Cup to torus

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The cup-to-torus transformation is a well-known diffeomorphism that maps a two-dimensional disk into a two-dimensional torus. This transformation has been extensively studied and applied in various fields, including topology, geometry, and physics. In essence, the cup-to-torus transformation represents a continuous deformation of a disk into a doughnut-shaped surface, which is topologically equivalent to a torus. This process can be visualized by imagining a rubber band being stretched and deformed into a circular ring shape. The mathematical representation of this transformation is given by the following parametric equations: x(u,v) = (2 + cos(v))(cos(u) + v sin(u)) y(u,v) = (2 + cos(v))(sin(u) - v cos(u)) z(u,v) = sin(v) where u and v are parameters that range from 0 to 2π. By varying the values of these parameters, we can generate a continuous family of tori with different radii and shapes. The cup-to-torus transformation is an important concept in topology because it demonstrates the existence of non-trivial diffeomorphisms between two-dimensional manifolds. It also has applications in physics, where it is used to model the behavior of membranes and surfaces under various types of deformation.