Let $\phi:D^3\to S^2$ be the composition $D^3\to S^3\to S^2$, the first map being the quotient by the boundary and the second map being the Hopf map. Then: $$f_t:x\mapsto(1-t)x+t\phi(x)$$ is a homotopy that takes every point in $D^3$ on a straight-line path towards its destination in $S^2$. Furthermore, everything on the boundary travels to the same point.
Can anyone help me visualize that homotopy? Preferably with an animation; I must confess that I have no idea how to make one myself.
Also, if there's another homotopy that's easier to visualize, where the points travel from $x$ to $\phi(x)$ but not necessarily on a straight-line path, that would be helpful too.
I apologize if this question isn't suitable for this site.