# Diffeomorphism between star-space and sphere-space

I am a robotics student who has very poor knowledge of topology, thus I hope my question is not ill-posed.

Studying the classical textbook [1], I found an interesting diffeomorphism from stars* to spheres. Both the book [1] and the original paper [2], report a formula to transform points form the star world to the sphere world. Since it is a diffeomorphism, I looked for the inverse smooth transformation but I cannot find it.

Does anyone know it?

I tried to compute the inverse algebraically but the map from stars to open balls is similar to $$f(x) = r \frac{x-x_0}{\|x-x_0\|}+c$$ where $$r,c$$ are the radius and the center of the ball, respectively, and $$x_0$$ is the vantage point of the star.

Thank you very much for your attention!

* a star-shaped set $$S$$ is a set where there exists at least one point (also called vantage point) that is within line of sight of all other points in the set.

REFERENCES

[1] Choset, Howie, et al. "Principles of robot motion: theory, algorithms, and implementations". MIT press, 2005.

[2] Koditschek, Daniel E, et al. "Robot Navigation Functions on Manifolds with Boundary". Advances in applied Mathematics 11, 412-442 (1990)

• The exact expression of the inverse map will depend on the shape of the star* set (those sets are often called star domains). The map from the star domain to the sphere probably maps the "vantage point" of the star to the center of the sphere and then linearly scales all segments starting from this point to the same length (radius of the sphere). To construct a reverse map you need to the scaling factor for each radius of the sphere to scale it back to its original length (basically you need the shape of the star domain) May 30, 2022 at 20:40
• *you need to know (last sentence) May 30, 2022 at 20:46
• But you should not call it a "sphere," the right terminology is an "open ball." For a proof, see here. May 30, 2022 at 22:01
• @david_sap Thank you for your answer. The map does what you have written: it scales the ray starting at the "center" point of the star set (the book and paper do not mention the vantage point but I think it is the same) through its unique intersection with the boundary in order to make it of the same lenght of the sphere radius. I also think that you need to know such a scaling factor to compute the inverse transformation. But then, since it is prooved to be a diffeomorphism, should we not be able to have such analytical inverse transformation? Thanks! May 31, 2022 at 9:51
• Yes, the analytical inverse transformation will always exist, but its exact expression will depend on the shape of the boundary of the star domain. If you have an expression for the direct map, finding the inverse should be doable: what you need is the distance between the "center" of the star set and the "unique intersection of every radius with the boundary". And, as @MoisheKohan correctly pointed out, we should talk of a ball instead of a sphere (a sphere is the external surface/boundary of a ball) May 31, 2022 at 17:50