance may not always provide an accurate measure of the Frechet distance, especially for non-convex surfaces.

Another approach to visualizing the Frechet distance between two surfaces is to use visualization techniques such as heatmaps or color maps to represent the distance between points on the surfaces. This can help in identifying regions where the distance is larger or smaller, and can provide a more intuitive understanding of the Frechet distance.

Lastly, it may also be helpful to consult with experts in the field of computational geometry or topology for further insights and techniques for calculating and visualizing the Frechet distance between surfaces.