What is the relationship between the unit simplex and the nonnegative orthant?

Question

I came upon this figure while reading something online.

Pictured above is the intersection of the unit sphere in the nonnegative orthant and the unit simplex.

Question: What is the relationship between these two sets? Can one set be continuously deformed into another? How about the other way around?

Update: How does Nash Embedding theorem make sense here?

Answer 1

They are homeomorphic by $$\varphi(\overline{x})=\frac{\overline{x}}{||\overline{x}||}$$ which is continuous and all, here $\overline{x}=(x,y,z)$ which is the coordinates in questin, as none can be all $0$s there is no issue of it being undefined.