Stereographic Projection from \( S^3\)

The previous visualization is flawed because circular boundaries appear as the non-boundary edge (away from the torus) of the outside subsurface 'tears apart' as we animate through \(\theta\). Since we require that the only boundary for our fibration surface must be exactly the knot , we identify points along this tear.

To acheive this, we define our surfaces as embedded in \(S^3 \in \mathbb{R}^4 \). Then we use stereographic projection \( \rho : S^3 \rightarrow\mathbb{R}^3 \) to visualize our surfaces by projecting from the point \(\{0,0,0,1\}\): $$\rho(x,y,u,v)=\{ \frac{x}{1-v}, \frac{y}{1-v}, \frac{u}{1-v}\}$$

By construction, this identifies all points on the non-boundary edge (away from the torus) of our subsurfaces. Now our visualization only 'tears' when the surface passes through the stereographic projection point, which is at infinity.