### 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.