$\mathbb{RP}^3$ is homeomorphic to the solid ball with antipodal points identified

$\mathbb{RP}^3$ is homeomorphic to the solid ball with antipodal points
identified

I am reading the book Application of Path integrals by Schulman, which has
a chapter on applications of homotopy theory to path integrals. In that he
says we can geometrically describe $SO(3)$ by a solid (3 -dimensional)
ball with radius $\pi$,and with antipodal points identified. Each point in
the ball at distance $\phi$ from the centre, represents a rotation about
the axis passing through that point and origin, and angle of rotation
$\phi$. Later he writes "Projective 3 space is homeomorphic to the solid
ball described earlier." I imagine $\mathbb{RP}^3$ to be the 3-sphere with
antipodal points identified.
How is this homeomorphic to the solid ball described above?