A Diffeomorphism from $T^2$ to $S^2$.

This repository explores the diffeomorphism between the torus and sphere, detailing their parametric equations, tangent vectors, and limits.

• Torus: Parametrically defined by two angles uuu and vvv, with tangent vectors derived from differentiation.
• Sphere: Parametrically represented by azimuthal angle θ\thetaθ and polar angle ϕ\phiϕ, with corresponding tangent vectors.
• Transition: As the major radius RRR of the torus approaches zero, the torus smoothly transforms into a sphere.
• Special Cases: Includes degenerate spheres (when r→0r \to 0r→0) and flat rings (when R→∞R \to \inftyR→∞).

Applications span topology, physics, and geometry. The framework is visualized using Python libraries like Matplotlib and Manim.